Elementary Number Theory: Primes, Congruences, and Secrets


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1 This is age i Printer: Oaque this Elementary Number Theory: Primes, Congruences, and Secrets William Stein November 16, 2011
2 To my wife Clarita Lefthand v
3 vi
4 Contents This is age vii Printer: Oaque this Preface ix 1 Prime Numbers Prime Factorization The Sequence of Prime Numbers Exercises The Ring of Integers Modulo n Congruences Modulo n The Chinese Remainder Theorem Quickly Comuting Inverses and Huge Powers Primality Testing The Structure of (Z/Z) Exercises Publickey Crytograhy Playing with Fire The DiffieHellman Key Exchange The RSA Crytosystem Attacking RSA Exercises Quadratic Recirocity Statement of the Quadratic Recirocity Law
5 viii Contents 4.2 Euler s Criterion First Proof of Quadratic Recirocity A Proof of Quadratic Recirocity Using Gauss Sums Finding Square Roots Exercises Continued Fractions The Definition Finite Continued Fractions Infinite Continued Fractions The Continued Fraction of e Quadratic Irrationals Recognizing Rational Numbers Sums of Two Squares Exercises Ellitic Curves The Definition The Grou Structure on an Ellitic Curve Integer Factorization Using Ellitic Curves Ellitic Curve Crytograhy Ellitic Curves Over the Rational Numbers Exercises Answers and Hints 149 References 155 Index 160
6 Preface This is age ix Printer: Oaque this This is a book about rime numbers, congruences, secret messages, and ellitic curves that you can read cover to cover. It grew out of undergraduate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B.C. when Euclid roved that there are infinitely many rime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every ositive integer factors uniquely as a roduct of rimes. Over a thousand years later (around 972A.D.) Arab mathematicians formulated the congruent number roblem that asks for a way to decide whether or not a given ositive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Diffie and Hellman introduced the first ever ublickey crytosystem, which enabled two eole to communicate secretely over a ublic communications channel with no redetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, ellitic curves revolutionized number theory, roviding striking new insights into the congruent number roblem, rimality testing, ublickey crytograhy, attacks on ublickey systems, and laying a central role in Andrew Wiles resolution of Fermat s Last Theorem. Today, ure and alied number theory is an exciting mix of simultaneously broad and dee theory, which is constantly informed and motivated by algorithms and exlicit comutation. Active research is underway that romises to resolve the congruent number roblem, deeen our understanding into the structure of rime numbers, and both challenge and imrove
7 x Preface our ability to communicate securely. The goal of this book is to bring the reader closer to this world. The reader is strongly encouraged to do every exercise in this book, checking their answers in the back (where many, but not all, solutions are given). Also, throughout the text there, are examles of calculations done using the owerful free oen source mathematical software system Sage (htt:// and the reader should try every such examle and exeriment with similar examles. Background. The reader should know how to read and write mathematical roofs and must have know the basics of grous, rings, and fields. Thus, the rerequisites for this book are more than the rerequisites for most elementary number theory books, while still being aimed at undergraduates. Notation and Conventions. We let N = {1, 2, 3,...} denote the natural numbers, and use the standard notation Z, Q, R, and C for the rings of integer, rational, real, and comlex numbers, resectively. In this book, we will use the words roosition, theorem, lemma, and corollary as follows. Usually a roosition is a less imortant or less fundamental assertion, a theorem is a deeer culmination of ideas, a lemma is something that we will use later in this book to rove a roosition or theorem, and a corollary is an easy consequence of a roosition, theorem, or lemma. More difficult exercises are marked with a (*). Acknowledgements. I would like to thank Brian Conrad, Carl Pomerance, and Ken Ribet for many clarifying comments and suggestions. Baurzhan Bektemirov, Lawrence Cabusora, and Keith Conrad read drafts of this book and made many comments, and Carl Witty commented extensively on the first two chaters. Frank Calegari used the course when teaching Math 124 at Harvard, and he and his students rovided much feedback. Noam Elkies made comments and suggested Exercise 4.6. Seth Kleinerman wrote a version of Section 5.4 as a class roject. Hendrik Lenstra made helful remarks about how to resent his factorization algorithm. Michael Abshoff, Sabmit Dasguta, David Joyner, Arthur Patterson, George Stehanides, Kevin Stern, Eve Thomson, TingYou Wang, and Heidi Williams all suggested corrections. I also benefited from conversations with Henry Cohn and David Savitt. I used Sage ([Sag08]), emacs, and L A TEX in the rearation of this book.
8 1 Prime Numbers This is age 1 Printer: Oaque this Every ositive integer can be written uniquely as a roduct of rime numbers, e.g., 100 = This is surrisingly difficult to rove, as we will see below. Even more astounding is that actually finding a way to write certain 1,000digit numbers as a roduct of rimes seems out of the reach of resent technology, an observation that is used by millions of eole every day when they buy things online. Since rime numbers are the building blocks of integers, it is natural to wonder how the rimes are distributed among the integers. There are two facts about the distribution of rime numbers. The first is that, [they are] the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can redict where the next one will srout. The second fact is even more astonishing, for it states just the oosite: that the rime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military recision. Don Zagier [Zag75] The Riemann Hyothesis, which is the most famous unsolved roblem in number theory, ostulates a very recise answer to the question of how the rime numbers are distributed. This chater lays the foundations for our study of the theory of numbers by weaving together the themes of rime numbers, integer factorization, and the distribution of rimes. In Section 1.1, we rigorously rove that the
9 2 1. Prime Numbers every ositive integer is a roduct of rimes, and give examles of secific integers for which finding such a decomosition would win one a large cash bounty. In Section 1.2, we discuss theorems about the set of rime numbers, starting with Euclid s roof that this set is infinite, and discuss the largest known rime. Finally we discuss the distribution of rimes via the rime number theorem and the Riemann Hyothesis. 1.1 Prime Factorization Primes The set of natural numbers is and the set of integers is N = {1, 2, 3, 4,...}, Z = {..., 2, 1, 0, 1, 2,...}. Definition (Divides). If a, b Z we say that a divides b, written a b, if ac = b for some c Z. In this case, we say a is a divisor of b. We say that a does not divide b, written a b, if there is no c Z such that ac = b. For examle, we have 2 6 and Also, all integers divide 0, and 0 divides only 0. However, 3 does not divide 7 in Z. Remark The notation b. : a for b is divisible by a is common in Russian literature on number theory. Definition (Prime and Comosite). An integer n > 1 is rime if the only ositive divisors of n are 1 and n. We call n comosite if n is not rime. The number 1 is neither rime nor comosite. The first few rimes of N are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,..., and the first few comosites are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34,.... Remark J. H. Conway argues in [Con97, viii] that 1 should be considered a rime, and in the 1914 table [Leh14], Lehmer considers 1 to be a rime. In this book, we consider neither 1 nor 1 to be rime. SAGE Examle We use Sage to comute all rime numbers between a and b 1.
10 1.1 Prime Factorization 3 sage: rime_range(10,50) [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] We can also comute the comosites in an interval. sage: [n for n in range(10,30) if not is_rime(n)] [10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28] Every natural number is built, in a unique way, out of rime numbers: Theorem (Fundamental Theorem of Arithmetic). Every natural number can be written as a roduct of rimes uniquely u to order. Note that rimes are the roducts with only one factor and 1 is the emty roduct. Remark Theorem 1.1.6, which we will rove in Section 1.1.4, is trickier to rove than you might first think. For examle, unique factorization fails in the ring Z[ 5] = {a + b 5 : a, b Z} C, where 6 factors in two different ways: 6 = 2 3 = (1 + 5) (1 5) The Greatest Common Divisor We will use the notion of the greatest common divisor of two integers to rove that if is a rime and ab, then a or b. Proving this is the key ste in our roof of Theorem Definition (Greatest Common Divisor). Let gcd(a, b) = max {d Z : d a and d b}, unless both a and b are 0 in which case gcd(0, 0) = 0. For examle, gcd(1, 2) = 1, gcd(6, 27) = 3, and for any a, gcd(0, a) = gcd(a, 0) = a. If a 0, the greatest common divisor exists because if d a then d a, and there are only a ositive integers a. Similarly, the gcd exists when b 0. Lemma For any integers a and b, we have gcd(a, b) = gcd(b, a) = gcd(±a, ±b) = gcd(a, b a) = gcd(a, b + a). Proof. We only rove that gcd(a, b) = gcd(a, b a), since the other cases are roved in a similar way. Suose d a and d b, so there exist integers c 1 and c 2 such that dc 1 = a and dc 2 = b. Then b a = dc 2 dc 1 = d(c 2 c 1 ),
11 4 1. Prime Numbers so d b a. Thus gcd(a, b) gcd(a, b a), since the set over which we are taking the max for gcd(a, b) is a subset of the set for gcd(a, b a). The same argument with a relaced by a and b relaced by b a, shows that gcd(a, b a) = gcd( a, b a) gcd( a, b) = gcd(a, b), which roves that gcd(a, b) = gcd(a, b a). Lemma Suose a, b, n Z. Then gcd(a, b) = gcd(a, b an). Proof. By reeated alication of Lemma 1.1.9, we have gcd(a, b) = gcd(a, b a) = gcd(a, b 2a) = = gcd(a, b an). Assume for the moment that we have already roved Theorem A naive way to comute gcd(a, b) is to factor a and b as a roduct of rimes using Theorem 1.1.6; then the rime factorization of gcd(a, b) can be read off from that of a and b. For examle, if a = 2261 and b = 1275, then a = and b = , so gcd(a, b) = 17. It turns out that the greatest common divisor of two integers, even huge numbers (millions of digits), is surrisingly easy to comute using Algorithm below, which comutes gcd(a, b) without factoring a or b. To motivate Algorithm , we comute gcd(2261, 1275) in a different way. First, we recall a helful fact. Proosition Suose that a and b are integers with b 0. Then there exists unique integers q and r such that 0 r < b and a = bq + r. Proof. For simlicity, assume that both a and b are ositive (we leave the general case to the reader). Let Q be the set of all nonnegative integers n such that a bn is nonnegative. Then Q is nonemty because 0 Q and Q is bounded because a bn < 0 for all n > a/b. Let q be the largest element of Q. Then r = a bq < b, otherwise q + 1 would also be in Q. Thus q and r satisfy the existence conclusion. To rove uniqueness, suose that q and r also satisfy the conclusion. Then q Q since r = a bq 0, so q q, and we can write q = q m for some m 0. If q q, then m 1 so r = a bq = a b(q m) = a bq + bm = r + bm b since r 0, a contradiction. Thus q = q and r = a bq = a bq = r, as claimed. For us, an algorithm is a finite sequence of instructions that can be followed to erform a secific task, such as a sequence of instructions in a comuter rogram, which must terminate on any valid inut. The word algorithm is sometimes used more loosely (and sometimes more recisely) than defined here, but this definition will suffice for us.
12 1.1 Prime Factorization 5 Algorithm (Division Algorithm). Suose a and b are integers with b 0. This algorithm comutes integers q and r such that 0 r < b and a = bq + r. We will not describe the actual stes of Algorithm , since it is just the familiar long division algorithm. Note that it might not be exactly the same as the standard long division algorithm you learned in school, because we make the remainder ositive even when dividing a negative number by a ositive number. We use the division algorithm reeatedly to comute gcd(2261, 1275). Dividing 2261 by 1275 we find that 2261 = , so q = 1 and r = 986. Notice that if a natural number d divides both 2261 and 1275, then d divides their difference 986 and d still divides On the other hand, if d divides both 1275 and 986, then it has to divide their sum 2261 as well! We have made rogress: gcd(2261, 1275) = gcd(1275, 986). This equality also follows by alying Lemma Reeating, we have 1275 = , so gcd(1275, 986) = gcd(986, 289). Kee going: 986 = = = Thus gcd(2261, 1275) = = gcd(51, 17), which is 17 because Thus gcd(2261, 1275) = 17. Aside from some tedious arithmetic, that comutation was systematic, and it was not necessary to factor any integers (which is something we do not know how to do quickly if the numbers involved have hundreds of digits). Algorithm (Greatest Common Division). Given integers a, b, this algorithm comutes gcd(a, b). 1. [Assume a > b > 0] We have gcd(a, b) = gcd( a, b ) = gcd( b, a ), so we may relace a and b by their absolute values and hence assume a, b 0. If a = b, outut a and terminate. Swaing if necessary, we assume a > b. If b = 0, we outut a. 2. [Quotient and Remainder] Using Algorithm , write a = bq + r, with 0 r < b and q Z.
13 6 1. Prime Numbers 3. [Finished?] If r = 0, then b a, so we outut b and terminate. 4. [Shift and Reeat] Set a b and b r, then go to Ste 2. Proof. Lemmas imly that gcd(a, b) = gcd(b, r) so the gcd does not change in Ste 4. Since the remainders form a decreasing sequence of nonnegative integers, the algorithm terminates. Examle Set a = 15 and b = = gcd(15, 6) = gcd(6, 3) 6 = gcd(6, 3) = gcd(3, 0) = 3 Note that we can just as easily do an examle that is ten times as big, an observation that will be imortant in the roof of Theorem below. Examle Set a = 150 and b = = gcd(150, 60) = gcd(60, 30) 60 = gcd(60, 30) = gcd(30, 0) = 30 SAGE Examle Sage uses the gcd command to comute the greatest common divisor of two integers. For examle, sage: gcd(97,100) 1 sage: gcd(97 * 10^15, 19^20 * 97^2) 97 Lemma For any integers a, b, n, we have gcd(an, bn) = gcd(a, b) n. Proof. The idea is to follow Examle ; we ste through Euclid s algorithm for gcd(an, bn) and note that at every ste the equation is the equation from Euclid s algorithm for gcd(a, b) but multilied through by n. For simlicity, assume that both a and b are ositive. We will rove the lemma by induction on a + b. The statement is true in the base case when a + b = 2, since then a = b = 1. Now assume a, b are arbitrary with a b. Let q and r be such that a = bq + r and 0 r < b. Then by Lemmas , we have gcd(a, b) = gcd(b, r). Multilying a = bq + r by n we see that an = bnq + rn, so gcd(an, bn) = gcd(bn, rn). Then b + r = b + (a bq) = a b(q 1) a < a + b, so by induction gcd(bn, rn) = gcd(b, r) n. Since gcd(a, b) = gcd(b, r), this roves the lemma. Lemma Suose a, b, n Z are such that n a and n b. Then n gcd(a, b).
14 1.1 Prime Factorization 7 Proof. Since n a and n b, there are integers c 1 and c 2, such that a = nc 1 and b = nc 2. By Lemma , gcd(a, b) = gcd(nc 1, nc 2 ) = n gcd(c 1, c 2 ), so n divides gcd(a, b). With Algorithm , we can rove that if a rime divides the roduct of two numbers, then it has got to divide one of them. This result is the key to roving that rime factorization is unique. Theorem (Euclid). Let be a rime and a, b N. If ab then a or b. You might think this theorem is intuitively obvious, but that might be because the fundamental theorem of arithmetic (Theorem 1.1.6) is deely ingrained in your intuition. Yet Theorem will be needed in our roof of the fundamental theorem of arithmetic. Proof of Theorem If a we are done. If a then gcd(, a) = 1, since only 1 and divide. By Lemma , gcd(b, ab) = b. Since b and, by hyothesis, ab, it follows (using Lemma ) that gcd(b, ab) = b gcd(, a) = b 1 = b Numbers Factor as Products of Primes In this section, we rove that every natural number factors as a roduct of rimes. Then we discuss the difficulty of finding such a decomosition in ractice. We will wait until Section to rove that factorization is unique. As a first examle, let n = The sum of the digits of n is divisible by 3, so n is divisible by 3 (see Proosition 2.1.9), and we have n = The number 425 is divisible by 5, since its last digit is 5, and we have 1275 = Again, dividing 85 by 5, we have 1275 = , which is the rime factorization of Generalizing this rocess roves the following roosition. Proosition Every natural number is a roduct of rimes. Proof. Let n be a natural number. If n = 1, then n is the emty roduct of rimes. If n is rime, we are done. If n is comosite, then n = ab with a, b < n. By induction, a and b are roducts of rimes, so n is also a roduct of rimes. Two questions immediately arise: (1) is this factorization unique, and (2) how quickly can we find such a factorization? Addressing (1), what if we had done something differently when breaking aart 1275 as a roduct of rimes? Could the rimes that show u be different? Let s try: we have
15 8 1. Prime Numbers 1275 = Now 255 = 5 51 and 51 = 17 3, and again the factorization is the same, as asserted by Theorem We will rove the uniqueness of the rime factorization of any integer in Section SAGE Examle The factor command in Sage factors an integer as a roduct of rimes with multilicities. For examle, sage: factor(1275) 3 * 5^2 * 17 sage: factor(2007) 3^2 * 223 sage: factor( ) 2 * 3 * 53 * 73 * 2531 * Regarding (2), there are algorithms for integer factorization. It is a major oen roblem to decide how fast integer factorization algorithms can be. We say that an algorithm to factor n is olynomial time if there is a olynomial f(x) such that for any n the number of stes needed by the algorithm to factor n is less than f(log 10 (n)). Note that log 10 (n) is an aroximation for the number of digits of the inut n to the algorithm. Oen Problem Is there an algorithm that can factor any integer n in olynomial time? Peter Shor [Sho97] devised a olynomial time algorithm for factoring integers on quantum comuters. We will not discuss his algorithm further, excet to note that in 2001 IBM researchers built a quantum comuter that used Shor s algorithm to factor 15 (see [LMG + 01, IBM01]). Building much larger quantum comuters aears to be extremely difficult. You can earn money by factoring certain large integers. Many crytosystems would be easily broken if factoring certain large integers was easy. Since nobody has roven that factoring integers is difficult, one way to increase confidence that factoring is difficult is to offer cash rizes for factoring certain integers. For examle, until recently there was a $10,000 bounty on factoring the following 174digit integer (see [RSA]): This number is known as RSA576 since it has 576 digits when written in binary (see Section for more on binary numbers). It was factored at the German Federal Agency for Information Technology Security in December 2003 (see [Wei03]):
16 1.1 Prime Factorization 9 The revious RSA challenge was the 155digit number It was factored on 22 August 1999 by a grou of sixteen researchers in four months on a cluster of 292 comuters (see [ACD + 99]). They found that RSA155 is the roduct of the following two 78digit rimes: = q = The next RSA challenge is RSA640: , and its factorization was worth $20,000 until November 2005 when it was factored by F. Bahr, M. Boehm, J. Franke, and T. Kleinjun. This factorization took five months. Here is one of the rime factors (you can find the other): (This team also factored a 663bit RSA challenge integer.) The smallest currently oen challenge is RSA704, worth $30,000: SAGE Examle Using Sage, we see that the above number has 212 decimal digits and is definitely comosite: sage: n = \ \ \ \ sage: len(n.str(2))
17 10 1. Prime Numbers 704 sage: len(n.str(10)) 212 sage: n.is_rime() False # this is instant These RSA numbers were factored using an algorithm called the number field sieve (see [LL93]), which is the bestknown general urose factorization algorithm. A descrition of how the number field sieve works is beyond the scoe of this book. However, the number field sieve makes extensive use of the ellitic curve factorization method, which we will describe in Section The Fundamental Theorem of Arithmetic We are ready to rove Theorem using the following idea. Suose we have two factorizations of n. Using Theorem , we cancel common rimes from each factorization, one rime at a time. At the end, we discover that the factorizations must consist of exactly the same rimes. The technical details are given below. Proof. If n = 1, then the only factorization is the emty roduct of rimes, so suose n > 1. By Proosition , there exist rimes 1,..., d such that Suose that n = 1 2 d. n = q 1 q 2 q m is another exression of n as a roduct of rimes. Since 1 n = q 1 (q 2 q m ), Euclid s theorem imlies that 1 = q 1 or 1 q 2 q m. By induction, we see that 1 = q i for some i. Now cancel 1 and q i, and reeat the above argument. Eventually, we find that, u to order, the two factorizations are the same. 1.2 The Sequence of Prime Numbers This section is concerned with three questions: 1. Are there infinitely many rimes? 2. Given a, b Z, are there infinitely many rimes of the form ax + b?
18 1.2 The Sequence of Prime Numbers How are the rimes saced along the number line? We first show that there are infinitely many rimes, then state Dirichlet s theorem that if gcd(a, b) = 1, then ax + b is a rime for infinitely many values of x. Finally, we discuss the Prime Number Theorem which asserts that there are asymtotically x/ log(x) rimes less than x, and we make a connection between this asymtotic formula and the Riemann Hyothesis There Are Infinitely Many Primes Each number on the left in the following table is rime. We will see soon that this attern does not continue indefinitely, but something similar works. 3 = = = = = Theorem (Euclid). There are infinitely many rimes. Proof. Suose that 1, 2,..., n are n distinct rimes. We construct a rime n+1 not equal to any of 1,..., n, as follows. If then by Proosition there is a factorization N = n + 1, (1.2.1) N = q 1 q 2 q m with each q i rime and m 1. If q 1 = i for some i, then i N. Because of (1.2.1), we also have i N 1, so i 1 = N (N 1), which is a contradiction. Thus the rime n+1 = q 1 is not in the list 1,..., n, and we have constructed our new rime. For examle, = = Multilying together the first six rimes and adding 1 doesn t roduce a rime, but it roduces an integer that is merely divisible by a new rime. Joke (Hendrik Lenstra). There are infinitely many comosite numbers. Proof. To obtain a new comosite number, multily together the first n comosite numbers and don t add 1.
19 12 1. Prime Numbers Enumerating Primes In this section we describe a sieving rocess that allows us to enumerate all rimes u to n. The sieve works by first writing down all numbers u to n, noting that 2 is rime, and crossing off all multiles of 2. Next, note that the first number not crossed off is 3, which is rime, and cross off all multiles of 3, etc. Reeating this rocess, we obtain a list of the rimes u to n. Formally, the algorithm is as follows: Algorithm (Prime Sieve). Given a ositive integer n, this algorithm comutes a list of the rimes u to n. 1. [Initialize] Let X = [3, 5,...] be the list of all odd integers between 3 and n. Let P = [2] be the list of rimes found so far. 2. [Finished?] Let be the first element of X. If n, aend each element of X to P and terminate. Otherwise aend to P. 3. [Cross Off] Set X equal to the sublist of elements in X that are not divisible by. Go to Ste 2. For examle, to list the rimes 40 using the sieve, we roceed as follows. First P = [2] and X = [3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]. We aend 3 to P and cross off all multiles of 3 to obtain the new list X = [5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37]. Next we aend 5 to P, obtaining P = [2, 3, 5], and cross off the multiles of 5, to obtain X = [7, 11, 13, 17, 19, 23, 29, 31, 37]. Because , we aend X to P and find that the rimes less than 40 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. Proof of Algorithm The art of the algorithm that is not clear is that when the first element a of X satisfies a n, then each element of X is rime. To see this, suose m is in X, so n m n and that m is divisible by no rime that is n. Write m = ei i with the i distinct rimes ordered so that 1 < 2 <.... If i > n for each i and there is more than one i, then m > n, a contradiction. Thus some i is less than n, which also contradicts our assumtions on m The Largest Known Prime Though Theorem imlies that there are infinitely many rimes, it still makes sense to ask the question What is the largest known rime?
20 1.2 The Sequence of Prime Numbers 13 A Mersenne rime is a rime of the form 2 q 1. According to [Cal] the largest known rime as of March 2007 is the 44th known Mersenne rime = , which has 9,808,358 decimal digits 1. This would take over 2000 ages to rint, assuming a age contains 60 lines with 80 characters er line. The Electronic Frontier Foundation has offered a $100,000 rize to the first erson who finds a 10,000,000 digit rime. Euclid s theorem imlies that there definitely are infinitely many rimes bigger than. Deciding whether or not a number is rime is interesting, as a theoretical roblem, and as a roblem with alications to crytograhy, as we will see in Section 2.4 and Chater 3. SAGE Examle We can comute the decimal exansion of in Sage, although watch out as this is a serious comutation that may take around a minute on your comuter. Also, do not rint out or s below, because both would take a very long time to scroll by. sage: = 2^ sage:.ndigits() Next we convert to a decimal string and look at some of the digits. sage: s =.str(10) # this takes a long time sage: len(s) # s is a very long string (long time) sage: s[:20] # the first 20 digits of (long time) sage: s[20:] # the last 20 digits (long time) Primes of the Form ax + b Next we turn to rimes of the form ax + b, where a and b are fixed integers with a > 1 and x varies over the natural numbers N. We assume that gcd(a, b) = 1, because otherwise there is no hoe that ax + b is rime infinitely often. For examle, 2x + 2 = 2(x + 1) is only rime if x = 0, and is not rime for any x N. Proosition There are infinitely many rimes of the form 4x 1. Why might this be true? We list numbers of the form 4x 1 and underline those that are rime. 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47,... 1 The 45th known Mersenne rime may have been found on August 23, 2008 as this book goes to ress.
21 14 1. Prime Numbers Not only is it lausible that underlined numbers will continue to aear indefinitely, it is something we can easily rove. Proof. Suose 1, 2,..., n are distinct rimes of the form 4x 1. Consider the number N = n 1. Then i N for any i. Moreover, not every rime N is of the form 4x + 1; if they all were, then N would be of the form 4x + 1. Since N is odd, each rime divisor i is odd so there is a N that is of the form 4x 1. Since i for any i, we have found a new rime of the form 4x 1. We can reeat this rocess indefinitely, so the set of rimes of the form 4x 1 cannot be finite. Note that this roof does not work if 4x 1 is relaced by 4x + 1, since a roduct of rimes of the form 4x 1 can be of the form 4x + 1. Examle Set 1 = 3, 2 = 7. Then is a rime of the form 4x 1. Next N = = 83 N = = 6971, which is again a rime of the form 4x 1. Again, N = = = This time 61 is a rime, but it is of the form 4x + 1 = However, is rime and = We are unstoable. N = = This time the small rime, 5591, is of the form 4x 1 and the large one is of the form 4x + 1. Theorem (Dirichlet). Let a and b be integers with gcd(a, b) = 1. Then there are infinitely many rimes of the form ax + b. Proofs of this theorem tyically use tools from advanced number theory, and are beyond the scoe of this book (see e.g., [FT93, VIII.4]) How Many Primes are There? We saw in Section that there are infinitely many rimes. In order to get a sense of just how many rimes there are, we consider a few warmu questions. Then we consider some numerical evidence and state the rime number theorem, which gives an asymtotic answer to our question,
22 1.2 The Sequence of Prime Numbers 15 and connect this theorem with a form of the famous Riemann Hyothesis. Our discussion of counting rimes in this section is very cursory; for more details, read Crandall and Pomerance s excellent book [CP01, 1.1.5]. The following vague discussion is meant to motivate a recise way to measure the number (or ercentage) of rimes. What ercentage of natural numbers are even? Answer: Half of them. What ercentage of natural numbers are of the form 4x 1? Answer: One fourth of them. What ercentage of natural numbers are erfect squares? Answer: Zero ercent of all natural numbers, in the sense that the limit of the roortion of erfect squares to all natural numbers converges to 0. More recisely, #{n N : n x and n is a erfect square} lim = 0, x x since the numerator is roughly x and lim x x x = 0. Likewise, it is an easy consequence of Theorem that zero ercent of all natural numbers are rime (see Exercise 1.4). We are thus led to ask another question: How many ositive integers x are erfect squares? Answer: Roughly x. In the context of rimes, we ask, Question How many natural numbers x are rime? Let π(x) = #{ N : x is a rime}. For examle, π(6) = #{2, 3, 5} = 3. Some values of π(x) are given in Table 1.1, and Figures 1.1 and 1.2 contain grahs of π(x). These grahs look like straight lines, which maybe bend down slightly. SAGE Examle To comute π(x) in Sage use the rime i(x) command: sage: rime_i(6) 3 sage: rime_i(100) 25 sage: rime_i( ) We can also draw a lot of π(x) using the lot command: sage: lot(rime_i, 1,1000, rgbcolor=(0,0,1)) Gauss was an inveterate comuter: he wrote in an 1849 letter that there are 216, 745 rimes less than 3, 000, 000 (this is wrong but close; the correct count is 216, 816).
23 16 1. Prime Numbers TABLE 1.1. Values of π(x) x π(x) (200, 46) (100, 25) (500, 95) (1000, 168) (900, 154) FIGURE 1.1. Grah of π(x) for x < 1000 Gauss conjectured the following asymtotic formula for π(x), which was later roved indeendently by Hadamard and Vallée Poussin in 1896 (but will not be roved in this book). Theorem (Prime Number Theorem). The function π(x) is asymtotic to x/ log(x), in the sense that lim x π(x) x/ log(x) = 1. We do nothing more here than motivate this dee theorem with a few further observations. The theorem imlies that so for any a, lim x π(x) lim x x = lim 1 x log(x) = 0, π(x) x/(log(x) a) = lim x π(x) x/ log(x) aπ(x) = 1. x Thus x/(log(x) a) is also asymtotic to π(x) for any a. See [CP01, 1.1.5] for a discussion of why a = 1 is the best choice. Table 1.2 comares π(x) and x/(log(x) 1) for several x < The record for counting rimes is π(10 23 ) = Note that such comutations are very difficult to get exactly right, so the above might be slightly wrong. For the reader familiar with comlex analysis, we mention a connection between π(x) and the Riemann Hyothesis. The Riemann zeta function ζ(s) is a comlex analytic function on C \ {1} that extends the function
24 1.2 The Sequence of Prime Numbers 17 TABLE 1.2. Comarison of π(x) and x/(log(x) 1) x π(x) x/(log(x) 1) (arox) FIGURE 1.2. Grahs of π(x) for x < and x <
25 18 1. Prime Numbers defined on a right half lane by n=1 n s. The Riemann Hyothesis is the conjecture that the zeros in C of ζ(s) with ositive real art lie on the line Re(s) = 1/2. This conjecture is one of the Clay Math Institute million dollar millennium rize roblems [Cla]. According to [CP01, 1.4.1], the Riemann Hyothesis is equivalent to the conjecture that x 1 Li(x) = log(t) dt is a good aroximation to π(x), in the following recise sense. Conjecture (Equivalent to the Riemann Hyothesis). For all x 2.01, π(x) Li(x) x log(x). 2 If x = 2, then π(2) = 1 and Li(2) = 0, but 2 log(2) = , so the inequality is not true for x 2, but 2.01 is big enough. We will do nothing more to exlain this conjecture, and settle for one numerical examle. Examle Let x = Then π(x) = , Li(x) = , π(x) Li(x) = , x log(x) = , x/(log(x) 1) = SAGE Examle We use Sage to grah π(x), Li(x), and x log(x). sage: P = lot(li, 2,10000, rgbcolor= urle ) sage: Q = lot(rime_i, 2,10000, rgbcolor= black ) sage: R = lot(sqrt(x)*log(x),2,10000,rgbcolor= red ) sage: show(p+q+r,xmin=0, figsize=[8,3]) The tomost line is Li(x), the next line is π(x), and the bottom line is x log(x).
26 1.3 Exercises 19 For more on the rime number theorem and the Riemann hyothesis see [Zag75] and [MS08]. 1.3 Exercises 1.1 Comute the greatest common divisor gcd(455, 1235) by hand. 1.2 Use the rime enumeration sieve to make a list of all rimes u to Prove that there are infinitely many rimes of the form 6x 1. π(x) 1.4 Use Theorem to deduce that lim x x = Let ψ(x) be the number of rimes of the form 4k 1 that are x. Use a comuter to make a conjectural guess about lim x ψ(x)/π(x). 1.6 So far 44 Mersenne rimes 2 1 have been discovered. Give a guess, backed u by an argument, about when the next Mersenne rime might be discovered (you will have to do some online research). 1.7 (a) Let y = Comute π(y) = #{rimes y}. (b) The rime number theorem imlies π(x) is asymtotic to How close is π(y) to y/ log(y), where y is as in (a)? 1.8 Let a, b, c, n be integers. Prove that (a) if a n and b n with gcd(a, b) = 1, then ab n. (b) if a bc and gcd(a, b) = 1, then a c. 1.9 Let a, b, c, d, and m be integers. Prove that (a) if a b and b c then a c. (b) if a b and c d then ac bd. (c) if m 0, then a b if and only if ma mb. (d) if d a and a 0, then d a. x log(x) In each of the following, aly the division algorithm to find q and r such that a = bq + r and 0 r < b : a = 300, b = 17, a = 729, b = 31, a = 300, b = 17, a = 389, b = (a) (Do this art by hand.) Comute the greatest common divisor of 323 and 437 using the algorithm described in class that involves quotients and remainders (i.e., do not just factor a and b).
27 20 1. Prime Numbers (b) Comute by any means the greatest common divisor of and (a) Suose a, b and n are ositive integers. Prove that if a n b n, then a b. (b) Suose is a rime and a and k are ositive integers. Prove that if a k, then k a k (a) Prove that if a ositive integer n is a erfect square, then n cannot be written in the form 4k + 3 for k an integer. (Hint: Comute the remainder uon division by 4 of each of (4m) 2, (4m + 1) 2, (4m + 2) 2, and (4m + 3) 2.) (b) Prove that no integer in the sequence 11, 111, 1111, 11111, ,... is a erfect square. (Hint: = = 4k +3.) 1.14 Prove that a ositive integer n is rime if and only if n is not divisible by any rime with 1 < n.
28 2 The Ring of Integers Modulo n This is age 21 Printer: Oaque this A startling fact about numbers is that it takes less than a second to decide with near certainty whether or not any given 1,000 digit number n is a rime, without actually factoring n. The algorithm for this involves doing some arithmetic with n that works differently deending on whether n is rime or comosite. In articular, we do arithmetic with the set (in fact, ring ) of integers {0, 1,..., n 1} using an innovative rule for addition and multilication, where the sum and roduct of two elements of that set is again in that set. Another surrising fact is that one can almost instantly comute the last 1,000 digits of a massive multibillion digit number like n = without exlicitly writing down all the digits of n. Again, this calculation involves arithmetic with the ring {0, 1,..., n 1}. This chater is about the ring Z/nZ of integers modulo n, the beautiful structure this ring has, and how to aly it to the above mentioned roblems, among others. It is foundational for the rest of this book. In Section 2.1, we discuss when linear equations modulo n have a solution, then introduce the Euler ϕ function and rove Euler s Theorem and Wilson s theorem. In Section 2.2, we rove the Chinese Remainer Theorem, which addresses simultaneous solubility of several linear equations modulo corime moduli. With these theoretical foundations in lace, in Section 2.3, we introduce algorithms for doing owerful comutations modulo n, including comuting large owers quickly, and solving linear equations. We finish in Section 2.4 with a discussion of recognizing rime numbers using arithmetic modulo n.
29 22 2. The Ring of Integers Modulo n 2.1 Congruences Modulo n Definition (Grou). A grou is a set G equied with a binary oeration G G G (denoted by multilication below) and an identity element 1 G such that: 1. For all a, b, c G, we have (ab)c = a(bc). 2. For each a G, we have 1a = a1 = a, and there exists b G such that ab = 1. Definition (Abelian Grou). An abelian grou is a grou G such that ab = ba for every a, b G. Definition (Ring). A ring R is a set equied with binary oerations + and and elements 0, 1 R such that R is an abelian grou under +, and for all a, b, c R we have 1a = a1 = a (ab)c = a(bc) a(b + c) = ab + ac. If, in addition, ab = ba for all a, b R, then we call R a commutative ring. In this section, we define the ring Z/nZ of integers modulo n, introduce the Euler ϕfunction, and relate it to the multilicative order of certain elements of Z/nZ. If a, b Z and n N, we say that a is congruent to b modulo n if n a b, and write a b (mod n). Let nz = (n) be the subset of Z consisting of all multiles of n (this is called the ideal of Z generated by n ). Definition (Integers Modulo n). The ring Z/nZ of integers modulo n is the set of equivalence classes of integers modulo n. It is equied with its natural ring structure: Examle For examle, (a + nz) + (b + nz) = (a + b) + nz (a + nz) (b + nz) = (a b) + nz. Z/3Z = {{..., 3, 0, 3,...}, {..., 2, 1, 4,...}, {..., 1, 2, 5,...}} SAGE Examle In Sage, we list the elements of Z/nZ as follows: sage: R = Integers(3) sage: list(r) [0, 1, 2]
30 2.1 Congruences Modulo n 23 We use the notation Z/nZ because Z/nZ is the quotient of the ring Z by the ideal nz of multiles of n. Because Z/nZ is the quotient of a ring by an ideal, the ring structure on Z induces a ring structure on Z/nZ. We often let a or a (mod n) denote the equivalence class a + nz of a. Definition (Field). A field K is a ring such that for every nonzero element a K there is an element b K such that ab = 1. For examle, if is a rime, then Z/Z is a field (see Exercise 2.12). Definition (Reduction Ma and Lift). We call the natural reduction ma Z Z/nZ, which sends a to a + nz, reduction modulo n. We also say that a is a lift of a + nz. Thus, e.g., 7 is a lift of 1 mod 3, since 7 + 3Z = 1 + 3Z. We can use that arithmetic in Z/nZ is well defined is to derive tests for divisibility by n (see Exercise 2.8). Proosition A number n Z is divisible by 3 if and only if the sum of the digits of n is divisible by 3. Proof. Write n = a + 10b + 100c +, where the digits of n are a, b, c, etc. Since 10 1 (mod 3), n = a + 10b + 100c + a + b + c + (mod 3), from which the roosition follows Linear Equations Modulo n In this section, we are concerned with how to decide whether or not a linear equation of the form ax b (mod n) has a solution modulo n. Algorithms for comuting solutions to ax b (mod n) are the toic of Section 2.3. First, we rove a roosition that gives a criterion under which one can cancel a quantity from both sides of a congruence. Proosition (Cancellation). If gcd(c, n) = 1 and ac bc (mod n), then a b (mod n). Proof. By definition n ac bc = (a b)c. Since gcd(n, c) = 1, it follows from Theorem that n a b, so a b (mod n), as claimed.
31 24 2. The Ring of Integers Modulo n When a has a multilicative inverse a in Z/nZ (i.e., aa 1 (mod n)) then the equation ax b (mod n) has a unique solution x a b (mod n). Thus, it is of interest to determine the units in Z/nZ, i.e., the elements which have a multilicative inverse. We will use comlete sets of residues to rove that the units in Z/nZ are exactly the a Z/nZ such that gcd(ã, n) = 1 for any lift ã of a to Z (it doesn t matter which lift). Definition (Comlete Set of Residues). We call a subset R Z of size n whose reductions modulo n are airwise distinct a comlete set of residues modulo n. In other words, a comlete set of residues is a choice of reresentative for each equivalence class in Z/nZ. For examle, R = {0, 1, 2,..., n 1} is a comlete set of residues modulo n. When n = 5, R = {0, 1, 1, 2, 2} is a comlete set of residues. Lemma If R is a comlete set of residues modulo n and a Z with gcd(a, n) = 1, then ar = {ax : x R} is also a comlete set of residues modulo n. Proof. If ax ax (mod n) with x, x R, then Proosition imlies that x x (mod n). Because R is a comlete set of residues, this imlies that x = x. Thus the elements of ar have distinct reductions modulo n. It follows, since #ar = n, that ar is a comlete set of residues modulo n. Proosition (Units). If gcd(a, n) = 1, then the equation ax b (mod n) has a solution, and that solution is unique modulo n. Proof. Let R be a comlete set of residues modulo n, so there is a unique element of R that is congruent to b modulo n. By Lemma , ar is also a comlete set of residues modulo n, so there is a unique element ax ar that is congruent to b modulo n, and we have ax b (mod n). Algebraically, this roosition asserts that if gcd(a, n) = 1, then the ma Z/nZ Z/nZ given by left multilication by a is a bijection. Examle Consider the equation 2x 3 (mod 7), and the comlete set R = {0, 1, 2, 3, 4, 5, 6} of coset reresentatives. We have so (mod 7). 2R = {0, 2, 4, 6, 8 1, 10 3, 12 5}, When gcd(a, n) 1, then the equation ax b (mod n) may or may not have a solution. For examle, 2x 1 (mod 4) has no solution, but 2x 2 (mod 4) does, and in fact it has more than one mod 4 (x = 1 and x = 3). Generalizing Proosition , we obtain the following more general criterion for solvability.
32 2.1 Congruences Modulo n 25 Proosition (Solvability). The equation ax b (mod n) has a solution if and only if gcd(a, n) divides b. Proof. Let g = gcd(a, n). If there is a solution x to the equation ax b (mod n), then n (ax b). Since g n and g a, it follows that g b. Conversely, suose that g b. Then n (ax b) if and only if ( n a g g x b ). g Thus ax b (mod n) has a solution if and only if a g x b g (mod n g ) has a solution. Since gcd(a/g, n/g) = 1, Proosition imlies this latter equation does have a solution. In Chater 4, we will study quadratic recirocity, which gives a nice criterion for whether or not a quadratic equation modulo n has a solution Euler s Theorem Let (Z/nZ) denote the set of elements [x] Z/nZ such that gcd(x, n) = 1. The set (Z/nZ) is a grou, called the grou of units of the ring Z/nZ; it will be of great interest to us. Each element of this grou has an order, and Lagrange s theorem from grou theory imlies that each element of (Z/nZ) has an order that divides the order of (Z/nZ). In elementary number theory, this fact goes by the monicker Fermat s Little Theorem when n is rime and Euler s Theorem in general, and we rerove it from basic rinciles in this section. Definition (Order of an Element). Let n N and x Z and suose that gcd(x, n) = 1. The order of x modulo n is the smallest m N such that x m 1 (mod n). To show that the definition makes sense, we verify that such an m exists. Consider x, x 2, x 3,... modulo n. There are only finitely many residue classes modulo n, so we must eventually find two integers i, j with i < j such that x j x i (mod n). Since gcd(x, n) = 1, Proosition imlies that we can cancel x s and conclude that x j i 1 (mod n). SAGE Examle Use x.multilicative order() to comute the order of an element of Z/nZ in Sage.
33 26 2. The Ring of Integers Modulo n sage: R = Integers(10) sage: a = R(3) # create an element of Z/10Z sage: a.multilicative_order() 4 Notice that the owers of a are eriodic with eriod 4, i.e., there are four owers and they reeat: sage: [a^i for i in range(15)] [1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9] The command range(n) we use above returns the list of integers between 0 and n 1, inclusive. Definition (Euler s ϕfunction). For n N, let For examle, ϕ(n) = #{a N : a n and gcd(a, n) = 1}. ϕ(1) = #{1} = 1, ϕ(2) = #{1} = 1, ϕ(5) = #{1, 2, 3, 4} = 4, ϕ(12) = #{1, 5, 7, 11} = 4. Also, if is any rime number then ϕ() = #{1, 2,..., 1} = 1. In Section 2.2.1, we rove that if gcd(m, r) = 1, then ϕ(mr) = ϕ(m)ϕ(r). This will yield an easy way to comute ϕ(n) in terms of the rime factorization of n. SAGE Examle Use the euler hi(n) command to comute ϕ(n) in Sage: sage: euler_hi(2007) 1332 Theorem (Euler s Theorem). If gcd(x, n) = 1, then x ϕ(n) 1 (mod n). Proof. As mentioned above, Euler s Theorem has the following groutheoretic interretation. The set of units in Z/nZ is a grou (Z/nZ) = {a Z/nZ : gcd(a, n) = 1} that has order ϕ(n). The theorem then asserts that the order of an element of (Z/nZ) divides the order ϕ(n) of (Z/nZ). This is a secial case of
34 2.1 Congruences Modulo n 27 the more general fact (Lagrange s Theorem) that if G is a finite grou and g G, then the order of g divides the cardinality of G. We now give an elementary roof of the theorem. Let P = {a : 1 a n and gcd(a, n) = 1}. In the same way that we roved Lemma , we see that the reductions modulo n of the elements of xp are the same as the reductions of the elements of P. Thus a P(xa) a (mod n), a P since the roducts are over the same numbers modulo n. Now cancel the a s on both sides to get x #P 1 (mod n), as claimed. SAGE Examle We illustrate Euler s Theorem using Sage. The Mod(x,n) command returns the equivalence class of x in Z/nZ. sage: n = 20 sage: k = euler_hi(n); k 8 sage: [Mod(x,n)^k for x in range(n) if gcd(x,n) == 1] [1, 1, 1, 1, 1, 1, 1, 1] Wilson s Theorem The following characterization of rime numbers, from the 1770s, is called Wilson s Theorem, though it was first roved by Lagrange. Proosition (Wilson s Theorem). An integer > 1 is rime if and only if ( 1)! 1 (mod ). For examle, if = 3, then ( 1)! = 2 1 (mod 3). If = 17, then But if = 15, then ( 1)! = (mod 17). ( 1)! = (mod 15), so 15 is comosite. Thus Wilson s theorem could be viewed as a rimality test, though, from a comutational oint of view, it is robably one of the world s least efficient rimality tests since comuting (n 1)! takes so many stes.
35 28 2. The Ring of Integers Modulo n Proof. The statement is clear when = 2, so henceforth we assume that > 2. We first assume that is rime and rove that ( 1)! 1 (mod ). If a {1, 2,..., 1}, then the equation ax 1 (mod ) has a unique solution a {1, 2,..., 1}. If a = a, then a 2 1 (mod ), so a 2 1 = (a 1)(a+1), so (a 1) or (a+1), so a {1, 1}. We can thus air off the elements of {2, 3,..., 2}, each with their inverse. Thus 2 3 ( 2) 1 (mod ). Multilying both sides by 1 roves that ( 1)! 1 (mod ). Next, we assume that ( 1)! 1 (mod ) and rove that must be rime. Suose not, so that 4 is a comosite number. Let l be a rime divisor of. Then l <, so l ( 1)!. Also, by assumtion, l (( 1)! + 1). This is a contradiction, because a rime can not divide a number a and also divide a + 1, since it would then have to divide (a + 1) a = 1. Examle We illustrate the key ste in the above roof in the case = 17. We have = (2 9) (3 6) (4 13) (5 7) (8 15) (10 12) (14 11) 1 (mod 17), where we have aired u the numbers a, b for which ab 1 (mod 17). SAGE Examle We use Sage to create a table of triles; the first column contains n, the second column contains (n 1)! modulo n, and the third contains 1 modulo n. Notice that the first columns contains a rime recisely when the second and third columns are equal. (The... notation indicates a multiline command in Sage; you should not tye the dots in exlicitly.) sage: for n in range(1,10):... rint n, factorial(n1) % n, 1 % n
36 2.2 The Chinese Remainder Theorem The Chinese Remainder Theorem In this section, we rove the Chinese Remainder Theorem, which gives conditions under which a system of linear equations is guaranteed to have a solution. In the 4th century a Chinese mathematician asked the following: Question There is a quantity whose number is unknown. Reeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What is the quantity? In modern notation, Question asks us to find a ositive integer solution to the following system of three equations: x 2 (mod 3) x 3 (mod 5) x 2 (mod 7) The Chinese Remainder Theorem asserts that a solution exists, and the roof gives a method to find one. (See Section 2.3 for the necessary algorithms.) Theorem (Chinese Remainder Theorem). Let a, b Z and n, m N such that gcd(n, m) = 1. Then there exists x Z such that x a x b (mod m), (mod n). Moreover x is unique modulo mn. Proof. If we can solve for t in the equation a + tm b (mod n), then x = a + tm will satisfy both congruences. To see that we can solve, subtract a from both sides and use Proosition together with our assumtion that gcd(n, m) = 1 to see that there is a solution. For uniqueness, suose that x and y solve both congruences. Then z = x y satisfies z 0 (mod m) and z 0 (mod n), so m z and n z. Since gcd(n, m) = 1, it follows that nm z, so x y (mod nm). Algorithm (Chinese Remainder Theorem). Given corime integers m and n and integers a and b, this algorithm find an integer x such that x a (mod m) and x b (mod n). 1. [Extended GCD] Use Algorithm below to find integers c, d such that cm + dn = [Answer] Outut x = a + (b a)cm and terminate.
37 30 2. The Ring of Integers Modulo n Proof. Since c Z, we have x a (mod m), and using that cm + dn = 1, we have a + (b a)cm a + (b a) b (mod n). Now we can answer Question First, we use Theorem to find a solution to the air of equations x 2 (mod 3), x 3 (mod 5). Set a = 2, b = 3, m = 3, n = 5. Ste 1 is to find a solution to t (mod 5). A solution is t = 2. Then x = a + tm = = 8. Since any x with x x (mod 15) is also a solution to those two equations, we can solve all three equations by finding a solution to the air of equations x 8 (mod 15) x 2 (mod 7). Again, we find a solution to t (mod 7). A solution is t = 1, so x = a + tm = = 23. Note that there are other solutions. Any x x (mod 3 5 7) is also a solution; e.g., = 128. SAGE Examle The CRT(a,b,m,n) command in Sage comutes an integer x such that x a (mod m) and x b (mod n). For examle, sage: CRT(2,3, 3, 5) 7 The CRT list command comutes a number that reduces to several numbers modulo corime moduli. We use it to answer Question 2.2.1: sage: CRT_list([2,3,2], [3,5,7]) Multilicative Functions Recall from Definition that the Euler ϕfunction is ϕ(n) = #{a : 1 a n and gcd(a, n) = 1}. Lemma Suose that m, n N and gcd(m, n) = 1. Then the ma ψ : (Z/mnZ) (Z/mZ) (Z/nZ). (2.2.1) defined by is a bijection. ψ(c) = (c mod m, c mod n)
38 2.3 Quickly Comuting Inverses and Huge Powers 31 Proof. We first show that ψ is injective. If ψ(c) = ψ(c ), then m c c and n c c, so nm c c because gcd(n, m) = 1. Thus c = c as elements of (Z/mnZ). Next we show that ψ is surjective, i.e., that every element of (Z/mZ) (Z/nZ) is of the form ψ(c) for some c. Given a and b with gcd(a, m) = 1 and gcd(b, n) = 1, Theorem imlies that there exists c with c a (mod m) and c b (mod n). We may assume that 1 c nm, and since gcd(a, m) = 1 and gcd(b, n) = 1, we must have gcd(c, nm) = 1. Thus ψ(c) = (a, b). Definition (Multilicative Function). A function f : N C is multilicative if, whenever m, n N and gcd(m, n) = 1, we have f(mn) = f(m) f(n). Proosition (Multilicativity of ϕ). The function ϕ is multilicative. Proof. The ma ψ of Lemma is a bijection, so the set on the left in (2.2.1) has the same size as the roduct set on the right in (2.2.1). Thus ϕ(mn) = ϕ(m) ϕ(n). The roosition is helful in comuting ϕ(n), at least if we assume we can comute the factorization of n (see Section for a connection between factoring n and comuting ϕ(n)). For examle, Also, for n 1, we have ϕ(12) = ϕ(2 2 ) ϕ(3) = 2 2 = 4. ϕ( n ) = n n = n n 1 = n 1 ( 1), (2.2.2) since ϕ( n ) is the number of numbers less than n minus the number of those that are divisible by. Thus, e.g., ϕ( ) = 388 ( ) = = Quickly Comuting Inverses and Huge Powers This section is about how to solve the equation ax 1 (mod n) when we know it has a solution, and how to efficiently comute a m (mod n). We also discuss a simle robabilistic rimality test that relies on our ability to comute a m (mod n) quickly. All three of these algorithms are of fundamental imortance to the crytograhy algorithms of Chater 3.
39 32 2. The Ring of Integers Modulo n How to Solve ax 1 (mod n) Suose a, n N with gcd(a, n) = 1. Then by Proosition the equation ax 1 (mod n) has a unique solution. How can we find it? Proosition (Extended Euclidean Reresentation). Suose a, b Z and let g = gcd(a, b). Then there exists x, y Z such that ax + by = g. Remark If e = cg is a multile of g, then cax + cby = cg = e, so e = (cx)a + (cy)b can also be written in terms of a and b. Proof of Proosition Let g = gcd(a, b). Then gcd(a/g, b/g) = 1, so by Proosition , the equation ( a g x 1 mod b ) (2.3.1) g has a solution x Z. Multilying (2.3.1) through by g yields ax g (mod b), so there exists y such that b ( y) = ax g. Then ax + by = g, as required. Given a, b and g = gcd(a, b), our roof of Proosition gives a way to exlicitly find x, y such that ax+by = g, assuming one knows an algorithm to solve linear equations modulo n. Since we do not know such an algorithm, we now discuss a way to exlicitly find x and y. This algorithm will in fact enable us to solve linear equations modulo n. To solve ax 1 (mod n) when gcd(a, n) = 1, use the Algorithm to find x and y such that ax + ny = 1. Then ax 1 (mod n). Examle Suose a = 5 and b = 7. The stes of Algorithm to comute gcd(5, 7) are as follows. Here we underline certain numbers, because it clarifies the subsequent back substitution we will use to find x and y. 7 = so 2 = = so 1 = = 5 2(7 5) = On the right, we have backsubstituted in order to write each artial remainder as a linear combination of a and b. In the last ste, we obtain gcd(a, b) as a linear combination of a and b, as desired. Examle That examle was not too comlicated, so we try another one. Let a = 130 and b = 61. We have 130 = = = = = = = = = =
40 2.3 Quickly Comuting Inverses and Huge Powers 33 Thus x = 23 and y = 49 is a solution to 130x + 61y = 1. Examle This examle is just like Examle above, excet we make the notation on the right more comact. 130 = = (1, 2) 61 = = ( 7, 15) = (0, 1) 7(1, 2) 8 = = (8, 17) = (1, 2) ( 7, 15) 5 = = ( 15, 32) = ( 7, 15) (8, 17) 3 = = (23, 49) = (8, 17) ( 15, 32) Notice at each ste that the vector on the right is just the vector from two stes ago minus a multile of the vector from one ste ago, where the multile is the cofficient of what we divide by. SAGE Examle The xgcd(a,b) command comutes the greatest common divisor g of a and b along with x, y such that ax + by = g. sage: xgcd(5,7) (1, 4, 3) sage: xgcd(130,61) (1, 23, 49) Algorithm (Extended Euclidean Algorithm). Suose a and b are integers and let g = gcd(a, b). This algorithm finds g, x and y such that ax + by = g. We describe only the stes when a > b 0, since one can easily reduce to this case. 1. [Initialize] Set x = 1, y = 0, r = 0, s = [Finished?] If b = 0, set g = a and terminate. 3. [Quotient and Remainder] Use Algorithm to write a = qb + c with 0 c < b. 4. [Shift] Set (a, b, r, s, x, y) = (b, c, x qr, y qs, r, s) and go to Ste 2. (This shift ste is nicely illustrated in Examle ) Proof. This algorithm is the same as Algorithm , excet that we kee track of extra variables x, y, r, s, so it terminates and when it terminates d = gcd(a, b). We omit the rest of the inductive roof that the algorithm is correct, and instead refer the reader to [Knu97, 1.2.1]. Algorithm (Inverse Modulo n). Suose a and n are integers and gcd(a, n) = 1. This algorithm finds an x such that ax 1 (mod n). 1. [Comute Extended GCD] Use Algorithm to comute integers x, y such that ax + ny = gcd(a, n) = [Finished] Outut x.
41 34 2. The Ring of Integers Modulo n Proof. Reduce ax+ny = 1 modulo n to see that x satisfies ax 1 (mod n). Examle Solve 17x 1 (mod 61). First, we use Algorithm to find x, y such that 17x + 61y = 1: 61 = = = = = = = = Thus ( 5) = 1 so x = 18 is a solution to 17x 1 (mod 61). SAGE Examle Sage imlements the above algorithm for quickly comuting inverses modulo n. For examle, sage: a = Mod(17, 61) sage: a^(1) How to Comute a m (mod n) Let a and n be integers, and m a nonnegative integer. In this section, we describe an efficient algorithm to comute a m (mod n). For the crytograhy alications in Chater 3, m will have hundreds of digits. The naive aroach to comuting a m (mod n) is to simly comute a m = a a a (mod n) by reeatedly multilying by a and reducing modulo m. Note that after each arithmetic oeration is comleted, we reduce the result modulo n so that the sizes of the numbers involved do not get too large. Nonetheless, this algorithm is horribly inefficient because it takes m 1 multilications, which is huge if m has hundreds of digits. A much more efficient algorithm for comuting a m (mod n) involves writing m in binary, then exressing a m as a roduct of exressions a 2i, for various i. These latter exressions can be comuted by reeatedly squaring a 2i. This more clever algorithm is not simler, but it is vastly more efficient since the number of oerations needed grows with the number of binary digits of m, whereas with the naive algorithm in the revious aragrah, the number of oerations is m 1. Algorithm (Write a number in binary). Let m be a nonnegative integer. This algorithm writes m in binary, so it finds ε i {0, 1} such that m = r i=0 ε i2 i with each ε i {0, 1}. 1. [Initialize] Set i = [Finished?] If m = 0, terminate. 3. [Digit] If m is odd, set ε i = 1, otherwise ε i = 0. Increment i.
42 2.3 Quickly Comuting Inverses and Huge Powers [Divide by 2] Set m = m 2, the greatest integer m/2. Goto Ste 2. SAGE Examle To write a number in binary using Sage, use the str command: sage: 100.str(2) Notice the above is the correct binary exansion: sage: 0*2^0 + 0*2^1 + 1*2^2 + 0*2^3 + 0*2^4 + 1*2^5 + 1*2^6 100 Algorithm (Comute Power). Let a and n be integers and m a nonnegative integer. This algorithm comutes a m modulo n. 1. [Write in Binary] Write m in binary using Algorithm , so a m = (mod n). ε i=1 a2i 2. [Comute Powers] Comute a, a 2, a 22 = (a 2 ) 2, a 23 = (a 22 ) 2, etc., u to a 2r, where r + 1 is the number of binary digits of m. 3. [Multily Powers] Multily together the a 2i such that ε i = 1, always working modulo n. Examle We can comute the last 2 digits of 7 91, by finding 7 91 (mod 100). First, because gcd(7, 100) = 1, we have by Theorem that 7 ϕ(100) 1 (mod 100). Because ϕ is multilicative, ϕ(100) = ϕ( ) = (2 2 2) (5 2 5) = 40. Thus (mod 100), hence (mod 100). We now comute 7 11 (mod 100) using the above algorithm. First, write 11 in binary by reeatedly dividing by = = = = So in binary, (11) 2 = 1011, which we check: 11 =
43 36 2. The Ring of Integers Modulo n Next, comute a, a 2, a 4, a 8 and outut a 8 a 2 a. We have a = 7 a 2 49 a a Note: it is easiest to square 49 by working modulo 4 and 25 and using the Chinese Remainder Theorem. Finally, a 8 a 2 a (mod 100). SAGE Examle Sage imlements the above algorithm for comuting owers efficiently. For examle, sage: Mod(7,100)^91 43 We can also, of course, directly comute 7 91 in Sage, though we would not want to do this by hand: sage: 7^ Primality Testing Theorem (Pseudorimality). An integer > 1 is rime if and only if for every a 0 (mod ), a 1 1 (mod ). Proof. If is rime, then the statement follows from Proosition If is comosite, then there is a divisor a of with 2 a <. If a 1 1 (mod ), then a 1 1. Since a, we have a a 1 1, hence there exists an integer k such that ak = a 1 1. Subtracting, we see that a 1 ak = 1, so a(a 2 k) = 1. This imlies that a 1, which is a contradiction since a 2. Suose n N. Using Theorem and Algorithm , we can either quickly rove that n is not rime, or convince ourselves that n is likely rime (but not quickly rove that n is rime). For examle, if 2 n 1 1 (mod n), then we have roved that n is not rime. On the other hand, if a n 1 1 (mod n) for a few a, it seems likely that n is rime, and we loosely refer to such a number that seems rime for several bases as a seudorime.
44 2.4 Primality Testing 37 There are comosite numbers n (called Carmichael numbers) with the amazing roerty that a n 1 1 (mod n) for all a with gcd(a, n) = 1. The first Carmichael number is 561, and it is a theorem that there are infinitely many such numbers ([AGP94]). Examle Is = 323 rime? We comute (mod 323). Making a table as above, we have i m ε i 2 2i mod Thus (mod 323), so 323 is not rime, though this comutation gives no information about how 323 factors as a roduct of rimes. In fact, one finds that 323 = SAGE Examle It s ossible to easily rove that a large number is comosite, but the roof does not easily yield a factorization. For examle if n = , then 2 n 1 1 (mod n), so n is comosite. sage: n = sage: Mod(2,n)^(n1) Note that factoring n actually takes much longer than the above comutation (which was essentially instant). sage: factor(n) # takes u to a few seconds * Another ractical rimality test is the MillerRabin test, which has the roerty that each time it is run on a number n it either correctly asserts that the number is definitely not rime, or that it is robably rime, and the robability of correctness goes u with each successive call. If Miller Rabin is called m times on n and in each case claims that n is robably rime, then one can in a recise sense bound the robability that n is comosite in terms of m.
45 38 2. The Ring of Integers Modulo n We state the MillerRabin algorithm recisely, but do not rove anything about the robability that it will succeed. Algorithm (MillerRabin Primality Test). Given an integer n 5 this algorithm oututs either true or false. If it oututs true, then n is robably rime, and if it oututs false, then n is definitely comosite. 1. [Slit Off Power of 2] Comute the unique integers m and k such that m is odd and n 1 = 2 k m. 2. [Random Base] Choose a random integer a with 1 < a < n. 3. [Odd Power] Set b = a m (mod n). If b ±1 (mod n) outut true and terminate. 4. [Even Powers] If b 2r 1 (mod n) for any r with 1 r k 1, outut true and terminate. Otherwise outut false. If MillerRabin oututs true for n, we can call it again with n and if it again oututs true then the robability that we have incorrectly determined that n is rime (when n is actually comosite) decreases. Proof. We will rove that the algorithm is correct, but will rove nothing about how likely the algorithm is to assert that a comosite is rime. We must rove that if the algorithm ronounces an integer n comosite, then n really is comosite. Thus suose n is rime, yet the algorithm ronounces n comosite. Then a m ±1 (mod n), and for all r with 1 r k 1 we have a 2rm 1 (mod n). Since n is rime and 2 k 1 m = (n 1)/2, Proosition imlies that a 2k 1m ±1 (mod n), so by our hyothesis a 2k 1m 1 (mod n). But then (a 2k 2m ) 2 1 (mod n), so by Proosition (which is roved right after it is stated, and whose roof does not deend on this argument), we have a 2k 2m ±1 (mod n). Again, by our hyothesis, this imlies a 2k 2m 1 (mod n). Reeating this argument inductively, we see that a m ±1 (mod n), which contradicts our hyothesis on a. Until recently it was an oen roblem to give an algorithm (with roof) that decides whether or not any integer is rime in time bounded by a olynomial in the number of digits of the integer. Agrawal, Kayal, and Saxena recently found the first olynomialtime rimality test (see [AKS02]). We will not discuss their algorithm further, because for our alications to crytograhy MillerRabin or seudorimality tests will be sufficient. See [Sho05, Ch. 21] for a book that gives a detailed exosition of this algorithm. SAGE Examle The is rime command uses a combination of techniques to determine (rovably correctly!) whether or not an integer is rime. sage: n = sage: is_rime(n) False
46 2.5 The Structure of (Z/Z) 39 We use the is rime function to make a table of the first few Mersenne rimes (see Section 1.2.3). sage: for in rimes(100):... if is_rime(2^  1):... rint, 2^ There is a secialized test for rimality of Mersenne numbers called the LucasLehmer test. This remarkably simle algorithm determines rovably correctly whether or not a number 2 1 is rime. We imlement it in a few lines of code and use the LucasLehmer test to check for rimality of two Mersenne numbers: sage: def is_rime_lucas_lehmer():... s = Mod(4, 2^  1)... for i in range(3, +1):... s = s^ return s == 0 sage: # Check rimality of 2^ sage: is_rime_lucas_lehmer(9941) True sage: # Check rimality of 2^next_rime(1000)1 sage: is_rime_lucas_lehmer(next_rime(1000)) False For more on Mersenne rimes, see the Great Internet Mersenne Prime Search (GIMPS) roject at htt:// 2.5 The Structure of (Z/Z) This section is about the structure of the grou (Z/Z) of units modulo a rime number. The main result is that this grou is always cyclic. We will use this result later in Chater 4 in our roof of quadratic recirocity. Definition (Primitive root). A rimitive root modulo an integer n is an element of (Z/nZ) of order ϕ(n).
47 40 2. The Ring of Integers Modulo n We will rove that there is a rimitive root modulo every rime. Since the unit grou (Z/Z) has order 1, this imlies that (Z/Z) is a cyclic grou, a fact that will be extremely useful, since it comletely determines the structure of (Z/Z) as a grou. If n is an odd rime ower, then there is a rimitive root modulo n (see Exercise 2.28), but there is no rimitive root modulo the rime ower 2 3, and hence none mod 2 n for n 3 (see Exercise 2.27). Section is the key inut to our roof that (Z/Z) is cyclic; here we show that for every divisor d of 1 there are exactly d elements of (Z/Z) whose order divides d. We then use this result in Section to roduce an element of (Z/Z) of order q r when q r is a rime ower that exactly divides 1 (i.e., q r divides 1, but q r+1 does not divide 1), and multily together these elements to obtain an element of (Z/Z) of order 1. SAGE Examle Use the rimitive root command to comute the smallest ositive integer that is a rimitive root modulo n. For examle, below we comute rimitive roots modulo for each rime < 20. sage: for in rimes(20):... rint, rimitive_root() Polynomials over Z/Z The olynomials x 2 1 has four roots in Z/8Z, namely 1, 3, 5, and 7. In contrast, the following roosition shows that a olynomial of degree d over a field, such as Z/Z, can have at most d roots. Proosition (Root Bound). Let f k[x] be a nonzero olynomial over a field k. Then there are at most deg(f) elements α k such that f(α) = 0.
48 2.5 The Structure of (Z/Z) 41 Proof. We rove the roosition by induction on deg(f). The cases in which deg(f) 1 are clear. Write f = a n x n + a 1 x + a 0. If f(α) = 0, then f(x) = f(x) f(α) = a n (x n α n ) + + a 1 (x α) + a 0 (1 1) = (x α)(a n (x n α n 1 ) + + a 2 (x + α) + a 1 ) = (x α)g(x), for some olynomial g(x) k[x]. Next, suose that f(β) = 0 with β α. Then (β α)g(β) = 0, so, since β α 0 and k is a field, we have g(β) = 0. By our inductive hyothesis, g has at most n 1 roots, so there are at most n 1 ossibilities for β. It follows that f has at most n roots. SAGE Examle We use Sage to find the roots of a olynomials over Z/13Z. sage: R.<x> = PolynomialRing(Integers(13)) sage: f = x^ sage: f.roots() [(12, 1), (10, 1), (4, 1)] sage: f(12) 0 The outut of the roots command above lists each root along with its multilicity (which is 1 in each case above). Proosition Let be a rime number and let d be a divisor of 1. Then f = x d 1 (Z/Z)[x] has exactly d roots in Z/Z. Proof. Let e = ( 1)/d. We have x 1 1 = (x d ) e 1 = (x d 1)((x d ) e 1 + (x d ) e ) = (x d 1)g(x), where g (Z/Z)[x] and deg(g) = de d = 1 d. Theorem imlies that x 1 1 has exactly 1 roots in Z/Z, since every nonzero element of Z/Z is a root! By Proosition 2.5.3, g has at most 1 d roots and x d 1 has at most d roots. Since a root of (x d 1)g(x) is a root of either x d 1 or g(x) and x 1 1 has 1 roots, g must have exactly 1 d roots and x d 1 must have exactly d roots, as claimed. SAGE Examle We use Sage to illustrate the roosition. sage: R.<x> = PolynomialRing(Integers(13)) sage: f = x^6 + 1 sage: f.roots() [(11, 1), (8, 1), (7, 1), (6, 1), (5, 1), (2, 1)]
49 42 2. The Ring of Integers Modulo n We ause to reemhasize that the analog of Proosition is false when is relaced by a comosite integer n, since a root mod n of a roduct of two olynomials need not be a root of either factor. For examle, f = x 2 1 = (x 1)(x + 1) Z/15Z[x] has the four roots 1, 4, 11, and Existence of Primitive Roots Recall from Section that the order of an element x in a finite grou is the smallest m 1 such that x m = 1. In this section, we rove that (Z/Z) is cyclic by using the results of Section to roduce an element of (Z/Z) of order d for each rime ower divisor d of 1, and then we multily these together to obtain an element of order 1. We will use the following lemma to assemble elements of each order dividing 1 to roduce an element of order 1. Lemma Suose a, b (Z/nZ) have orders r and s, resectively, and that gcd(r, s) = 1. Then ab has order rs. Proof. This is a general fact about commuting elements of any grou; our roof only uses that ab = ba and nothing secial about (Z/nZ). Since (ab) rs = a rs b rs = 1, the order of ab is a divisor of rs. Write this divisor as r 1 s 1 where r 1 r and s 1 s. Raise both sides of the equation to the ower r 2 = r/r 1 to obtain a r1s1 b r1s1 = (ab) r1s1 = 1 a r1r2s1 b r1r2s1 = 1. Since a r1r2s1 = (a r1r2 ) s1 = 1, we have b r1r2s1 = 1, so s r 1 r 2 s 1. Since gcd(s, r 1 r 2 ) = gcd(s, r) = 1, it follows that s = s 1. Similarly r = r 1, so the order of ab is rs. Theorem (Primitive Roots). There is a rimitive root modulo any rime. In articular, the grou (Z/Z) is cyclic. Proof. The theorem is true if = 2, since 1 is a rimitive root, so we may assume > 2. Write 1 as a roduct of distinct rime owers q ni i : 1 = q n1 1 qn2 2 qnr r. By Proosition 2.5.5, the olynomial x qn i i the olynomial x qn i 1 i 1 has exactly q ni 1 i 1 has exactly q ni i roots. There are q ni i roots, and q ni 1 i =
50 2.5 The Structure of (Z/Z) 43 q ni 1 i (q i 1) elements a Z/Z such that a qn i i = 1 but a qn i 1 i 1; each of these elements has order q ni i. Thus for each i = 1,..., r, we can choose an a i of order q ni i. Then, using Lemma reeatedly, we see that a = a 1 a 2 a r has order q n1 1 qnr r = 1, so a is a rimitive root modulo. Examle We illustrate the roof of Theorem when = 13. We have 1 = 12 = The olynomial x 4 1 has roots {1, 5, 8, 12} and x 2 1 has roots {1, 12}, so we may take a 1 = 5. The olynomial x 3 1 has roots {1, 3, 9}, and we set a 2 = 3. Then a = 5 3 = 15 2 is a rimitive root. To verify this, note that the successive owers of 2 (mod 13) are 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1. Examle Theorem is false if, for examle, is relaced by a ower of 2 bigger than 4. For examle, the four elements of (Z/8Z) each have order dividing 2, but ϕ(8) = 4. Theorem (Primitive Roots mod n ). Let n be a ower of an odd rime. Then there is a rimitive root modulo n. The roof is left as Exercise Proosition (Number of rimitive roots). If there is a rimitive root modulo n, then there are exactly ϕ(ϕ(n)) rimitive roots modulo n. Proof. The rimitive roots modulo n are the generators of (Z/nZ), which by assumtion is cyclic of order ϕ(n). Thus they are in bijection with the generators of any cyclic grou of order ϕ(n). In articular, the number of rimitive roots modulo n is the same as the number of elements of Z/ϕ(n)Z with additive order ϕ(n). An element of Z/ϕ(n)Z has additive order ϕ(n) if and only if it is corime to ϕ(n). There are ϕ(ϕ(n)) such elements, as claimed. Examle For examle, there are ϕ(ϕ(17)) = ϕ(16) = = 8 rimitive roots mod 17, namely 3, 5, 6, 7, 10, 11, 12, 14. The ϕ(ϕ(9)) = ϕ(6) = 2 rimitive roots modulo 9 are 2 and 5. There are no rimitive roots modulo 8, even though ϕ(ϕ(8)) = ϕ(4) = 2 > Artin s Conjecture Conjecture (Emil Artin). Suose a Z is not 1 or a erfect square. Then there are infinitely many rimes such that a is a rimitive root modulo.
51 44 2. The Ring of Integers Modulo n There is no single integer a such that Artin s conjecture is known to be true. For any given a, Pieter [Mor93] roved that there are infinitely many such that the order of a is divisible by the largest rime factor of 1. Hooley [Hoo67] roved that something called the Generalized Riemann Hyothesis imlies Conjecture Remark Artin conjectured more recisely that if N(x, a) is the number of rimes x such that a is a rimitive root modulo, then N(x, a) is asymtotic to C(a)π(x), where C(a) is a ositive constant that deends only on a and π(x) is the number of rimes u to x Comuting Primitive Roots Theorem does not suggest an efficient algorithm for finding rimitive roots. To actually find a rimitive root mod in ractice, we try a = 2, then a = 3, etc., until we find an a that has order 1. Comuting the order of an element of (Z/Z) requires factoring 1, which we do not know how to do quickly in general, so finding a rimitive root modulo for large seems to be a difficult roblem. Algorithm (Primitive Root). Given a rime, this algorithm comutes the smallest ositive integer a that generates (Z/Z). 1. [ = 2?] If = 2 outut 1 and terminate. Otherwise set a = [Prime Divisors] Comute the rime divisors 1,..., r of [Generator?] If for every i, we have a ( 1)/i 1 (mod ), then a is a generator of (Z/Z), so outut a and terminate. 4. [Try next] Set a = a + 1 and go to Ste 3. Proof. Let a (Z/Z). The order of a is a divisor d of the order 1 of the grou (Z/Z). Write d = ( 1)/n, for some divisor n of 1. If a is not a generator of (Z/Z), then since n ( 1), there is a rime divisor i of 1 such that i n. Then a ( 1)/i = (a ( 1)/n ) n/i 1 (mod ). Conversely, if a is a generator, then a ( 1)/i 1 (mod ) for any i. Thus the algorithm terminates with Ste 3 if and only if the a under consideration is a rimitive root. By Theorem 2.5.8, there is at least one rimitive root, so the algorithm terminates. 2.6 Exercises 2.1 Prove that for any ositive integer n, the set (Z/nZ) under multilication modulo n is a grou.
52 2.2 Comute the following gcd s using Algorithm : 2.6 Exercises 45 gcd(15, 35) gcd(247, 299) gcd(51, 897) gcd(136, 304) 2.3 Use Algorithm to find x, y Z such that 2261x y = Prove that if a and b are integers and is a rime, then (a + b) a + b (mod ). You may assume that the binomial coefficient is an integer.! r!( r)! 2.5 (a) Prove that if x, y is a solution to ax+by = d, with d = gcd(a, b), then for all c Z, x = x + c b d, is also a solution to ax + by = d. y = y c a d (b) Find two distinct solutions to 2261x y = 17. (c) Prove that all solutions are of the form (2.6.1) for some c. (2.6.1) 2.6 Let f(x) = x 2 + ax + b Z[x] be a quadratic olynomial with integer coefficients, for examle, f(x) = x 2 + x + 6. Formulate a conjecture about when the set {f(n) : n Z and f(n) is rime} is infinite. Give numerical evidence that suorts your conjecture. 2.7 Find four comlete sets of residues modulo 7, where the ith set satisfies the ith condition: (1) nonnegative, (2) odd, (3) even, (4) rime. 2.8 Find rules in the sirit of Proosition for divisibility of an integer by 5, 9, and 11, and rove each of these rules using arithmetic modulo a suitable n. 2.9 (*) (The following roblem is from the 1998 Putnam Cometition.) Define a sequence of decimal integers a n as follows: a 1 = 0, a 2 = 1, and a n+2 is obtained by writing the digits of a n+1 immediately followed by those of a n. For examle, a 3 = 10, a 4 = 101, and a 5 = Determine the n such that a n is a multile of 11, as follows: (a) Find the smallest integer n > 1 such that a n is divisible by 11. (b) Prove that a n is divisible by 11 if and only if n 1 (mod 6) Find an integer x such that 37x 1 (mod 101).
53 46 2. The Ring of Integers Modulo n 2.11 What is the order of 2 modulo 17? 2.12 Let be a rime. Prove that Z/Z is a field Find an x Z such that x 4 (mod 17) and x 3 (mod 23) Prove that if n > 4 is comosite then 2.15 For what values of n is ϕ(n) odd? (n 1)! 0 (mod n) (a) Prove that ϕ is multilicative as follows. Suose m, n are ositive integers and gcd(m, n) = 1. Show that the natural ma ψ : Z/mnZ Z/mZ Z/nZ is an injective homomorhism of rings, hence bijective by counting, then look at unit grous. (b) Prove conversely that if gcd(m, n) > 1, then the natural ma ψ : Z/mnZ Z/mZ Z/nZ is not an isomorhism Seven cometitive math students try to share a huge hoard of stolen math books equally between themselves. Unfortunately, six books are left over, and in the fight over them, one math student is exelled. The remaining six math students, still unable to share the math books equally since two are left over, again fight, and another is exelled. When the remaining five share the books, one book is left over, and it is only after yet another math student is exelled that an equal sharing is ossible. What is the minimum number of books that allows this to haen? 2.18 Show that if is a ositive integer such that both and are rime, then = Let ϕ : N N be the Euler ϕ function. (a) Find all natural numbers n such that ϕ(n) = 1. (b) Do there exist natural numbers m and n such that ϕ(mn) ϕ(m) ϕ(n)? 2.20 Find a formula for ϕ(n) directly in terms of the rime factorization of n (a) Prove that if ϕ : G H is a grou homomorhism, then ker(ϕ) is a subgrou of G. (b) Prove that ker(ϕ) is normal, i.e., if a G and b ker(ϕ), then a 1 ba ker(ϕ) Is the set Z/5Z = {0, 1, 2, 3, 4} with binary oeration multilication modulo 5 a grou?
54 2.6 Exercises Find all four solutions to the equation x (mod 35) Prove that for any ositive integer n the fraction (12n + 1)/(30n + 2) is in reduced form Suose a and b are ositive integers. (a) Prove that gcd(2 a 1, 2 b 1) = 2 gcd(a,b) 1. (b) Does it matter if 2 is relaced by an arbitrary rime? (c) What if 2 is relaced by an arbitrary ositive integer n? 2.26 For every ositive integer b, show that there exists a ositive integer n such that the olynomial x 2 1 (Z/nZ)[x] has at least b roots (a) Prove that there is no rimitive root modulo 2 n for any n 3. (b) (*) Prove that (Z/2 n Z) is generated by 1 and Let be an odd rime. (a) (*) Prove that there is a rimitive root modulo 2. (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a rimitive root modulo n. (c) Exlicitly find a rimitive root modulo (*) In terms of the rime factorization of n, characterize the integers n such that there is a rimitive root modulo n Comute the last two digits of Find the integer a such that 0 a < 113 and a 37 (mod 113) Find the roortion of rimes < 1000 such that 2 is a rimitive root modulo Find a rime such that the smallest rimitive root modulo is 37.
55 48 2. The Ring of Integers Modulo n
56 3 Publickey Crytograhy This is age 49 Printer: Oaque this In the 1970s, techniques from number theory changed the world forever by roviding, for the first time ever, a way for two eole to communicate secret messages under the assumtion that all of their communication is interceted and read by an adversary. This idea has stood the test of time. In fact, whenever you buy something online, you use such a system, which tyically involves working in the ring of integers modulo n. This chater tells the story of several such systems. 3.1 Playing with Fire I recently watched a TV show called La Femme Nikita about a woman named Nikita who is forced to be an agent for a shady antiterrorist organization called Section One. Nikita has strong feelings for fellow agent Michael, and she most trusts Walter, Section One s exbiker gadgets and exlosives exert. Often Nikita s worst enemies are her sueriors and coworkers at Section One. A synosis for a Season Three eisode is as follows: PLAYING WITH FIRE On a mission to secure detonation chis from a terrorist organization s heavily armed base cam, Nikita is catured as a hostage by the enemy. Or so it is made to look. Michael and Nikita have actually created the scenario in order to secretly rendezvous with each other. The ruse works, but when Birkoff
57 50 3. Publickey Crytograhy FIGURE 3.1. Diffie and Hellman (hotos from [Sin99]) [Section One s master hacker] accidentally discovers encryted messages between Michael and Nikita sent with Walter s hel, Birkoff is forced to tell Madeline. Susecting that Michael and Nikita may be lanning a cou d e tat, Oerations and Madeline use a second team of oeratives to track Michael and Nikita s next secret rendezvous... killing them if necessary. What sort of encrytion might Walter have heled them to use? I let my imagination run free, and this is what I came u with. After being catured at the base cam, Nikita is given a hone by her cators in hoes that she ll use it and they ll be able to figure out what she is really u to. Everyone is eagerly listening in on her calls. Remark In this book, we will assume a method is available for roducing random integers. Methods for generating random integers are involved and interesting, but we will not discuss them in this book. For an indeth treatment of random numbers, see [Knu98, Ch. 3]. Nikita remembers a conversation with Walter about a ublickey crytosystem called the DiffieHellman key exchange. She remembers that it allows two eole to agree on a secret key in the resence of eavesdroers. Moreover, Walter mentioned that though DiffieHellman was the first ever ublickey exchange system, it is still in common use today (for examle, in OenSSH rotocol version 2, see htt:// Nikita ulls out her handheld comuter and hone, calls u Michael, and they do the following, which is wrong (try to figure out what is wrong as you read it). 1. Together they choose a big rime number and a number g with 1 < g <. 2. Nikita secretly chooses an integer n.
58 3. Michael secretly chooses an integer m. 4. Nikita tells Michael ng (mod ). 5. Michael tells mg (mod ) to Nikita. 3.2 The DiffieHellman Key Exchange The secret key is s = nmg (mod ), which both Nikita and Michael can easily comute. Here s a very simle examle with small numbers that illustrates what Michael and Nikita do. (They really used much larger numbers.) 1. = 97, g = 5 2. n = m = ng 58 (mod 97) 5. mg 87 (mod 97) 6. s = nmg = 78 (mod 97) Nikita and Michael are foiled because everyone easily figures out s: 1. Everyone knows, g, ng (mod ), and mg (mod ). 2. Using Algorithm 2.3.7, anyone can easily find a, b Z such that ag + b = 1, which exists because gcd(g, ) = Then, ang n (mod ), so everyone knows Nikita s secret key n, and hence can easily comute the shared secret s. To taunt her, Nikita s cators give her a aragrah from a review of Diffie and Hellman s 1976 aer New Directions in Crytograhy [DH76]: The authors discuss some recent results in communications theory [...] The first [method] has the feature that an unauthorized eavesdroer will find it comutationally infeasible to deciher the message [...] They roose a coule of techniques for imlementing the system, but the reviewer was unconvinced. 3.2 The DiffieHellman Key Exchange As night darkens Nikita s cell, she reflects on what has haened. Uon realizing that she misremembered how the system works, she hones Michael and they do the following:
59 52 3. Publickey Crytograhy 1. Together Michael and Nikita choose a 200digit integer that is likely to be rime (see Section 2.4), and choose a number g with 1 < g <. 2. Nikita secretly chooses an integer n. 3. Michael secretly chooses an integer m. 4. Nikita comutes g n (mod ) on her handheld comuter and tells Michael the resulting number over the hone. 5. Michael tells Nikita g m (mod ). 6. The shared secret key is then s (g n ) m (g m ) n g nm which both Nikita and Michael can comute. (mod ), Here is a simlified examle that illustrates what they did, that involves only relatively simle arithmetic. 1. = 97, g = 5 2. n = m = g n 7 (mod ) 5. g m 39 (mod ) 6. s (g n ) m 14 (mod ) The Discrete Log Problem Nikita communicates with Michael by encryting everything using their agreed uon secret key (for examle, using a standard symmetric ciher such as AES, Arcfour, Cast128, 3DES, or Blowfish). In order to understand the conversation, the eavesdroer needs s, but it takes a long time to comute s given only, g, g n, and g m. One way would be to comute n from knowledge of g and g n ; this is ossible, but aears to be comutationally infeasible, in the sense that it would take too long to be ractical. Let a, b, and n be real numbers with a, b > 0 and n 0. Recall that the log to the base b function is characterized by log b (a) = n if and only if a = b n. We use the log b function in algebra to solve the following roblem: Given a base b and a ower a of b, find an exonent n such that a = b n.
60 That is, given a = b n and b, find n. 3.2 The DiffieHellman Key Exchange 53 SAGE Examle The number a = is the nth ower of b = 3 for some n. We quickly find that n = log 3 (19683) = log(19683)/ log(3) = 9. sage: log( ) sage: log(3.0) sage: log( ) / log(3.0) Sage can quickly comute a numerical aroximation for log(x), for any x, by comuting a artial sum of an aroriate raidlyconverging infinite series (at least for x in a certain range). The discrete log roblem is the analog of comuting log b (a) but where both b and a are elements of a finite grou. Problem (Discrete Log Problem). Let G be a finite grou, for examle, G = (Z/Z). Given b G and a ower a of b, find a ositive integer n such that b n = a. As far as we know, finding discrete logarithms in (Z/Z) when is large is very difficult in ractice. Over the years, many eole have been very motivated to try. For examle, if Nikita s cators could efficiently solve Problem 3.2.2, then they could read the messages she exchanges with Michael. Unfortunately, we have no formal roof that comuting discrete logarithms on a classical comuter is difficult. Also, Peter Shor [Sho97] showed that if one could build a sufficiently comlicated quantum comuter, it could solve the discrete logarithm roblem in time bounded by a olynomial function of the number of digits of #G. It is easy to give an inefficient algorithm that solves the discrete log roblem. Simly try b 1, b 2, b 3, etc., until we find an exonent n such that b n = a. For examle, suose a = 18, b = 5, and = 23. Working modulo 23, we have b 1 = 5, b 2 = 2, b 3 = 10,..., b 12 = 18, so n = 12. When is large, comuting the discrete log this way soon becomes imractical, because increasing the number of digits of the modulus makes the comutation take vastly longer. SAGE Examle Perhas art of the reason that comuting discrete logarithms is difficult, is that the logarithm in the real numbers is continuous, but the (minimum) logarithm of a number mod n bounces around at random. We illustrate this exotic behavior in Figure 3.2. This draws the continuous lot.
61 54 3. Publickey Crytograhy FIGURE 3.2. Grahs of the continuous log and of the discrete log modulo 53. Which icture looks easier to redict? sage: lot(log, 0.1,10, rgbcolor=(0,0,1)) This draws the discrete lot. sage: = 53 sage: R = Integers() sage: a = R.multilicative_generator() sage: v = sorted([(a^n, n) for n in range(1)]) sage: G = lot(oint(v,ointsize=50,rgbcolor=(0,0,1))) sage: H = lot(line(v,rgbcolor=(0.5,0.5,0.5))) sage: G + H Realistic DiffieHellman Examle In this section, we resent an examle that uses bigger numbers. First, we rove a roosition that we can use to choose a rime in such a way that it is easy to find a g (Z/Z) with order 1. We have already seen in Section 2.5 that for every rime there exists an element g of order 1, and we gave Algorithm for finding a rimitive root for any rime. The significance of Proosition below is that it suggests an algorithm for finding a rimitive root that is easier to use in ractice when is large, because it does not require factoring 1. Of course, one could also just use a random g for DiffieHellman; it is not essential that g generates (Z/Z). Proosition Suose is a rime such that ( 1)/2 is also rime. Then each element of (Z/Z) has order one of 1, 2, ( 1)/2, or 1. Proof. Since is rime, the grou (Z/Z) is of order 1. By assumtion, the rime factorization of 1 is 2 (( 1)/2). Let a (Z/Z). Then by Theorem , a 1 = 1, so the order of a is a divisor of 1, which roves the roosition.
62 3.2 The DiffieHellman Key Exchange 55 Given a rime with ( 1)/2 rime, find an element of order 1 as follows. If 2 has order 1, we are done. If not, 2 has order ( 1)/2 since 2 does not have order either 1 or 2. Then 2 has order 1. Let = Then is rime, but ( 1)/2 is not. So we kee adding 2 to and testing seudorimality using algorithms from Section 2.4 until we find that the next seudorime after is q = It turns out that q seudorime and (q 1)/2 is also seudorime. We find that 2 has order (q 1)/2, so g = 2 has order q 1 modulo q, and is hence a generator of (Z/qZ), at least assuming that q is really rime. The secret random numbers generated by Nikita and Michael are and Nikita sends n = m = g n = (Z/Z) to Michael, and Michael sends g m = (Z/Z) to Nikita. They agree on the secret key g nm = (Z/Z). SAGE Examle We illustrate the above comutations using Sage. sage: q = sage: q.is_rime() True sage: is_rime((q1)//2) True sage: g = Mod(2, q) sage: g.multilicative_order() sage: n = sage: m = sage: g^n sage: g^m sage: (g^n)^m sage: (g^m)^n
63 56 3. Publickey Crytograhy The Man in the Middle Attack Since their first system was broken, instead of talking on the hone, Michael and Nikita can now only communicate via text messages. One of her cators, The Man, is watching each of the transmissions; moreover, he can intercet messages and send false messages. When Nikita sends a message to Michael announcing g n (mod ), The Man intercets this message, and sends his own number g t (mod ) to Michael. Eventually, Michael and The Man agree on the secret key g tm (mod ), and Nikita and The Man agree on the key g tn (mod ). When Nikita sends a message to Michael she unwittingly uses the secret key g tn (mod ); The Man then intercets it, decryts it, changes it, and reencryts it using the key g tm (mod ), and sends it on to Michael. This is bad because now The Man can read every message sent between Michael and Nikita, and moreover, he can change them in transmission in subtle ways. One way to get around this attack is to use a digital signature scheme based on the RSA crytosystem. We will not discuss digital signatures further in this book, but will discuss RSA in the next section. 3.3 The RSA Crytosystem The DiffieHellman key exchange has drawbacks. As discussed in Section 3.2.3, it is suscetible to the man in the middle attack. This section is about the RSA ublickey crytosystem of Rivest, Shamir, and Adleman [RSA78], which is an alternative to DiffieHellman that is more flexible in some ways. We first describe the RSA crytosystem, then discuss several ways to attack it. It is imortant to be aware of such weaknesses, in order to avoid foolish mistakes when imlementing RSA. We barely scratched the surface here of the many ossible attacks on secific imlementations of RSA or other crytosystems How RSA works The fundamental idea behind RSA is to try to construct a tradoor or oneway function on a set X. This is an invertible function E : X X such that it is easy for Nikita to comute E 1, but extremely difficult for anybody else to do so. Here is how Nikita makes a oneway function E on the set of integers modulo n. 1. Using a method hinted at in Section 2.4, Nikita icks two large rimes and q, and lets n = q.
64 2. It is then easy for Nikita to comute 3.3 The RSA Crytosystem 57 ϕ(n) = ϕ() ϕ(q) = ( 1) (q 1). 3. Nikita next chooses a random integer e with 1 < e < ϕ(n) and gcd(e, ϕ(n)) = Nikita uses the algorithm from Section to find a solution x = d to the equation ex 1 (mod ϕ(n)). 5. Finally, Nikita defines a function E : Z/nZ Z/nZ by E(x) = x e Z/nZ. Note that anybody can comute E fairly quickly using the reeatedsquaring algorithm from Section Nikita s ublic key is the air of integers (n, e), which is just enough information for eole to easily comute E. Nikita knows a number d such that ed 1 (mod ϕ(n)), so, as we will see, she can quickly comute E 1. To send Nikita a message, roceed as follows. Encode your message, in some way, as a sequence of numbers modulo n (see Section 3.3.2) then send m 1,..., m r Z/nZ, E(m 1 ),..., E(m r ) to Nikita. (Recall that E(m) = m e for m Z/nZ.) When Nikita receives E(m i ), she finds each m i by using that E 1 (m) = m d, a fact that follows from Proosition Proosition (Decrytion Key). Let n be an integer that is a roduct of distinct rimes and let d, e N be such that 1 de 1 for each rime n. Then a de a (mod n) for all a Z. Proof. Since n a de a, if and only if a de a for each rime divisor of n, it suffices to rove that a de a (mod ) for each rime divisor of n. If gcd(a, ) 1, then a 0 (mod ), so a de a (mod ). If gcd(a, ) = 1, then Theorem asserts that a 1 1 (mod ). Since 1 de 1, we have a de 1 1 (mod ) as well. Multilying both sides by a shows that a de a (mod ). Thus to decryt E(m i ) Nikita comutes E(m i ) d = (m e i ) d = m i.
65 58 3. Publickey Crytograhy SAGE Examle We imlement the RSA crytosystem using Sage. The rsa function creates a key with (at most) the given number of bits, i.e., if bits equals 20, it creates a key n = q such that n is aroximately Tyical reallife crytosystems would choose keys that are 512, 1024, or 2048 bits long. Try generating large keys yourself using Sage; how long does it take? sage: def rsa(bits):... # only rove correctness u to 1024 bits... roof = (bits <= 1024)... = next_rime(zz.random_element(2**(bits//2 +1)),... roof=roof)... q = next_rime(zz.random_element(2**(bits//2 +1)),... roof=roof)... n = * q... hi_n = (1) * (q1)... while True:... e = ZZ.random_element(1,hi_n)... if gcd(e,hi_n) == 1: break... d = lift(mod(e,hi_n)^(1))... return e, d, n... sage: def encryt(m,e,n):... return lift(mod(m,n)^e)... sage: def decryt(c,d,n):... return lift(mod(c,n)^d)... sage: e,d,n = rsa(20) sage: c = encryt(123, e, n) sage: decryt(c, d, n) Encoding a Phrase in a Number In order to use the RSA crytosystem to encryt messages, it is necessary to encode them as a sequence of numbers of size less than n = q. We now describe a simle way to do this. Note that in any actual deloyed imlementation, it is crucial that you add extra random characters ( salt ) at the beginning of each block of the message, so that the same lain text encodes differently each time. This hels thwart chosen lain text attacks. Suose s is a sequence of caital letters and saces, and that s does not begin with a sace. We encode s as a number in base 27 as follows: a single sace corresonds to 0, the letter A to 1, B to 2,..., Z to 26. Thus RUN
66 3.3 The RSA Crytosystem 59 NIKITA is a number written in base 27. RUN NIKITA = (in decimal). To recover the letters from the decimal number, reeatedly divide by 27 and read off the letter corresonding to each remainder = A = T = I = K = I = N = = N 507 = U 18 = R If 27 k n, then any sequence of k letters can be encoded as above using a ositive integer n. Thus if we can encryt integers of size at most n, then we must break our message u into blocks of size at most log 27 (n). SAGE Examle We use Sage to imlement conversion between a string and a number, though in a bit more generally than in the toy illustration above (which used only base 27). The inut string s on a comuter is stored in a format called ASCII, so each letter corresonds to an integer between 0 and 255, inclusive. This number is obtained from the letter using the ord command. sage: def encode(s):... s = str(s) # make inut a string... return sum(ord(s[i])*256^i for i in range(len(s))) sage: def decode(n):... n = Integer(n) # make inut an integer... v = []... while n!= 0:... v.aend(chr(n % 256))... n //= 256 # this relaces n by floor(n/256).... return.join(v) sage: m = encode( Run Nikita! ); m sage: decode(m) Run Nikita!
67 60 3. Publickey Crytograhy Some Comlete Examles To make the arithmetic easier to follow, we use small rime numbers and q and encryt the single letter X using the RSA crytosystem. First, we comute the arameters of an RSA crytosystem. 1. Choose and q: Let = 17, q = 19, so n = q = Comute ϕ(n): ϕ(n) = ϕ( q) = ϕ() ϕ(q) = ( 1)(q 1) = q q + 1 = = Randomly choose an e < 288: We choose e = Solve 95x 1 (mod 288). Using the GCD algorithm, we find that d = 191 solves the equation. We have thus comuted the arameters of an RSA ublic key crytosystem. The ublic key is (323, 95), so the encrytion function is E(x) = x 95, and the decrytion function is D(x) = x 191. Next, we encryt the letter X. It is encoded as the number 24, since X is the 24th letter of the alhabet. We have To decryt, we comute E 1 : E(24) = = 294 Z/323Z. E 1 (294) = = 24 Z/323Z. This next examle illustrates RSA but with bigger numbers. Let = , q = Then, n = q = and ϕ(n) = ( 1)(q 1) =
68 3.4 Attacking RSA 61 Using a seudorandom number generator on a comuter, the author randomly chose the integer Then, e = d = Since log 27 (n) 38.04, we can encode then encryt single blocks of u to 38 letters. Let s encryt the string RUN NIKITA, which encodes as m = We have E(m) = m e = Remark In ractice, one usually choses e to be small, since that does not seem to reduce the security of RSA, and makes the key size smaller. For examle, in the OenSSL documentation (see htt:// about their imlementation of RSA, it states that The exonent is an odd number, tyically 3, 17 or Attacking RSA Suose Nikita s ublic key is (n, e) and her decrytion key is d, so ed 1 (mod ϕ(n)). If somehow we comute the factorization n = q, then we can comute ϕ(n) = ( 1)(q 1) and hence comute d. Thus, if we can factor n then we can break the corresonding RSA ublickey crytosystem Factoring n Given ϕ(n) Suose n = q. Given ϕ(n), it is very easy to comute and q. We have ϕ(n) = ( 1)(q 1) = q ( + q) + 1, so we know both q = n and + q = n + 1 ϕ(n). Thus, we know the olynomial x 2 ( + q)x + q = (x )(x q) whose roots are and q. These roots can be found using the quadratic formula. Examle The number n = q = is a roduct of two rimes, and ϕ(n) = We have f = x 2 (n + 1 ϕ(n))x + n = x x = (x )(x ),
69 62 3. Publickey Crytograhy where the factorization ste is easily accomlished using the quadratic formula: b + b 2 4ac 2a = = We conclude that n = SAGE Examle The following Sage function factors n = q given n and ϕ(n). sage: def crack_rsa(n, hi_n):... R.<x> = PolynomialRing(QQ)... f = x^2  (n+1 hi_n)*x + n... return [b for b, _ in f.roots()] sage: crack_rsa( , ) [ , ] When and q are Close Suose that and q are close to each other. Then it is easy to factor n using a factorization method of Fermat called the Fermat Factorization Method. Suose n = q with > q. Then, Since and q are close, is small, ( ) 2 ( ) 2 + q q n =. 2 2 s = q 2 t = + q 2 is only slightly larger than n, and t 2 n = s 2 is a erfect square. So, we just try t = n, t = n + 1, t = n + 2,... until t 2 n is a erfect square s 2. (Here x denotes the least integer n x.) Then = t + s, q = t s. Examle Suose n = Then n =
70 3.4 Attacking RSA 63 If t = , then t 2 n = If t = , then t 2 n = If t = , then t 2 n = 804 Z. Thus s = 804. We find that = t + s = and q = t s = SAGE Examle We imlement the above algorithm for factoring an RSA modulus n = q, when one of and q is close to n. sage: def crack_when_q_close(n):... t = Integer(ceil(sqrt(n)))... while True:... k = t^2  n... if k > 0:... s = Integer(int(round(sqrt(t^2  n))))... if s^2 + n == t^2:... return t+s, ts t += 1... sage: crack_when_q_close( ) (153649, ) For examle, you might think that choosing a random rime, and the next rime after would be a good idea, but instead it creates an easytocrack crytosystem. sage: = next_rime(2^128); sage: q = next_rime() sage: crack_when_q_close(*q) ( , ) Factoring n Given d In this section, we show that finding the decrytion key d for an RSA crytosystem is, in ractice, at least as difficult as factoring n. We give a robabilistic algorithm that given a decrytion key determines the factorization of n. Consider an RSA crytosystem with modulus n and encrytion key e. Suose we somehow finding an integer d such that a ed a (mod n) for all a. Then m = ed 1 satisfies a m 1 (mod n) for all a that are corime to n. As we saw in Section 3.4.1, knowing ϕ(n) leads directly to a factorization of n. Unfortunately, knowing d does not seem to lead easily to
71 64 3. Publickey Crytograhy a factorization of n. However, there is a robabilistic rocedure that, given an m such that a m 1 (mod n), will find a factorization of n with high robability (we will not analyze the robability here). Algorithm (Probabilistic Algorithm to Factor n). Let n = q be the roduct of two distinct odd rimes, and suose m is an integer such that a m 1 (mod n) for all a corime to n. This robabilistic algorithm factors n with high robability. In the stes below, a always denotes an integer corime to n = q. 1. [Divide out owers of 2] If m is even and a m/2 1 (mod n) for several randomly chosen a, set m = m/2, and go to Ste 1, otherwise let a be such that a m/2 1 (mod n). 2. [Comute GCD] Choose a random a and comute g = gcd(a m/2 1, n). 3. [Terminate?] If g is a roer divisor of n, outut g and terminate. Otherwise go to Ste 2. Before giving the roof, we introduce some more terminology from algebra. Definition (Grou Homomorhism). Let G and H be grous. A ma ϕ : G H is a grou homomorhism if for all a, b G we have ϕ(ab) = ϕ(a)ϕ(b). A grou homomorhism is called surjective if for every c H there is a G such that ϕ(a) = c. The kernel of a grou homomorhism ϕ : G H is the set ker(ϕ) of elements a G such that ϕ(a) = 1. A grou homomorhism is injective if ker(ϕ) = {1}. Definition (Subgrou). If G is a grou and H is a subset of G, then H is a subgrou if H is a grou under the grou oeration on G. For examle, if ϕ : G H is a grou homomorhism, then ker(ϕ) is a subgrou of G (see Exercise 2.21). We now return to discussing Algorithm In Ste 1, note that m is even since ( 1) m 1 (mod n), so it makes sense to consider m/2. It is not ractical to determine whether or not a m/2 1 (mod n) for all a, because it would require doing a comutation for too many a. Instead, we try a few random a; if a m/2 1 (mod n) for the a we check, we divide m by 2. Also note that if there exists even a single a such that a m/2 1 (mod n), then half the a have this roerty, since then a a m/2 is a surjective homomorhism (Z/nZ) {±1} and the kernel has index 2. Proosition imlies that if x 2 1 (mod ) then x = ±1 (mod ). In Ste 2, since (a m/2 ) 2 1 (mod n), we also have (a m/2 ) 2 1 (mod ) and (a m/2 ) 2 1 (mod q), so a m/2 ±1 (mod ) and a m/2 ±1 (mod q). Since a m/2 1 (mod n), there are three ossibilities for these signs, so with ositive robability one of the following two ossibilities occurs: 1. a m/2 +1 (mod ) and a m/2 1 (mod q)
72 3.4 Attacking RSA a m/2 1 (mod ) and a m/2 +1 (mod q). The only other ossibility is that both signs are 1. In the first case, a m/2 1 but q a m/2 1, so gcd(a m/2 1, q) =, and we have factored n. Similarly, in the second case, gcd(a m/2 1, q) = q, and we again factor n. Examle Somehow we discover that the RSA crytosystem with n = and e = has decrytion key d = We use this information and Algorithm to factor n. If m = ed 1 = , then ϕ(q) m, so a m 1 (mod n) for all a corime to n. For each a 20 we find that a m/2 1 (mod n), so we relace m with m 2 = Again, we find with this new m that for each a 20, a m/2 1 (mod n), so we relace m by Yet again, for each a 20, a m/2 1 (mod n), so we relace m by This is enough, since 2 m/ (mod n). Then, gcd(2 m/2 1, n) = gcd( , ) = , and we have found a factor of n. Dividing, we find that n = SAGE Examle We imlement Algorithm in Sage. sage: def crack_given_decryt(n, m):... n = Integer(n); m = Integer(m); # some tye checking... # Ste 1: divide out owers of 2... while True:... if is_odd(m): break... divide_out = True... for i in range(5):... a = randrange(1,n)... if gcd(a,n) == 1:... if Mod(a,n)^(m//2)!= 1:... divide_out = False... break
73 66 3. Publickey Crytograhy... if divide_out:... m = m//2... else:... break... # Ste 2: Comute GCD... while True:... a = randrange(1,n)... g = gcd(lift(mod(a, n)^(m//2))  1, n)... if g!= 1 and g!= n:... return g... We show how to verify Examle using Sage. sage: n= ; e= ; d= sage: crack_given_decryt(n, e*d  1) sage: factor(n) * We try a much larger examle. sage: e = sage: d = sage: n = sage: = crack_given_decryt(n, e*d  1) sage: # random outut (could be other rime divisor) sage: n % Further Remarks If one were to imlement an actual RSA crytosystem, there are many additional tricks and ideas to kee in mind. For examle, one can add some extra random letters to each block of text, so that a given string will encryt differently each time it is encryted. This makes it more difficult for an attacker who knows the encryted and laintext versions of one message to gain information about subsequent encryted messages. In any articular imlementation, there might be attacks that would be devastating in ractice, but which would not require factorization of the RSA modulus. RSA is in common use, for examle, it is used in OenSSH rotocol version 1 (see htt:// We will consider the ElGamal crytosystem in Sections It has a similar flavor to RSA, but is more flexible in some ways.
74 3.5 Exercises 67 Probably the best general urose attack on RSA is the number field sieve, which is a general algorithm for factoring integers of the form q. A descrition of the sieve is beyond the scoe of this book. The ellitic curve method is another related general algorithm that we will discuss in detail in Section 6.3. SAGE Examle Here is a simle examle of using a variant of the number field sieve (called the quadratic sieve) in Sage to factor an RSA key with about 192 bits: sage: set_random_seed(0) sage: = next_rime(randrange(2^96)) sage: q = next_rime(randrange(2^97)) sage: n = * q sage: qsieve(n) ([ , ], ) 3.5 Exercises 3.1 This roblem concerns encoding hrases using numbers using the encoding of Section What is the longest that an arbitrary sequence of letters (no saces) can be if it must fit in a number that is less than 10 20? 3.2 Suose Michael creates an RSA crytosystem with a very large modulus n for which the factorization of n cannot be found in a reasonable amount of time. Suose that Nikita sends messages to Michael by reresenting each alhabetic character as an integer between 0 and 26 (A corresonds to 1, B to 2, etc., and a sace to 0), then encryts each number searately using Michael s RSA crytosystem. Is this method secure? Exlain your answer. 3.3 For any n N, let σ(n) be the sum of the divisors of n; for examle, σ(6) = = 12 and σ(10) = = 18. Suose that n = qr with, q, and r distinct rimes. Devise an efficient algorithm that given n, ϕ(n) and σ(n), comutes the factorization of n. For examle, if n = 105, then = 3, q = 5, and r = 7, so the inut to the algorithm would be n = 105, ϕ(n) = 48, and σ(n) = 192, and the outut would be 3, 5, and You and Nikita wish to agree on a secret key using the DiffieHellman key exchange. Nikita announces that = 3793 and g = 7. Nikita
75 68 3. Publickey Crytograhy secretly chooses a number n < and tells you that g n 454 (mod ). You choose the random number m = What is the secret key? 3.5 You see Michael and Nikita agree on a secret key using the Diffie Hellman key exchange. Michael and Nikita choose = 97 and g = 5. Nikita chooses a random number n and tells Michael that g n 3 (mod 97), and Michael chooses a random number m and tells Nikita that g m 7 (mod 97). Brute force crack their code: What is the secret key that Nikita and Michael agree uon? What is n? What is m? 3.6 In this roblem, you will crack an RSA crytosystem. What is the secret decoding number d for the RSA crytosystem with ublic key (n, e) = ( , )? 3.7 Nikita creates an RSA crytosystem with ublic key (n, e) = ( , ). In the following two roblems, show the stes you take to factor n. (Don t simly factor n directly using a comuter.) (a) Somehow you discover that d = Show how to use the robabilistic algorithm of Section to factor n. (b) In art (a) you found that the factors and q of n are very close. Show how to use the Fermat Factorization Method of Section to factor n.
76 4 Quadratic Recirocity This is age 69 Printer: Oaque this A linear equation ax b (mod n) has a solution if and only if gcd(a, n) divides b (see Proosition ). This chater is about some amazing mathematics motivated by the search for a criterion for whether or not a given quadratic equation ax 2 + bx + c 0 (mod n) has a solution. In many cases, the Chinese Remainder Theorem and the quadratic formula reduce this to the key question of whether a given integer a is a erfect square modulo a rime. The Quadratic Recirocity Law of Gauss rovides a recise answer to the following question: For which rimes is the image of a in (Z/Z) a erfect square? A dee fact, which we will comletely rove in this chater, is that the answer deends only on the reduction of modulo 4a. Thus to decide if a is a square modulo, one only needs to consider the residue of modulo 4a, which is extremely surrising. It turns out that this recirocity law goes to the heart of modern number theory and touches on advanced toics such as class field theory and the Langlands rogram. There are over a hundred roofs of the Quadratic Recirocity Law (see [Lem] for a long list). In this chater, we give two roofs. The first, which we give in Section 4.3, is comletely elementary and involves keeing track of integer oints in intervals. It is satisfying because one can understand every detail without much abstraction, but it might be unsatisfying if you find it difficult to concetualize what is going on. In contrast, our second
77 70 4. Quadratic Recirocity roof, which we give in Section 4.4, is more abstract and uses a concetual develoment of roerties of Gauss sums. You should read Sections 4.1 and 4.2, then at least one of Section 4.3 or Section 4.4, deending on your taste and how much abstract algebra you know. In Section 4.5, we return to the comutational question of actually finding square roots and solving quadratic equations in ractice. 4.1 Statement of the Quadratic Recirocity Law In this section, we state the Quadratic Recirocity Law. Definition (Quadratic Residue). Fix a rime. An integer a not divisible by is a quadratic residue modulo if a is a square modulo ; otherwise, a is a quadratic nonresidue. For examle, the squares modulo 5 are 1 2 = 1, 2 2 = 4, 3 2 = 4, 4 2 = 1, (mod 5) so 1 and 4 are both quadratic residues and 2 and 3 are quadratic nonresidues. The quadratic recirocity theorem is the deeest theorem that we will rove in this book. It connects the question of whether or not a is a quadratic residue modulo to the question of whether is a quadratic residue modulo each of the rime divisors of a. To exress it recisely, we introduce some new notation. Definition (Legendre Symbol). Let be an odd rime and let a be an integer. Set ( ) a = 0 if gcd(a, ) 1, +1 if a is a quadratic residue, and 1 if a is a quadratic nonresidue. We call this symbol the Legendre Symbol. For examle, we have ( ) 1 = 1, 5 ( ) 2 = 1, 5 ( ) 3 = 1, 5 ( ) 4 = 1, 5 ( ) 5 = 1. 5 This notation is well entrenched in the literature even though it is also the notation for a divided by ; be careful not to confuse the two. SAGE Examle Use the legendre symbol command to comute the Legendre symbol in Sage.
78 sage: legendre_symbol(2,3) 1 sage: legendre_symbol(1,3) 1 sage: legendre_symbol(3,5) 1 sage: legendre_symbol(mod(3,5), 5) Statement of the Quadratic Recirocity Law 71 ( ) ( a Since only deends on a (mod ), it makes sense to define ( ) a Z/Z to be for any lift ã of a to Z. ã a ) for Recall (see Definition 3.4.6) that a grou homomorhism ϕ : G H is a ma such that for every a, b G we have ϕ(ab) = ϕ(a)ϕ(b). Moreover, we say that ϕ is surjective if for every c H there is an a G with ϕ(a) = c. The next lemma exlains how the quadratic residue symbol defines a surjective grou homomorhism. ( Lemma The ma ψ : (Z/Z) {±1} given by ψ(a) = surjective grou homomorhism. a ) is a Proof. By Theorem 2.5.8, rimitive roots exist, so there is g (Z/Z) such that the elements of (Z/Z) are g, g 2,..., g ( 1)/2, g (+1)/2,..., g 1 = 1. Since 1 is even, the squares of elements of (Z/Z) are g 2, g 4,..., g ( 1)/2 2 = 1, g +1 = g 2,..., g 2( 1). Note that the owers of g starting with g +1 = g 2 all aeared earlier on the list. Thus, the erfect squares in (Z/Z) are exactly the owers g n with n = 2, 4,..., 1, even, and the nonsquares the owers g n with n = 1, 3,..., 2, odd. It follows that ψ is a homomorhism since an odd lus an odd is even, the sum of two evens is even, and odd lus an even is odd. Moreover, since g is not a square, ψ(g) = 1, so ψ is surjective. Remark We rehrase the above roof in the language of grou theory. The grou G = (Z/Z) of order 1 is a cyclic grou. Since is odd, 1 is even, so the subgrou H of squares of elements of G has index 2 ( in G. (See Exercise 4.2 for why H is a subgrou.) Since a ) = 1 if and only if a H, we see that ψ is the comosition G G/H = {±1}, where we identify the nontrivial element of G/H with 1. Remark We can alternatively rove that ψ is surjective without using that (Z/Z) is cyclic, as follows. If a (Z/Z) is a square, say a b 2
79 72 4. Quadratic Recirocity TABLE 4.1. When is 5 a square modulo? ( ) ( ) 5 5 mod 5 mod (mod ), then a ( 1)/2 = b 1 1 (mod ), so a is a root of f = x ( 1)/2 1. By Proosition 2.5.3, the olynomial f has at most ( 1)/2 roots. Thus, there must be ( an ) a (Z/Z) that is not a root of f, and for that a, we a have ψ(a) = = 1, and trivially ψ(1) = 1, so the ma ψ is surjective. Note that this argument does not rove that ψ is a homomorhism. ( ) The symbol only deends on the residue class of a modulo, so a making a table of values ( ) a 5 ( for ) many values of a would be easy. Would it 5 be easy to make a table of for many? Perhas, since there aears ( ) 5 to be a simle attern in Table 4.1. It seems that ( ) deends only on 5 the congruence class of modulo 5. More recisely, ( ) = 1 if and only if 1, 4 (mod 5), i.e., = 1 if and only if is a square modulo 5. 5 Based on similar observations, in the 18th century various mathematicians found a conjectural exlanation for the mystery suggested by Table 4.1. Finally, on Aril 8, 1796, at the age of 19, Gauss roved the following theorem. Theorem (Gauss s Quadratic Recirocity Law). Suose and q are distinct odd rimes. Then Also ( ) 1 = ( 1) ( 1)/2 and ( ) = ( 1) 1 2 q 1 2 q ( ) q. We will give two roofs of Gauss s formula relating ( ) { 2 1 if ±1 (mod 8) = 1 if ±3 (mod 8). ( ) q to ( ) q. The first elementary roof is in Section 4.3, and the second more algebraic roof is in Section 4.4.
80 In our examle, Gauss s theorem imlies that ( ) 5 ( 1 2 ( = ( 1) 2 = = 5) 5) 4.2 Euler s Criterion 73 { +1 if 1, 4 (mod 5) 1 if 2, 3 (mod 5). As an alication, the following examle illustrates how to answer questions like is a a square modulo b using Theorem Examle Is 69 a square modulo the rime 389? We have ( ) ( ) ( ) ( ) = = = ( 1) ( 1) = Here ( ) 3 = 389 and ( ) 23 = 389 Thus 69 is a square modulo 389. ( ) 389 = 3 ( ) ( ) = = ( ) ( ) 1 2 = ( ) 2 = 1, 3 ( ) 2 23 = ( 1) = 1. SAGE Examle We could also do this comutation in Sage as follows: sage: legendre_symbol(69,389) 1 Though we know that 69 is a square modulo 389, we don t know an exlicit x such that x 2 69 (mod 389)! This is reminiscent of how we roved using Theorem that certain numbers are comosite without knowing a factorization. Remark The Jacobi symbol is an extension of the Legendre symbol to comosite moduli. For more details, see Exercise Euler s Criterion Let be an odd rime and a an integer not ( divisible ) by. Euler used a the existence of rimitive roots to show that is congruent to a ( 1)/2 modulo. We will use this fact reeatedly below in both roofs of Theorem ( ) Proosition (Euler s Criterion). We have = 1 if and only if a a ( 1)/2 1 (mod ).
81 74 4. Quadratic Recirocity Proof. The ma ϕ : (Z/Z) (Z/Z) given by ϕ(a) = a ( 1)/2 is a grou homomorhism, since owering is a grou homomorhism of any abelian grou (see Exercise 4.2). Let ψ : (Z/Z) {±1} be the homo ( morhism ψ(a) = b (Z/Z), so a ) of Lemma If a ker(ψ), then a = b 2 for some ϕ(a) = a ( 1)/2 = (b 2 ) ( 1)/2 = b 1 = 1. Thus ker(ψ) ker(ϕ). By Lemma 4.1.4, ker(ψ) has index 2 in (Z/Z), i.e., #(Z/Z) = 2 # ker(ψ). Since the kernel of a homomorhism is a grou, and the order of a subgrou divides the order of the grou, we have either ker(ϕ) = ker(ψ) or ϕ = 1. If ϕ = 1, the olynomial x ( 1)/2 1 has 1 roots in the field Z/Z, which contradicts Proosition Thus ker(ϕ) = ker(ψ), which roves the roosition. SAGE Examle From a comutational ( ) oint of view, Corollary rovides a convenient way to comute, which we illustrate in Sage: sage: def kr(a, ):... if Mod(a,)^((1)//2) == 1:... return 1... else:... return 1 sage: for a in range(1,5):... rint a, kr(a,5) Corollary The equation x 2 ( a ) (mod ) has no solution if and only if a ( 1)/2 1 (mod ). Thus a ( 1)/2 (mod ). Proof. This follows from Proosition and the fact that the olynomial x 2 1 has no roots besides +1 and 1 (which follows from Proosition 2.5.5). As additional comutational ( ) motivation for the value of Corollary 4.2.3, a note that to evaluate using Theorem would not be ractical if a and are both very large, because it would require ( ) factoring a. However, Corollary rovides a method for evaluating without factoring a. Examle Suose = 11. By squaring each element of (Z/11Z), we see that the squares modulo 11 are {1, 3, 4, 5, 9}. We comute a ( 1)/2 = a 5 a a a
82 for each a (Z/11Z) and get 4.3 First Proof of Quadratic Recirocity = 1, 2 5 = 1, 3 5 = 1, 4 5 = 1, 5 5 = 1, 6 5 = 1, 7 5 = 1, 8 5 = 1, 9 5 = 1, 10 5 = 1. Thus the a with a 5 = 1 are {1, 3, 4, 5, 9}, just as Proosition redicts. Examle We determine whether or not 3 is a square modulo the rime = sage: = sage: Mod(3, )^((1)//2) so 3 ( 1)/2 1 (mod ). Thus 3 is not a square modulo. This comutation wasn t difficult, but it would have been tedious by hand. Since 3 is small, the Quadratic Recirocity Law rovides a way to answer this question, which could easily be carried out by hand: ( ) ( ) = ( 1) (3 1)/2 ( )/2 3 ( ) 1 = ( 1) = First Proof of Quadratic Recirocity Our first roof of quadratic recirocity is elementary. The roof involves keeing track of integer oints in( intervals. ) Proving Gauss s lemma is the first ste; this lemma comutes in terms of the number of integers of a a certain tye that lie in a certain interval. We next rove Lemma 4.3.3, which controls how the arity of the number of integer oints in an interval changes when an endoint of the interval is changed. We then rove ( that a ) deends only on modulo 4a by alying Gauss s Lemma and keeing careful track of intervals as they are rescaled and their endoints are changed. Finally, in Section 4.3.2, we use some basic algebra to deduce the Quadratic Recirocity Law using the tools we ve just develoed. Our roof follows the one given in [Dav99] closely. Lemma (Gauss s Lemma). Let be an odd rime and let a be an integer 0 (mod ). Form the numbers a, 2a, 3a,..., 1 2 a
83 76 4. Quadratic Recirocity and reduce them modulo to lie in the interval ( 2, 2 ), i.e., for each of the above roducts k a find a number in the interval ( 2, 2 ) that is congruent to k a modulo. Let ν be the number of negative numbers in the resulting set. Then ( ) a = ( 1) ν. Proof. In defining ν, we exressed each number in { S = a, 2a,..., 1 } 2 a as congruent to a number in the set { 1, 1, 2, 2,..., 1 2, 1 }. 2 No number 1, 2,..., 1 2 aears more than once, with either choice of sign, because if it did then either two elements of S are congruent modulo or 0 is the sum of two elements of S, and both events are imossible (the former case cannot occur because of cancellation modulo, and in the latter case we would have ka + ja 0 (mod ) for 1 k, j ( 1)/2, so k + j 0 (mod ), a contradiction). The resulting set must be of the form { T = ε 1 1, ε 2 2,..., ε ( 1)/2 1 }, 2 where each ε i is either +1 or 1. Multilying together the elements of S and of T, we see that ( ) 1 (1a) (2a) (3a) 2 a ( (ε 1 1) (ε 2 2) ε ( 1)/2 1 ) (mod ), 2 so a ( 1)/2 ε 1 ε 2 ε ( 1)/2 (mod ). ( ) The lemma then follows from Proosition 4.2.1, since = a ( 1)/2. SAGE Examle We illustrate Gauss s Lemma using Sage. The gauss function below rints out a list of the normalized numbers aearing in the statement of Gauss s Lemma, and returns ( 1) ν. In each case below, ( ( 1) ν = a ). sage: def gauss(a, ):... # make the list of numbers reduced modulo a
84 4.3 First Proof of Quadratic Recirocity v = [(n*a)% for n in range(1, (1)//2 + 1)]... # normalize them to be in the range /2 to /2... v = [(x if (x < /2) else x  ) for x in v]... # sort and rint the resulting numbers... v.sort()... rint v... # count the number that are negative... num_neg = len([x for x in v if x < 0])... return (1)^num_neg sage: gauss(2, 13) [5, 3, 1, 2, 4, 6] 1 sage: legendre_symbol(2,13) 1 sage: gauss(4, 13) [6, 5, 2, 1, 3, 4] 1 sage: legendre_symbol(4,13) 1 sage: gauss(2,31) [15, 13, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14] 1 sage: legendre_symbol(2,31) Euler s Proosition For rational numbers a, b Q, let (a, b) Z = {x Z : a x b} be the set of integers between a and b. The following lemma will hel us to kee track of how many integers lie in certain intervals. Lemma Let a, b Q. Then for any integer n, # ((a, b) Z) # ((a, b + 2n) Z) (mod 2) and # ((a, b) Z) # ((a 2n, b) Z) (mod 2), rovided that each interval involved in the congruence is nonemty. Note that if one of the intervals is emty, then the statement may be false; for examle, if (a, b) = ( 1/2, 1/2) and n = 1, then #((a, b) Z) = 1 but #(a, b 2) Z = 0.
85 78 4. Quadratic Recirocity Proof. Let x denotes the least integer x. Since n > 0, (a, b + 2n) = (a, b) [b, b + 2n), where the union is disjoint. There are 2n integers b, b + 1,..., b + 2n 1 in the interval [b, b + 2n), so the first congruence of the lemma is true in this case. We also have (a, b 2n) = (a, b) minus [b 2n, b) and [b 2n, b) contains exactly 2n integers, so the lemma is also true when n is negative. The statement about # ((a 2n, b) Z) is roved in a similar manner. Once we have roved the following roosition, it will be easy to deduce the Quadratic Recirocity Law. Proosition (Euler). Let be an odd rime and let a ( be a) ositive ( ) integer with a. If q is a rime with q ± (mod 4a), then =. ( Proof. We will aly Lemma to comute and I = S = a { a, 2a, 3a,..., 1 } 2 a ). Let ( ) ( ) (( 1 3 2,, 2 b 1 ) ), b, 2 2 where b = 1 2 a or 1 2 (a 1), whichever is an integer. We check that every element of S that is equivalent modulo to something in the interval ( 2, 0) lies in I. First suose that b = 1 2a. Then b = 1 2 a = 2 a > 1 2 a, so each element of S that is equivalent modulo to an element of ( 2, 0) lies in I. Next suose that b = 1 2 (a 1). Then b + 2 = a = 1 + a > a, so ((b 1 2 ), b) is the last interval that could contain an element of S that reduces to ( 2, 0). Note that the integer endoints of I are not in S, since a a q
86 4.3 First Proof of Quadratic Recirocity 79 those endoints are divisible by, but no element of S is divisible by. Thus, by Lemma 4.3.1, ( ) a = ( 1) #(S I). To comute #(S I), first rescale by a to see that ( 1 #(S I) = # a S 1 ) a I = # (Z 1a ) I, where ( 1 ( a I = 2a, ) ( 3 a 2a, 2 ) ( (2b 1), b )), a 2a a 1 a S = {1, 2, 3, 4,..., ( 1)/2}, and the second equality is because 1 a I (0, ( 1)/2 + 1/2], since Write = 4ac + r, and let ( ( r J = 2a, r ) a b a a 2 a = 2 = ( 3r 2a, 2r a ) ( (2b 1)r, br )). 2a a The only difference between 1 ai and J is that the endoints of intervals are changed by addition of an even integer, since r 2a 2a = 2a 2c 2a = 2c. By Lemma 4.3.3, ν = # (Z 1a ) I #(Z J) (mod 2). ( ) a Thus = ( 1) ν deends only on r and a, i.e., only on modulo 4a. ( ) ( ) Thus if q (mod 4a), then =. a If q (mod 4a), then the only change in the above comutation is that r is relaced by 4a r. This changes J into ( K = 2 r 2a, 4 r ) ( 6 3r a 2a, 8 2r ) a ( ) 4b 2. a q (2b 1)r, 4b br 2a a
87 80 4. Quadratic Recirocity Thus K is the same as J, excet even integers have been added to the endoints. By Lemma 4.3.3, ( ) 1 #(K Z) # a I Z (mod 2), ( ) ( ) so = again, which comletes the roof. a a q The following more careful analysis in the secial case when a = 2 hels illustrate the roof of the above lemma, and the result is frequently useful in comutations. For an alternative roof of the roosition, see Exercise 4.6. Proosition (Legendre Lymbol of 2). Let be an odd rime. Then ( ) { 2 1 if ±1 (mod 8) = 1 if ±3 (mod 8). Proof. When a = 2, the set S = {a, 2a,..., } is {2, 4, 6,..., 1}. We must count the arity of the number of elements of S that lie in the interval I = ( 2, ). Writing = 8c + r, we have ) ( 1 # (I S) = # = # 2 I Z (( 2c + r 4, 4c + r 2 (( = # 4, 2) ) Z ) ) Z # (( r 4 2), r ) Z (mod 2), where the last equality comes from Lemma The ossibilities for r are 1, 3, 5, 7. When r = 1, the cardinality is 0; when r = 3, 5 it is 1; and when r = 7 it is Proof of Quadratic Recirocity It is now straightforward to deduce the Quadratic Recirocity Law. First Proof of Theorem First suose that q (mod 4). By swaing and q if necessary, we may assume that > q, and write q = 4a. Since = 4a + q, ( ) q ( ) 4a + q = = q ( ) 4a = q ( 4 q ) ( ) a = q ( ) a, q and ( ) q = ( ) ( ) 4a 4a = = ( ) 1 ( ) a.
88 4.4 A Proof of Quadratic Recirocity Using Gauss Sums 81 ( Proosition imlies that ( ) q ( ) q = a q ) ( = a ), since q (mod 4a). Thus ( ) 1 = ( 1) 1 2 = ( 1) 1 2 q 1 2, where the last equality is because 1 2 is even if and only if q 1 2 is even. Next suose that q (mod 4), so q (mod 4). Write + q = 4a. We have ( ) ( ) ( ) ( ) ( ) ( ) 4a q a q 4a a = =, and = =. q q q ( ) ( ) Since q (mod 4a), Proosition imlies that =. Since ( 1) 1 2 q 1 2 = 1, the roof is comlete. a q a 4.4 A Proof of Quadratic Recirocity Using Gauss Sums In this section, we resent a beautiful roof of Theorem using algebraic identities satisfied by sums of roots of unity. The objects we introduce in the roof are of indeendent interest, and rovide a owerful tool to rove higherdegree analogs of quadratic recirocity. (For more on higher recirocity, see [IR90]. See also Section 6 of [IR90], on which the roof below is modeled.) Definition (Root of Unity). An nth root of unity is a comlex number ζ such that ζ n = 1. A root of unity ζ is a rimitive nth root of unity if n is the smallest ositive integer such that ζ n = 1. For examle, 1 is a rimitive second root of unity, and ζ = is a rimitive cube root of unity. More generally, for any n N the comlex number ζ n = cos(2π/n) + i sin(2π/n) is a rimitive nth root of unity (this follows from the identity e iθ = cos(θ)+ i sin(θ)). For the rest of this section, we fix an odd rime and the rimitive th root ζ = ζ of unity. SAGE Examle In Sage, use the CyclotomicField command to create an exact th root of ζ unity. Exressions in ζ are always reexressed as olynomials in ζ of degree at most 1. sage: K.<zeta> = CyclotomicField(5) sage: zeta^5 1
89 82 4. Quadratic Recirocity sage: 1/zeta zeta^3  zeta^2  zeta  1 Definition (Gauss Sum). Fix an odd rime. The Gauss sum associated to an integer a is 1 ( ) n g a = ζ an, n=1 where ζ = ζ = cos(2π/) + i sin(2π/) = e 2πi/. Note that is imlicit in the definition of g a. If we were to change, then the Gauss sum g a associated to a would be different. The definition of g a also deends on our choice of ζ; we ve chosen ζ = ζ, but could have chosen a different ζ and then g a could be different. SAGE Examle We define a gauss sum function and comute the Gauss sum g 2 for = 5: sage: def gauss_sum(a,):... K.<zeta> = CyclotomicField()... return sum(legendre_symbol(n,) * zeta^(a*n)... for n in range(1,)) sage: g2 = gauss_sum(2,5); g2 2*zeta^3 + 2*zeta^2 + 1 sage: g2.comlex_embedding() e16*I sage: g2^2 5 Here, g 2 is initially outut as a olynomial in ζ 5, so there is no loss of recision. The comlex embedding command shows some embedding of g 2 into the comlex numbers, which is only correct to about the first 15 digits. Note that g 2 2 = 5, so g 2 = 5. We comute a grahical reresentation of the Gauss sum g 2 as follows (see Figure 4.1): zeta = CDF(ex(2*i*I/5)) v = [legendre_symbol(n,5) * zeta^(2*n) for n in range(1,5)] S = sum([oint(tule(z), ointsize=100) for z in v]) show(s + oint(tule(sum(v)), ointsize=100, rgbcolor= red )) Figure 4.1 illustrates the Gauss sum g 2 for = 5. The Gauss sum is obtained by adding the oints on the unit circle, with signs as indicated, to obtain the real number 5. This suggests the following roosition, whose roof will require some work. Proosition (Gauss Sum). For any a not divisible by, g 2 a = ( 1) ( 1)/2.
90 4.4 A Proof of Quadratic Recirocity Using Gauss Sums FIGURE 4.1. The red dot is the Gauss sum g 2 for = 5 SAGE Examle We illustrate using Sage that the roosition is correct for = 7 and = 13: sage: [gauss_sum(a, 7)^2 for a in range(1,7)] [7, 7, 7, 7, 7, 7] sage: [gauss_sum(a, 13)^2 for a in range(1,13)] [13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13] In order to rove the roosition, we introduce a few lemmas. Lemma For any integer a, 1 ζ an = n=0 { if a 0 (mod ), 0 otherwise. Proof. If a 0 (mod ), then ζ a = 1, so the sum equals the number of summands, which is. If a 0 (mod ), then we use the identity x 1 = (x 1)(x x + 1) with x = ζ a. We have ζ a 1, so ζ a 1 0 and 1 ζ an = ζa 1 ζ a 1 = 1 1 ζ a 1 = 0. n=0
91 84 4. Quadratic Recirocity Lemma If x and y are arbitrary integers, then { 1 ζ (x y)n if x y (mod ), = 0 otherwise. n=0 Proof. This follows from Lemma by setting a = x y. Lemma We have g 0 = 0. Proof. By definition 1 ( ) n g 0 =. (4.4.1) n=0 By Lemma 4.1.4, the ma ( ) : (Z/Z) {±1} is a surjective homomorhism of grous. Thus, half the elements of (Z/Z) ma to +1 ( and ) half ma to 1 (the subgrou that mas to +1 has index 2). Since = 0, the sum (4.4.1) is 0. 0 Lemma For any integer a, ( ) a g a = g 1. Proof. When a 0 (mod ), the lemma follows from Lemma 4.4.9, so suose that a 0 (mod ). Then, ( ) a g a = ( a ) 1 n=0 ( ) n ζ an = 1 n=0 ( ) an 1 ζ an = m=0 ( ) m ζ m = g 1. Here, we use that multilication by a is an automorhism of Z/Z. Finally, ( ) ( ) 2 a a multily both sides by and use that = 1. We have enough lemmas to rove Proosition Proof of Proosition We evaluate the sum 1 a=0 g ag a in two different ways. By Lemma , since a 0 (mod ) we have g a g a = ( a ) g 1 ( a ) g 1 = ( 1 ) ( a ) 2 g 2 1 = ( 1) ( 1)/2 g 2 1, ( ) a where the last ste follows from Proosition and that {±1}. Thus 1 g a g a = ( 1)( 1) ( 1)/2 g1. 2 (4.4.2) a=0
92 4.4 A Proof of Quadratic Recirocity Using Gauss Sums 85 On the other hand, by definition 1 ( ) n g a g a = = = n=0 1 1 n=0 m=0 1 1 n=0 m=0 ζ an ( n ( n 1 m=0 ) ( m ( ) m ζ am ) ζ an ζ am ) ( ) m ζ an am. Let δ(n, m) = 1 if n m (mod ) and 0 otherwise. By Lemma 4.4.8, g a g a = a=0 = = 1 a=0 n=0 m=0 1 1 n=0 m=0 1 1 n=0 m=0 1 = n=0 ( n = ( 1). ( n ( n ) 2 ( n ) ( ) m ζ an am ) ( ) m 1 ) ( m a=0 ζ an am ) δ(n, m) Equate (4.4.2) and the above equality, then cancel ( 1) to see that Since a 0 (mod ), we have g 2 1 = ( 1) ( 1)/2. g 2 a = and the roosition is roved. ( a ) 2 = 1, so by Lemma , ( ) 2 a g1 2 = g 2 1, Proof of Quadratic Recirocity We are now ready to rove Theorem using Gauss sums. Proof. Let q be an odd rime with q. Set = ( 1) ( 1)/2 and recall that Proosition asserts that = g 2, where g = g 1 = ( ) 1 n=0 ζ n. n
93 86 4. Quadratic Recirocity Proosition imlies that ( ) ( ) (q 1)/2 q (mod q). We have g q 1 = (g 2 ) (q 1)/2 = ( ) (q 1)/2, so multilying both sides of the dislayed equation by g yields a congruence ( ) g q g (mod q). (4.4.3) q But wait, what does this congruence mean, ( ) given that g q is not an integer? It means that the difference g q g is a multile of q in the ring Z[ζ] of all olynomials in ζ with coefficients in Z. The ring Z[ζ]/(q) has characteristic q, so if x, y Z[ζ], then (x + y) q x q + y q (mod q). Alying this to (4.4.3), we see that g q = ( 1 n=0 By Lemma , ( ) ) q n ζ n 1 n=0 g q g q q ( ) q n ζ nq ( ) q g Combining this with (4.4.3) yields ( ) ( ) q g g q 1 n=0 (mod q). (mod q). ( ) n ζ nq g q (mod q). Since ( ) g( 2 = ) and q, we can cancel g from both sides to find that q q (mod q). Since both residue symbols are ±1 and q is odd, it ) ( ) follows that =. Finally, we note using Corollary that ( q ( ) ( ( 1) ( 1)/2 ) = = q q q ( 1 q ) ( 1)/2 ( ) = ( 1) q q ( ). q 4.5 Finding Square Roots We return in this section to the question of comuting square roots. If K is a field in which 2 0, and a, b, c K, with a 0, then the two solutions to the quadratic equation ax 2 + bx + c = 0 are x = b ± b 2 4ac. 2a
94 4.5 Finding Square Roots 87 Now assume K = Z/Z, with an odd rime. Using Theorem 4.1.7, we can decide whether or not b 2 4ac is a erfect square in Z/Z, and hence whether or not ax 2 + bx + c = 0 has a solution in Z/Z. However, Theorem says nothing about how to actually find a solution when there is one. Also note that for this roblem we do not need the full Quadratic Recirocity Law; in ractice, deciding whether an element of Z/Z is a erfect square with Proosition is quite fast, in view of Section 2.3. Suose a Z/Z is a nonzero quadratic residue. If 3 (mod 4), then b = a +1 4 is a square root of a because b 2 = a +1 2 = a = a 1 2 a = ( ) a a = a. We can comute b in time olynomial in the number of digits of using the owering algorithm of Section 2.3. Suose next that 1 (mod 4). Unfortunately, we do not know a deterministic algorithm that takes a and as inut, oututs a square root of a modulo when one exists, and is olynomialtime in log(). Remark There is an algorithm due to Schoof [Sch85] that comutes the square root of a in time O(( ( a ) 1/2+ε log()) 9 ). This beautiful algorithm (which makes use of ellitic curves) is not olynomial time in the sense described above, since for large a it takes exonentially longer than for small a. We next describe a robabilistic algorithm to comute a square root of a modulo, which is very quick in ractice. Recall the notion of ring from Definition We will also need the notion of ring homomorhism and isomorhism. Definition (Homomorhism of Rings). Let R and S be rings. A homomorhism of rings ϕ : R S is a ma such that for all a, b R, we have ϕ(ab) = ϕ(a)ϕ(b), ϕ(a + b) = ϕ(a) + ϕ(b), and ϕ(1) = 1. An isomorhism ϕ : R S of rings is a ring homomorhism that is bijective. Consider the ring defined as follows. We have R = (Z/Z)[x]/(x 2 a) R = {u + vα : u, v Z/Z}
95 88 4. Quadratic Recirocity with multilication defined by (u + vα)(z + wα) = (uz + awv) + (uw + vz)α. Here α corresonds to the class of x in R. SAGE Examle We define and work with the ring R above in Sage as follows (for = 13): sage: S.<x> = PolynomialRing(GF(13)) sage: R.<alha> = S.quotient(x^23) sage: (2+3*alha)*(1+2*alha) 7*alha + 7 Let b and c be the square roots of a in Z/Z (though we cannot easily comute b and c yet, we can consider them in order to deduce an algorithm to find them). We have ring homomorhisms f : R Z/Z and g : R Z/Z given by f(u + vα) = u + vb and g(u + vα) = u + vc. Together, these define a ring isomorhism ϕ : R Z/Z Z/Z given by ϕ(u + vα) = (u + vb, u + vc). Choose in some way a random element z of (Z/Z), and define u, v Z/Z by u + vα = (1 + zα) 1 2, where we comute (1+zα) 1 2 quickly using an analog of the binary owering algorithm of Section If v = 0, we try again with another random z. If v 0, we can quickly find the desired square roots b and c as follows. The quantity u+vb is a ( 1)/2 ower in Z/Z, so it equals either 0, 1, or 1, so b = u/v, (1 u)/v, or ( 1 u)/v, resectively. Since we know u and v, we can try each of u/v, (1 u)/v, and ( 1 u)/v and see which is a square root of a. Examle Continuing Examle 4.1.8, we find a square root of 69 modulo 389. We aly the algorithm described above in the case 1 (mod 4). We first choose the random z = 24 and find that (1 + 24α) 194 = 1. The coefficient of α in the ower is 0, and we try again with z = 51. This time, we have (1 + 51α) 194 = 239α = u + vα. The inverse of 239 in Z/389Z is 153, so we consider the following three ossibilities for a square root of 69: u v = 0 1 u v = u v = 153. Thus, 153 and 153 are the square roots of 69 in Z/389Z. SAGE Examle We imlement the above algorithm in Sage and illustrate it with some examles.
96 4.6 Exercises 89 sage: def find_sqrt(a, ):... assert (1)%4 == 0... assert legendre_symbol(a,) == 1... S.<x> = PolynomialRing(GF())... R.<alha> = S.quotient(x^2  a)... while True:... z = GF().random_element()... w = (1 + z*alha)^((1)//2)... (u, v) = (w[0], w[1])... if v!= 0: break... if (u/v)^2 == a: return u/v... if ((1u)/v)^2 == a: return (1u)/v... if ((1u)/v)^2 == a: return (1u)/v... sage: b = find_sqrt(3,13) sage: b # random: either 9 or 3 9 sage: b^2 3 sage: b = find_sqrt(3,13) sage: b # see, it s random 4 sage: find_sqrt(5,389) # random: either 303 or sage: find_sqrt(5,389) # see, it s random Exercises 4.1 Calculate the following by hand: ( ) ( 3 97, ), ( ) (, and 5! ) Let G be an abelian grou, and let n be a ositive integer. (a) Prove that the ma ϕ : G G given by ϕ(x) = x n is a grou homomorhism. (b) Prove that the subset H of G of squares of elements of G is a subgrou. 4.3 Use Theorem to show that for 5 rime, ( ) { 3 1 if 1, 11 (mod 12), = 1 if 5, 7 (mod 12).
97 90 4. Quadratic Recirocity ( ) 4.4 (*) Use that (Z/Z) 3 is cyclic to give a direct roof that = 1 when 1 (mod 3). (Hint: There is an element c (Z/Z) of order 3. Show that (2c + 1) 2 = 3.) ( ) 4.5 (*) If 1 (mod 5), show directly that = 1 by the method of Exercise 4.4. (Hint: Let c (Z/Z) be an element of order 5. Show that (c + c 4 ) 2 + (c + c 4 ) 1 = 0, etc.) ( ) 4.6 (*) Let be an odd rime. In this exercise, you will rove that = 1 if and only if ±1 (mod 8). (a) Prove that 5 x = 1 t2 1 + t 2, y = 2t 1 + t 2 is a arameterization of the set of solutions to x 2 + y 2 1 (mod ), in the sense that the solutions (x, y) Z/Z are in bijection with the t Z/Z { } such that 1+t 2 0 (mod ). Here, t = corresonds to the oint ( 1, 0). (Hint: if (x 1, y 1 ) is a solution, consider the line y = t(x + 1) through (x 1, y 1 ) and ( 1, 0), and solve for x 1, y 1 in terms of t.) (b) Prove that the number of solutions to x 2 + y 2 1 (mod ) is + 1 if 3 (mod 4) and 1 if 1 (mod 4). (c) Consider the set ( S) of airs ( ) (a, b) (Z/Z) (Z/Z) such that a + b = 1 and = = 1. Prove that #S = ( + 1 4)/4 a b if 3 (mod 4) and #S = ( 1 4)/4 if 1 (mod 4). Conclude that #S is odd if and only if ±1 (mod 8). (d) The ma σ(a, b) = (b, a) that swas coordinates is a bijection of the set S. It has exactly one fixed oint ( ) if and only if there is a an a Z/Z such that 2a = 1 and = 1. Also, rove that ( ) a 2a = 1 has a solution a Z/Z with = 1 if and only if ( ) = 1. 2 (e) Finish by showing that σ has exactly one fixed oint if and only if #S is odd, i.e., if and only if ±1 (mod 8). Remark: The method of roof of this exercise can be generalized to give a roof of the full Quadratic Recirocity Law. 4.7 How many natural numbers x < 2 13 satisfy the equation x 2 5 (mod )? You may assume that is rime. 2
98 4.6 Exercises Find the natural number x < 97 such that x 4 48 (mod 97). Note that 97 is rime. 4.9 In this roblem, we ( will ) formulate an analog of quadratic recirocity for a symbol like, but without the restriction that q be a rime. a q Suose n is an odd ositive integer, which we factor as k We define the Jacobi symbol ( a n) as follows: ( a = n) k ( ) ei a. i=1 i i=1 ei i. (a) Give an examle to show that ( a n) = 1 need not imly that a is a erfect square modulo n. (b) (*) Let n be odd and a and b be integers. Prove that the following holds: i. ( ( a b ) ( n) n = ab ) ( n. (Thus a a ) n induces a homomorhism from (Z/nZ) to {±1}.) ii. ( ) 1 n n (mod 4). iii. ( 2 n) = 1 if n ±1 (mod 8) and 1 otherwise. iv. Assume a is ositive and odd. Then ( ) a 1 ( a n = ( 1) 2 n 1 2 n ) a 4.10 (*) Prove that for any n Z, the integer n 2 + n + 1 does not have any divisors of the form 6k 1.
99 92 4. Quadratic Recirocity
100 5 Continued Fractions This is age 93 Printer: Oaque this The golden ratio is equal to the infinite fraction and the fraction , = is an excellent aroximation to π. Both of these observations are exlained by continued fractions. Continued fractions are theoretically beautiful and rovide tools that yield owerful algorithms for solving roblems in number theory. For examle, continued fractions rovide a fast way to write a rime even a hundred digit rime as a sum of two squares, when ossible. Continued fractions are thus a beautiful algorithmic and concetual tool in number theory that has many alications. For examle, they rovide a surrisingly efficient way to recognize a rational number given just the first few digits of its decimal exansion, and they give a sense in which e is less comlicated than π (see Examle and Section 5.4). In Section 5.2, we study continued fractions of finite length and lay the foundations for our later investigations. In Section 5.3, we give the continued fraction rocedure, which associates to a real number x a continued fraction that converges to x. In Section 5.5, we characterize (eventually)
101 94 5. Continued Fractions eriodic continued fractions as the continued fractions of nonrational roots of quadratic olynomials, then discuss an unsolved mystery concerning continued fractions of roots of irreducible olynomials of degree greater than 2. We conclude the chater with alications of continued fractions to recognizing aroximations to rational numbers (Section 5.6) and writing integers as sums of two squares (Section 5.7). The reader is encouraged to read more about continued fractions in [HW79, Ch. X], [Khi63], [Bur89, 13.3], and [NZM91, Ch. 7]. 5.1 The Definition A continued fraction is an exression of the form 1 a a a 2 + a 3 +. In this book, we will assume that the a i are real numbers and a i > 0 for i 1, and the exression may or may not go on indefinitely. More general notions of continued fractions have been extensively studied, but they are beyond the scoe of this book. We will be most interested in the case when the a i are all integers. We denote the continued fraction dislayed above by [a 0, a 1, a 2,...]. For examle, [1, 2] = = 3 2, 1 [3, 7, 15, 1, 292] = = = ,
102 5.2 Finite Continued Fractions 95 and 1 [2, 1, 2, 1, 1, 4, 1, 1, 6] = = = The second two examles were chosen to foreshadow that continued fractions can be used to obtain good rational aroximations to irrational numbers. Note that the first aroximates π, and the second e. 5.2 Finite Continued Fractions This section is about continued fractions of the form [a 0, a 1,..., a m ] for some m 0. We give an inductive definition of numbers n and q n such that for all n m [a 0, a 1,..., a n ] = n q n. (5.2.1) We ( then give related formulas for the determinants of the 2 2 matrices n n 1 ) ( n q n q n 1 and n 2 ) q n q n 2, which we will reeatedly use to deduce roerties of the sequence of artial convergents [a 0,..., a k ]. We will use Algorithm to rove that every rational number is reresented by a continued fraction, as in (5.2.1). Definition (Finite Continued Fraction). A finite continued fraction is an exression 1 a a a a n where each a m is a real number and a m > 0 for all m 1. Definition (Simle Continued Fraction). A simle continued fraction is a finite or infinite continued fraction in which the a i are all integers.
103 96 5. Continued Fractions To get a feeling for continued fractions, observe that [a 0 ] = a 0, [a 0, a 1 ] = a = a 0a 1 + 1, a 1 a 1 1 [a 0, a 1, a 2 ] = a 0 + a = a 0a 1 a 2 + a 0 + a 2. a 1 a a 2 Also, [a 0, a 1,..., a n 1, a n ] = [ a 0, a 1,..., a n 2, a n = a [a 1,..., a n ] = [a 0, [a 1,..., a n ]]. a n ] SAGE Examle The continued fraction command comutes continued fractions: sage: continued_fraction(17/23) [0, 1, 2, 1, 5] sage: continued_fraction(e) [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 11] Use the otional second argument bits = n to determine the recision (in bits) of the inut number that is used to comute the continued fraction. sage: continued_fraction(e, bits=20) [2, 1, 2, 1, 1, 4, 1, 1, 6] sage: continued_fraction(e, bits=30) [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1] You can obtain the value of a continued fraction and even do arithmetic with continued fractions: sage: a = continued_fraction(17/23); a [0, 1, 2, 1, 5] sage: a.value() 17/23 sage: b = continued_fraction(6/23); b [0, 3, 1, 5] sage: a + b [1]
104 5.2.1 Partial Convergents 5.2 Finite Continued Fractions 97 Fix a finite continued fraction [a 0,..., a m ]. We do not assume at this oint that the a i are integers. Definition (Partial convergents). For 0 n m, the nth convergent of the continued fraction [a 0,..., a m ] is [a 0,..., a n ]. These convergents for n < m are also called artial convergents. For each n with 2 n m, define real numbers n and q n as follows: 2 = 0, 1 = 1, 0 = a 0, n = a n n 1 + n 2, q 2 = 1, q 1 = 0, q 0 = 1, q n = a n q n 1 + q n 2. Proosition (Partial Convergents). For n 0 with n m we have [a 0,..., a n ] = n q n. Proof. We use induction. The assertion is obvious when n = 0, 1. Suose the roosition is true for all continued fractions of length n 1. Then [a 0,..., a n ] = [a 0,..., a n 2, a n ] a ( ) n a n a n n 2 + n 3 = ( ) a n a n q n 2 + q n 3 = (a n 1a n + 1) n 2 + a n n 3 (a n 1 a n + 1)q n 2 + a n q n 3 = a n(a n 1 n 2 + n 3 ) + n 2 a n (a n 1 q n 2 + q n 3 ) + q n 2 = a n n 1 + n 2 a n q n 1 + q n 2 = n q n. SAGE Examle If c is a continued fraction, use c.convergents() to comute a list of the artial convergents of c. sage: c = continued_fraction(i,bits=33); c [3, 7, 15, 1, 292, 2] sage: c.convergents() [3, 22/7, 333/106, 355/113, /33102, /66317] As we will see, the convergents of a continued fraction are the best rational aroximations to the value of the continued fraction. In the examle above, the listed convergents are the best rational aroximations of π with given denominator size.
105 98 5. Continued Fractions Proosition For n 0 with n m we have and Equivalently, and n q n 1 q n n 1 = ( 1) n 1 (5.2.2) n q n 2 q n n 2 = ( 1) n a n. (5.2.3) n q n n 1 q n 1 = ( 1) n 1 1 q n q n 1 n n 2 = ( 1) n a n. q n q n 2 q n q n 2 Proof. The case for n = 0 is obvious from the definitions. Now suose n > 0 and the statement is true for n 1. Then n q n 1 q n n 1 = (a n n 1 + n 2 )q n 1 (a n q n 1 + q n 2 ) n 1 = n 2 q n 1 q n 2 n 1 = ( n 1 q n 2 n 2 q n 1 ) = ( 1) n 2 = ( 1) n 1. This comletes the roof of (5.2.2). For (5.2.3), we have n q n 2 n 2 q n = (a n n 1 + n 2 )q n 2 n 2 (a n q n 1 + q n 2 ) = a n ( n 1 q n 2 n 2 q n 1 ) = ( 1) n a n. Remark Exressed in terms of matrices, the roosition asserts that the determinant of ( n n 1 ) q n q n 1 is ( 1) n 1, and of ( n n 2 ) q n q n 2 is ( 1) n a n. SAGE Examle We use Sage to verify Proosition for the first few terms of the continued fraction of π. sage: c = continued_fraction(i); c [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3] sage: for n in range(1, len(c)):... rint c.n(n)*c.qn(n1)  c.qn(n)*c.n(n1), sage: for n in range(len(c)):... rint c.n(n)*c.qn(n2)  c.qn(n)*c.n(n2), Corollary (Convergents in lowest terms). If [a 0, a 1,..., a m ] is a simle continued fraction, so each a i is an integer, then the n and q n are integers and the fraction n /q n is in lowest terms.
106 5.2 Finite Continued Fractions 99 Proof. It is clear that the n and q n are integers, from the formula that defines them. If d is a ositive divisor of both n and q n, then d ( 1) n 1, so d = 1. SAGE Examle We illustrate Corollary using Sage. sage: c = continued_fraction([1,2,3,4,5]) sage: c.convergents() [1, 3/2, 10/7, 43/30, 225/157] sage: [c.n(n) for n in range(len(c))] [1, 3, 10, 43, 225] sage: [c.qn(n) for n in range(len(c))] [1, 2, 7, 30, 157] The Sequence of Partial Convergents Let [a 0,..., a m ] be a continued fraction and for n m let c n = [a 0,..., a n ] = n q n denote the nth convergent. Recall that by definition of continued fraction, a n > 0 for n > 0, which gives the artial convergents of a continued fraction additional structure. For examle, the artial convergents of [2, 1, 2, 1, 1, 4, 1, 1, 6] are 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465. To make the size of these numbers clearer, we aroximate them using decimals. We also underline every other number, to illustrate some extra structure. 2, 3, , , , , , , The underlined numbers are smaller than all of the nonunderlined numbers, and the sequence of underlined numbers is strictly increasing, whereas the nonunderlined numbers strictly decrease. SAGE Examle Figure 5.1 illustrates the above attern on another continued fraction using Sage. sage: c = continued_fraction([1,1,1,1,1,1,1,1]) sage: v = [(i, c.n(i)/c.qn(i)) for i in range(len(c))] sage: P = oint(v, rgbcolor=(0,0,1), ointsize=40) sage: L = line(v, rgbcolor=(0.5,0.5,0.5)) sage: L2 = line([(0,c.value()),(len(c)1,c.value())], \... thickness=0.5, rgbcolor=(0.7,0,0)) sage: (L+L2+P).show(xmin=0,ymin=1)
107 Continued Fractions FIGURE 5.1. Grah of a Continued Fraction We next rove that this extra structure is a general henomenon. Proosition (How Convergents Converge). The even indexed convergents c 2n increase strictly with n, and the odd indexed convergents c 2n+1 decrease strictly with n. Also, the odd indexed convergents c 2n+1 are greater than all of the even indexed convergents c 2m. Proof. The a n are ositive for n 1, so the q n are ositive. By Proosition 5.2.7, for n 2, c n c n 2 = ( 1) n a n q n q n 2, which roves the first claim. Suose for the sake of contradiction that there exist integers r and m such that c 2m+1 < c 2r. Proosition imlies that for n 1, c n c n 1 = ( 1) n 1 1 q n q n 1 has sign ( 1) n 1, so for all s 0 we have c 2s+1 > c 2s. Thus it is imossible that r = m. If r < m, then by what we roved in the first aragrah, c 2m+1 < c 2r < c 2m, a contradiction (with s = m). If r > m, then c 2r+1 < c 2m+1 < c 2r, which is also a contradiction (with s = r) Every Rational Number is Reresented Proosition (Rational Continued Fractions). Every nonzero rational number can be reresented by a simle continued fraction.
108 5.3 Infinite Continued Fractions 101 Proof. Without loss of generality, we may assume that the rational number is a/b, with b 1 and gcd(a, b) = 1. Algorithm gives: a = b a 0 + r 1, 0 < r 1 < b b = r 1 a 1 + r 2, 0 < r 2 < r 1 r n 2 = r n 1 a n 1 + r n, r n 1 = r n a n < r n < r n 1 Note that a i > 0 for i > 0 (also r n = 1, since gcd(a, b) = 1). Rewrite the equations as follows: a/b = a 0 + r 1 /b = a 0 + 1/(b/r 1 ), b/r 1 = a 1 + r 2 /r 1 = a 1 + 1/(r 1 /r 2 ), r 1 /r 2 = a 2 + r 3 /r 2 = a 2 + 1/(r 2 /r 3 ), r n 1 /r n = a n. It follows that a b = [a 0, a 1,..., a n ]. The roof of Proosition leads to an algorithm for comuting the continued fraction of a rational number. A nonzero rational number can be reresented in exactly two ways; for examle, 2 = [1, 1] = [2] (see Exercise 5.2). 5.3 Infinite Continued Fractions This section begins with the continued fraction rocedure, which associates a sequence a 0, a 1,... of integers to a real number x. After giving several examles, we rove that x = lim n [a 0, a 1,..., a n ] by roving that the odd and even artial convergents become arbitrarily close to each other. We also show that if a 0, a 1,... is any infinite sequence of ositive integers, then the sequence of c n = [a 0, a 1,..., a n ] converges. More generally, if a n is an arbitrary sequence of ositive reals such that n=0 a n diverges then (c n ) converges The Continued Fraction Procedure Let x R and write x = a 0 + t 0
109 Continued Fractions with a 0 Z and 0 t 0 < 1. We call the number a 0 the floor of x, and we also sometimes write a 0 = x. If t 0 0, write 1 t 0 = a 1 + t 1 with a 1 N and 0 t 1 < 1. Thus t 0 = 1 a 1+t 1 = [0, a 1 + t 1 ], which is a continued fraction exansion of t 0, which need not be simle. Continue in this manner so long as t n 0 writing 1 t n = a n+1 + t n+1 with a n+1 N and 0 t n+1 < 1. We call this rocedure, which associates to a real number x the sequence of integers a 0, a 1, a 2,..., the continued fraction rocess. Examle Let x = 8 3. Then x = , so a 0 = 2 and t 0 = 2 3. Then 1 t 0 = 3 2 = , so a 1 = 1 and t 1 = 1 2. Then 1 t 1 = 2, so a 2 = 2, t 2 = 0, and the sequence terminates. Notice that 8 = [2, 1, 2], 3 so the continued fraction rocedure roduces the continued fraction of 8 3. Examle Let x = Then so a 0 = 1 and t 0 = We have 1 t 0 = x = , = = 1 + 5, 2 so a 1 = 1 and t 1 = Likewise, a n = 1 for all n. As we will see below, the following exciting equality makes sense = SAGE Examle The equality of Examle is consistent with the following Sage calculation:
110 5.3 Infinite Continued Fractions 103 sage: def cf(bits):... x = (1 + sqrt(realfield(bits)(5))) / 2... return continued_fraction(x) sage: cf(10) [1, 1, 1, 1, 1, 1, 1, 3] sage: cf(30) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2] sage: cf(50) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Examle Suose x = e = Using the continued fraction rocedure, we find that a 0, a 1, a 2,... = 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,... For examle, a 0 = 2 is the floor of 2. Subtracting 2 and inverting, we obtain 1/ = , so a 1 = 1. Subtracting 1 and inverting yields 1/ = , so a 2 = 2. We will rove in Section 5.4 that the continued fraction of e obeys a simle attern. The 5th artial convergent of the continued fraction of e is [a 0, a 1, a 2, a 3, a 4, a 5 ] = = , which is a good rational aroximation to e, in the sense that e = Note that < 1/32 2 = , which illustrates the bound in Corollary Let s do the same thing with π = Alying the continued fraction rocedure, we find that the continued fraction of π is a 0, a 1, a 2,... = 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14,... The first few artial convergents are 3, 22 7, , , , These are good rational aroximations to π; for examle, = Notice that the continued fraction of e exhibits a nice attern (see Section 5.4 for a roof), whereas the continued fraction of π exhibits no attern
111 Continued Fractions that is obvious to the author. The continued fraction of π has been extensively studied, and over 20 million terms have been comuted. The data suggests that every integer aears infinitely often as a artial convergent. For much more about the continued fraction of π, or of any other sequence in this book, tye the first few terms of the sequence into [Slo] Convergence of Infinite Continued Fractions Lemma For every n such that a n is defined, we have x = [a 0, a 1,..., a n + t n ], and if t n 0, then x = [a 0, a 1,..., a n, 1 t n ]. Proof. We use induction. The statements are both true when n = 0. If the second statement is true for n 1, then x = [ a 0, a 1,..., a n 1, 1 t n 1 = [a 0, a 1,..., a n 1, a n + t n ] [ = a 0, a 1,..., a n 1, a n, 1 ]. t n Similarly, the first statement is true for n if it is true for n 1. Theorem (Continued Fraction Limit). Let a 0, a 1,... be a sequence of integers such that a n > 0 for all n 1, and for each n 0, set c n = [a 0, a 1,... a n ]. Then lim n c n exists. Proof. For any m n, the number c n is a artial convergent of [a 0,..., a m ]. By Proosition , the even convergents c 2n form a strictly increasing sequence and the odd convergents c 2n+1 form a strictly decreasing sequence. Moreover, the even convergents are all c 1 and the odd convergents are all c 0. Hence α 0 = lim n c 2n and α 1 = lim n c 2n+1 both exist, and α 0 α 1. Finally, by Proosition so α 0 = α 1. c 2n c 2n 1 = 1 1 q 2n q 2n 1 2n(2n 1) 0, ] We define [a 0, a 1,...] = lim n c n.
112 5.3 Infinite Continued Fractions 105 Examle We illustrate the theorem with x = π. As in the roof of Theorem 5.3.6, let c n be the nth artial convergent to π. The c n with n odd converge down to π c 1 = , c 3 = , c 5 = whereas the c n with n even converge u to π c 2 = , c 4 = , c 6 = Theorem Let a 0, a 1, a 2,... be a sequence of real numbers such that a n > 0 for all n 1, and for each n 0, set c n = [a 0, a 1,... a n ]. Then lim c n exists if and only if the sum n n=0 a n diverges. Proof. We only rove that if a n diverges, then lim n c n exists. A roof of the converse can be found in [Wal48, Ch. 2, Thm. 6.1]. Let q n be the sequence of denominators of the artial convergents, as defined in Section 5.2.1, so q 2 = 1, q 1 = 0, and for n 0, we have q n = a n q n 1 + q n 2. As we saw in the roof of Theorem 5.3.6, the limit lim n c n exists rovided that the sequence {q n q n 1 } diverges to ositive infinity. For n even, q n = a n q n 1 + q n 2 = a n q n 1 + a n 2 q n 3 + q n 4 = a n q n 1 + a n 2 q n 3 + a n 4 q n 5 + q n 6 = a n q n 1 + a n 2 q n a 2 q 1 + q 0 and for n odd, q n = a n q n 1 + a n 2 q n a 1 q 0 + q 1. Since a n > 0 for n > 0, the sequence {q n } is increasing, so q i 1 for all i 0. Alying this fact to the above exressions for q n, we see that for n even q n a n + a n a 2, and for n odd q n a n + a n a 1. If a n diverges, then at least one of a 2n or a 2n+1 must diverge. The above inequalities then imly that at least one of the sequences {q 2n } or {q 2n+1 } diverge to infinity. Since {q n } is an increasing sequence, it follows that {q n q n 1 } diverges to infinity.
113 Continued Fractions Examle Let a n = 1 n log(n) for n 2 and a 0 = a 1 = 0. By the integral test, a n diverges, so by Theorem 5.3.8, the continued fraction [a 0, a 1, a 2,...] converges. This convergence is very slow, since, e.g. yet [a 0, a 1,..., a 9999 ] = [a 0, a 1,..., a ] = Theorem Let x R be a real number. Then x is the value of the (ossibly infinite) simle continued fraction [a 0, a 1, a 2,...] roduced by the continued fraction rocedure. Proof. If the sequence is finite, then some t n = 0 and the result follows by Lemma Suose the sequence is infinite. By Lemma 5.3.5, x = [a 0, a 1,..., a n, 1 t n ]. By Proosition (which we aly in a case when the artial quotients of the continued fraction are not integers), we have x = Thus, if c n = [a 0, a 1,..., a n ], then 1 n + n 1 t n. 1 q n + q n 1 t n Thus x c n = x n = q n 1 t n n q n + n 1 q n 1 t n n q n n q n 1 ( ). 1 q n t n q n + q n 1 = n 1q n n q ( n 1 ) 1 q n t n q n + q n 1 ( 1) n = ( ). 1 q n t n q n + q n 1 1 x c n = ( ) 1 q n t n q n + q n 1 1 < q n (a n+1 q n + q n 1 ) 1 1 = q n q n+1 n(n + 1) 0.
114 5.4 The Continued Fraction of e In the inequality, we use that a n+1 is the integer art of t n, and is hence 1 t n < 1, since t n < 1. This corollary follows from the roof of Theorem Corollary (Convergence of continued fraction). Let a 0, a 1,... define a simle continued fraction, and let x = [a 0, a 1,...] R be its value. Then for all m, x m < 1. q m q m+1 q m Proosition If x is a rational number, then the sequence a 0, a 1,... roduced by the continued fraction rocedure terminates. Proof. Let [b 0, b 1,..., b m ] be the continued fraction reresentation of x that we obtain using Algorithm , so the b i are the artial quotients at each ste. If m = 0, then x is an integer, so we may assume m > 0. Then x = b 0 + 1/[b 1,..., b m ]. If [b 1,..., b m ] = 1, then m = 1 and b 1 = 1, which will not haen using Algorithm , since it would give [b 0 +1] for the continued fraction of the integer b Thus [b 1,..., b m ] > 1, so in the continued fraction algorithm we choose a 0 = b 0 and t 0 = 1/[b 1,..., b m ]. Reeating this argument enough times roves the claim. 5.4 The Continued Fraction of e The continued fraction exansion of e begins [2, 1, 2, 1, 1, 4, 1, 1, 6,...]. The obvious attern in fact does continue, as Euler roved in 1737 (see [Eul85]), and we will rove in this section. As an alication, Euler gave a roof that e is irrational by noting that its continued fraction is infinite. The roof we give below draws heavily on the roof in [Coh], which describes a slight variant of a roof of Hermite (see [Old70]). The continued fraction reresentation of e is also treated in the German book [Per57], but the roof requires substantial background from elsewhere in that text Preliminaries First, we write the continued fraction of e in a slightly different form. Instead of [2, 1, 2, 1, 1, 4,...], we can start the sequence of coefficients [1, 0, 1, 1, 2, 1, 1, 4,...] to make the attern the same throughout. (Everywhere else in this chater we assume that the artial quotients a n for n 1 are ositive, but
115 Continued Fractions temorarily relax that condition here and allow a 1 = 0.) The numerators and denominators of the convergents given by this new sequence satisfy a simle recurrence. Using r i as a standin for i or q i, we have r 3n = r 3n 1 + r 3n 2 r 3n 1 = r 3n 2 + r 3n 3 r 3n 2 = 2(n 1)r 3n 3 + r 3n 4. Our first goal is to collase these three recurrences into one recurrence that only makes mention of r 3n, r 3n 3, and r 3n 6. We have r 3n = r 3n 1 + r 3n 2 = (r 3n 2 + r 3n 3 ) + (2(n 1)r 3n 3 + r 3n 4 ) = (4n 3)r 3n 3 + 2r 3n 4. This same method of simlification also shows us that r 3n 3 = 2r 3n 7 + (4n 7)r 3n 6. To get rid of 2r 3n 4 in the first equation, we make the substitutions 2r 3n 4 = 2(r 3n 5 + r 3n 6 ) = 2((2(n 2)r 3n 6 + r 3n 7 ) + r 3n 6 ) = (4n 6)r 3n 6 + 2r 3n 7. Substituting for 2r 3n 4 and then 2r 3n 7, we finally have the needed collased recurrence, r 3n = 2(2n 1)r 3n 3 + r 3n Two Integral Sequences We define the sequences x n = 3n, y n = q 3n. Since the 3nconvergents will converge to the same real number that the n convergents do, x n /y n also converges to the limit of the continued fraction. Each sequence {x n }, {y n } will obey the recurrence relation derived in the revious section (where z n is a standin for x n or y n ): z n = 2(2n 1)z n 1 + z n 2, for all n 2. (5.4.1) The two sequences can be found in Table 5.1. (The initial conditions x 0 = 1, x 1 = 3, y 0 = y 1 = 1 are taken straight from the first few convergents of the original continued fraction.) Notice that since we are skiing several convergents at each ste, the ratio x n /y n converges to e very quickly.
116 5.4 The Continued Fraction of e 109 TABLE 5.1. Convergents n x n y n x n /y n A Related Sequence of Integrals Now, we define a sequence of real numbers T 0, T 1, T 2,... by the following integrals: T n = 1 0 t n (t 1) n n! e t dt. Below, we comute the first two terms of this sequence exlicitly. (When we comute T 1, we are doing the integration by arts u = t(t 1), dv = e t dt. Since the integral runs from 0 to 1, the boundary condition is 0 when evaluated at each of the endoints. This vanishing will be helful when we do the integral in the general case.) T 0 = T 1 = = 0 1 e t dt = e 1, t(t 1)e t dt 0 ((t 1) + t)e t dt 1 1 = (t 1)e t te t = 1 e + 2(e 1) = e 3. e t dt The reason that we defined this series now becomes aarent: T 0 = y 0 e x 0 and T 1 = y 1 e x 1. In general, it will be true that T n = y n e x n. We will now rove this fact. It is clear that if T n were to satisfy the same recurrence that the x i and y i do in (5.4.1), then the above statement holds by induction. (The initial conditions are correct, as needed.) So, we simlify T n by integrating by
117 Continued Fractions arts twice in succession: T n = 1 = = t n (t 1) n n! e t dt t n 1 (t 1) n + t n (t 1) n 1 ( t n 2 (t 1) n (n 2)! (n 1)! + n tn 1 (t 1) n 1 (n 1)! e t dt + n tn 1 (t 1) n 1 + tn (t 1) n 2 (n 1)! (n 2)! 1 t n 2 (t 1) n 2 = 2nT n 1 + (2t 2 2t + 1) e t dt (n 2)! = 2nT n t n 1 (t 1) n 1 (n 2)! = 2nT n 1 + 2(n 1)T n 1 + T n 2 = 2(2n 1)T n 1 + T n 2, ) e t dt 1 e t t n 2 (t 1) n 2 dt + 0 (n 2)! e t dt which is the desired recurrence. Therefore, T n = y n e x n. To conclude the roof, we consider the limit as n aroaches infinity: lim n by insection, and therefore 1 0 t n (t 1) n n! e t dt = 0, x n lim = lim n y (e T n ) = e. n n y n Therefore, the ratio x n /y n aroaches e, and the continued fraction exansion [2, 1, 2, 1, 1, 4, 1, 1,...] does in fact converge to e Extensions of the Argument The method of roof of this section generalizes to show that the continued fraction exansion of e 1/n is [1, (n 1), 1, 1, (3n 1), 1, 1, (5n 1), 1, 1, (7n 1),...] for all n N (see Exercise 5.6). 5.5 Quadratic Irrationals The main result of this section is that the continued fraction exansion of a number is eventually reeating if and only if the number is a quadratic
118 5.5 Quadratic Irrationals 111 irrational. This can be viewed as an analog for continued fractions of the familiar fact that the decimal exansion of x is eventually reeating if and only if x is rational. The roof that continued fractions of quadratic irrationals eventually reeats is surrisingly difficult and involves an interesting finiteness argument. Section emhasizes our striking ignorance about continued fractions of real roots of irreducible olynomials over Q of degree bigger than 2. Definition (Quadratic Irrational). A quadratic irrational is a real number α R that is irrational and satisfies a quadratic olynomial with coefficients in Q. Thus, for examle, (1 + 5)/2 is a quadratic irrational. Recall that = [1, 1, 1,...]. The continued fraction of 2 is [1, 2, 2, 2, 2, 2,...], and the continued fraction of 389 is [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,...]. Does the [1, 2, 1, 1, 1, 1, 2, 1, 38] attern reeat over and over again? SAGE Examle We comute more terms of the continued fraction exansion of 389 using Sage: sage: def cf_sqrt_d(d, bits):... x = sqrt(realfield(bits)(d))... return continued_fraction(x) sage: cf_sqrt_d(389,50) [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38] sage: cf_sqrt_d(389,100) [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1] Periodic Continued Fractions Definition (Periodic Continued Fraction). A eriodic continued fraction is a continued fraction [a 0, a 1,..., a n,...] such that a n = a n+h for some fixed ositive integer h and all sufficiently large n. We call the minimal such h the eriod of the continued fraction.
119 Continued Fractions Examle Consider the eriodic continued fraction [1, 2, 1, 2,...] = [1, 2]. What does it converge to? We have so if α = [1, 2] then α = [1, 2] = α Thus 2α 2 2α 1 = 0, so = α + 1 α α = = 1 + α 2α + 1 = 3α + 1 2α + 1 Theorem (Periodic Characterization). An infinite simle continued fraction is eriodic if and only if it reresents a quadratic irrational. Proof. (= ) First suose that [a 0, a 1,..., a n, a n+1,..., a n+h ] is a eriodic continued fraction. Set α = [a n+1, a n+2,...]. Then so by Proosition α = [a n+1,..., a n+h, α], α = α n+h + n+h 1 αq n+h + q n+h 1. Here we use that α is the last artial quotient. Thus, α satisfies a quadratic equation with coefficients in Q. Comuting as in Examle and rationalizing the denominators, and using that the a i are all integers, shows that [a 0, a 1,...] = [a 0, a 1,..., a n, α] 1 = a a 1 + a α is of the form c + dα, with c, d Q, so [a 0, a 1,...] also satisfies a quadratic olynomial over Q.,
120 5.5 Quadratic Irrationals 113 The continued fraction rocedure alied to the value of an infinite simle continued fraction yields that continued fraction back, so by Proosition , α Q because it is the value of an infinite continued fraction. ( =) Suose α R is an irrational number that satisfies a quadratic equation aα 2 + bα + c = 0 (5.5.1) with a, b, c Z and a 0. Let [a 0, a 1,...] be the continued fraction exansion of α. For each n, let so r n = [a n, a n+1,...], α = [a 0, a 1,..., a n 1, r n ]. We will rove eriodicity by showing that the set of r n s is finite. If we have shown finiteness, then there exists n, h > 0 such that r n = r n+h, so [a 0,..., a n 1, r n ] = [a 0,..., a n 1, a n,..., a n+h 1, r n+h ] = [a 0,..., a n 1, a n,..., a n+h 1, r n ] = [a 0,..., a n 1, a n,..., a n+h 1, a n,..., a n+h 1, r n+h ] = [a 0,..., a n 1, a n,..., a n+h 1 ]. It remains to show there are only finitely many distinct r n. We have α = n q n = r n n 1 + n 2 r n q n 1 + q n 2. Substituting this exression for α into the quadratic equation (5.5.1), we see that A n r 2 n + B n r n + C n = 0, where A n = a 2 n 1 + b n 1 q n 1 + cq 2 n 1, B n = 2a n 1 n 2 + b( n 1 q n 2 + n 2 q n 1 ) + 2cq n 1 q n 2, and C n = a 2 n 2 + b n 2 q n 2 + cq 2 n 2. Note that A n, B n, C n Z, that C n = A n 1, and that B 2 n 4A n C n = (b 2 4ac)( n 1 q n 2 q n 1 n 2 ) 2 = b 2 4ac. Recall from the roof of Theorem that α n 1 < 1. q n q n 1 q n 1
121 Continued Fractions Thus, so Hence, Thus, ( A n = a αq n 1 + αq n 1 n 1 < 1 q n < 1 q n 1, n 1 = αq n 1 + δ q n 1 δ q n 1 with δ < 1. ) 2 + b ( αq n 1 + δ q n 1 = (aα 2 + bα + c)qn aαδ + a δ2 qn bδ = 2aαδ + a δ2 qn bδ. δ2 A n = 2aαδ + a q 2 n 1 ) q n 1 + cq 2 n 1 + bδ < 2 aα + a + b. We conclude that there are only finitely many ossibilities for the integer A n. Also, C n = A n 1 and B n = b 2 4(ac A n C n ), so there are only finitely many triles (A n, B n, C n ), and hence only finitely many ossibilities for r n as n varies, which comletes the roof. (The roof above closely follows [HW79, Thm. 177, g ].) Continued Fractions of Algebraic Numbers of Higher Degree Definition (Algebraic Number). An algebraic number is a root of a olynomial f Q[x]. Oen Problem Give a simle descrition of the comlete continued fractions exansion of the algebraic number 3 2. It begins [1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1,...] The author does not see a attern, and the 534 reduces his confidence that he will. Lang and Trotter (see [LT72]) analyzed many terms of the continued fraction of 3 2 statistically, and their work suggests that 3 2 has an unusual continued fraction; later work in [LT74] suggests that maybe it does not.
122 Khintchine (see [Khi63, g. 59]) 5.6 Recognizing Rational Numbers 115 No roerties of the reresenting continued fractions, analogous to those which have just been roved, are known for algebraic numbers of higher degree [as of 1963]. [...] It is of interest to oint out that u till the resent time no continued fraction develoment of an algebraic number of higher degree than the second is known [emhasis added]. It is not even known if such a develoment has bounded elements. Generally seaking the roblems associated with the continued fraction exansion of algebraic numbers of degree higher than the second are extremely difficult and virtually unstudied. Richard Guy (see [Guy94, g. 260]) Is there an algebraic number of degree greater than two whose simle continued fraction has unbounded artial quotients? Does every such number have unbounded artial quotients? Baum and Sweet [BS76] answered the analog of Richard Guy s question, but with algebraic numbers relaced by elements of a field K other than Q. (The field K is F 2 ((1/x)), the field of Laurent series in the variable 1/x over the finite field with two elements. An element of K is a olynomial in x lus a formal ower series in 1/x.) They found an α of degree 3 over K whose continued fraction has all terms of bounded degree, and other elements of various degrees greater than 2 over K whose continued fractions have terms of unbounded degree. 5.6 Recognizing Rational Numbers Suose that somehow you can comute aroximations to some rational number, and want to figure what the rational number robably is. Comuting the aroximation to high enough recision to find a eriod in the decimal exansion is not a good aroach, because the eriod can be huge (see below). A much better aroach is to comute the simle continued fraction of the aroximation, and truncate it before a large artial quotient a n, then comute the value of the truncated continued fraction. This results in a rational number that has a relatively small numerator and denominator, and is close to the aroximation of the rational number, since the tail end of the continued fraction is at most 1/a n. We begin with a contrived examle, which illustrates how to recognize a rational number. Let x = 9495/3847 =
123 Continued Fractions The continued fraction of the truncation is [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1, , 2, 1, 1, 1,...] We have [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1] = Notice that no reetition is evident in the digits of x given above, though we know that the decimal exansion of x must be eventually eriodic, since all decimal exansions of rational numbers are eventually eriodic. In fact, the length of the eriod of the decimal exansion of 1/3847 is 3846, which is the order of 10 modulo 3847 (see Exercise 5.7). For a slightly less contrived alication of this idea, suose f(x) Z[x] is a olynomial with integer coefficients, and we know for some reason that one root of f is a rational number. We can find that rational number, by using Newton s method to aroximate each root, and continued fractions to decide whether each root is a rational number (we can substitute the value of the continued fraction aroximation into f to see if it is actually a root). One could also use the wellknown Rational Root Theorem, which asserts that any rational root n/d of f, with n, d Z corime, has the roerty that n divides the constant term of f and d the leading coefficient of f. However, using that theorem to find n/d would require factoring the constant and leading terms of f, which could be comletely imractical if they have a few hundred digits (see Section 1.1.3). In contrast, Newton s method and continued fractions should quickly find n/d, assuming the degree of f isn t too large. For examle, suose f = 3847x x To aly Newton s method, let x 0 be a guess for a root of f. Iterate using the recurrence x n+1 = x n f(x n) f (x n ). Choosing x 0 = 0, aroximations of the first two iterates are and x 1 = , x 2 = The continued fraction of the aroximations x 1 and x 2 are and [2, 2, 6, 1, 47, 2, 1, 4, 3, 1, 5, 8, 2, 3] [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1, 103, 8, 1, 2, 3,...]. Truncating the continued fraction of x 2 before 103 gives [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1], which evaluates to 9495/3847, which is a rational root of f.
124 5.7 Sums of Two Squares 117 SAGE Examle We do the above calculation using SAGE. First we imlement the Newton iteration: sage: def newton_root(f, iterates=2, x0=0, rec=53):... x = RealField(rec)(x0)... R = PolynomialRing(ZZ, x )... f = R(f)... g = f.derivative()... for i in range(iterates):... x = x  f(x)/g(x)... return x Next we run the Newton iteration, and comute the continued fraction of the result: sage: a = newton_root(3847*x^ *x ); a sage: cf = continued_fraction(a); cf [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1, 103, 8, 1, 2, 3, 1, 1] We truncate the continued fraction and comute its value. sage: c = cf[:12]; c [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1] sage: c.value() 9495/3847 Another comutational alication of continued fractions, which we can only hint at, is that there are functions in certain arts of advanced number theory (that are beyond the scoe of this book) that take rational values at certain oints, and which can only be comuted efficiently via aroximations; using continued fractions as illustrated above to evaluate such functions is crucial. 5.7 Sums of Two Squares In this section, we aly continued fractions to rove the following theorem. Theorem A ositive integer n is a sum of two squares if and only if all rime factors of n such that 3 (mod 4) have even exonent in the rime factorization of n. We first consider some examles. Notice that 5 = is a sum of two squares, but 7 is not a sum of two squares. Since 2001 is divisible by 3 (because is divisible by 3), but not by 9 (since is not), Theorem imlies that 2001 is not a sum of two squares. The theorem also imlies that is a sum of two squares.
125 Continued Fractions SAGE Examle We use Sage to write a short rogram that naively determines whether or not an integer n is a sum of two squares, and if so returns a, b such that a 2 + b 2 = n. sage: def sum_of_two_squares_naive(n):... for i in range(int(sqrt(n))):... if is_square(n  i^2):... return i, (Integer(ni^2)).sqrt()... return "%s is not a sum of two squares"%n We next use our function in a coule of cases. sage: sum_of_two_squares_naive(23) 23 is not a sum of two squares sage: sum_of_two_squares_naive(389) (10, 17) sage: sum_of_two_squares_naive(2007) 2007 is not a sum of two squares sage: sum_of_two_squares_naive(2008) 2008 is not a sum of two squares sage: sum_of_two_squares_naive(2009) (28, 35) sage: 28^2 + 35^ sage: sum_of_two_squares_naive(2*3^4*5*7^2*13) (189, 693) Definition (Primitive). A reresentation n = x 2 + y 2 is rimitive if x and y are corime. Lemma If n is divisible by a rime 3 (mod 4), then n has no rimitive reresentations. Proof. Suose n has a rimitive reresentation, n = x 2 + y 2, and let be any rime factor of n. Then x 2 + y 2 and gcd(x, y) = 1, so x and y. Since Z/Z is a field, we may divide by y 2 in the equation x 2 + y 2 ( 0 )(mod ) to see that (x/y) 2 1 (mod ). Thus the Legendre symbol equals +1. However, by Proosition 4.2.1, 1 ( ) 1 = ( 1) ( 1)/2 ( ) 1 so = 1 if and only if ( 1)/2 is even, which is to say 1 (mod 4).
126 5.7 Sums of Two Squares 119 Proof of Theorem (= ). Suose that 3 (mod 4) is a rime, that r n but r+1 n with r odd, and that n = x 2 + y 2. Letting d = gcd(x, y), we have with gcd(x, y ) = 1 and x = dx, y = dy, and n = d 2 n (x ) 2 + (y ) 2 = n. Because r is odd, n, so Lemma imlies that gcd(x, y ) > 1, which is a contradiction. To reare for our roof of the imlication ( =) of Theorem 5.7.1, we reduce the roblem to the case when n is rime. Write n = n 2 1n 2, where n 2 has no rime factors 3 (mod 4). It suffices to show that n 2 is a sum of two squares, since (x y 2 1)(x y 2 2) = (x 1 x 2 y 1 y 2 ) 2 + (x 1 y 2 + x 2 y 1 ) 2, (5.7.1) so a roduct of two numbers that are sums of two squares is also a sum of two squares. Since 2 = is a sum of two squares, it suffices to show that any rime 1 (mod 4) is a sum of two squares. Lemma If x R and n N, then there is a fraction a b terms such that 0 < b n and x a 1 b b(n + 1). in lowest Proof. Consider the continued fraction [a 0, a 1,...] of x. By Corollary , for each m x m < 1. q m q m+1 q m Since q m+1 q m + 1 and q 0 = 1, either there exists an m such that q m n < q m+1, or the continued fraction exansion of x is finite and n is larger than the denominator of the rational number x, in which case we take a b = x and are done. In the first case, so a b = m q m x m < 1 q m q m+1 q m satisfies the conclusion of the lemma. 1 q m (n + 1), Proof of Theorem ( =). As discussed above, it suffices to rove that any rime 1 (mod 4) is a sum of two squares. Since 1 (mod 4), ( 1) ( 1)/2 = 1,
127 Continued Fractions Proosition imlies that 1 is a square modulo ; i.e., there exists r Z such that r 2 1 (mod ). Lemma 5.7.5, with n = and x = r, imlies that there are integers a, b such that 0 < b < and r a b 1 b(n + 1) < 1 b. Letting c = rb + a, we have that so But c rb (mod ), so Thus b 2 + c 2 =. c < b b = = 0 < b 2 + c 2 < 2. b 2 + c 2 b 2 + r 2 b 2 b 2 (1 + r 2 ) 0 (mod ). Remark Our roof of Theorem leads to an efficient algorithm to comute a reresentation of any 1 (mod 4) as a sum of two squares. SAGE Examle We next use Sage and Theorem to give an efficient algorithm for writing a rime 1 (mod 4) as a sum of two squares. First we imlement the algorithm that comes out of the roof of the theorem. sage: def sum_of_two_squares():... = Integer()... assert %4 == 1, " must be 1 modulo 4"... r = Mod(1,).sqrt().lift()... v = continued_fraction(r/)... n = floor(sqrt())... for x in v.convergents():... c = r*x.denominator() + *x.numerator()... if n <= c and c <= n:... return (abs(x.denominator()),abs(c)) Next we use the algorithm to write the first 10digit rime 1 (mod 4) as a sum of two squares: sage: = next_rime(next_rime(10^10)) sage: sum_of_two_squares() (55913, 82908) The above calculation was essentially instantanoues. If instead we use the naive algorithm from before, it takes several seconds to write as a sum of two squares. sage: sum_of_two_squares_naive() (55913, 82908)
128 5.8 Exercises Exercises 5.1 If c n = n /q n is the nth convergent of [a 0, a 1,..., a n ] and a 0 > 0, show that [a n, a n 1,..., a 1, a 0 ] = n n 1 and (Hint: In the first case, notice that [a n, a n 1,..., a 2, a 1 ] = q n q n 1. n = a n + n 2 = a n + 1 n 1 n 1.) n 1 n Show that every nonzero rational number can be reresented in exactly two ways by a finite simle continued fraction. (For examle, 2 can be reresented by [1, 1] and [2], and 1/3 by [0, 3] and [0, 2, 1].) 5.3 Evaluate the infinite continued fraction [2, 1, 2, 1]. 5.4 Determine the infinite continued fraction of Let a 0 R and a 1,..., a n and b be ositive real numbers. Prove that if and only if n is odd. [a 0, a 1,..., a n + b] < [a 0, a 1,..., a n ] 5.6 (*) Extend the method resented in the text to show that the continued fraction exansion of e 1/k is [1, (k 1), 1, 1, (3k 1), 1, 1, (5k 1), 1, 1, (7k 1),...] for all k N. (a) Comute 0, 3, q 0, and q 3 for the above continued fraction. Your answers should be in terms of k. (b) Condense three stes of the recurrence for the numerators and denominators of the above continued fraction. That is, roduce a simle recurrence for r 3n in terms of r 3n 3 and r 3n 6 whose coefficients are olynomials in n and k. (c) Define a sequence of real numbers by T n (k) = 1 1/k (kt) n (kt 1) n k n 0 n! e t dt. i. Comute T 0 (k), and verify that it equals q 0 e 1/k 0. ii. Comute T 1 (k), and verify that it equals q 3 e 1/k 3.
129 Continued Fractions iii. Integrate T n (k) by arts twice in succession, as in Section 5.4, and verify that T n (k), T n 1 (k), and T n 2 (k) satisfy the recurrence roduced in art 6b, for n 2. (d) Conclude that the continued fraction [1, (k 1), 1, 1, (3k 1), 1, 1, (5k 1), 1, 1, (7k 1),...] reresents e 1/k. 5.7 Let d be an integer that is corime to 10. Prove that the decimal exansion of 1 d has a eriod equal to the order of 10 modulo d. (Hint: 1 For every ositive integer r, we have 1 10 = r n 1 10 rn.) 5.8 Find a ositive integer that has at least three different reresentations as the sum of two squares, disregarding signs and the order of the summands. 5.9 Show that if a natural number n is the sum of two rational squares it is also the sum of two integer squares (*) Let be an odd rime. Show that 1, 3 (mod 8) if and only if can be written as = x 2 +2y 2 for some choice of integers x and y Prove that of any four consecutive integers, at least one is not reresentable as a sum of two squares.
130 6 Ellitic Curves This is age 123 Printer: Oaque this Ellitic curves are number theoretic objects that are central to both ure and alied number theory. Dee roblems in number theory such as the congruent number roblem which integers are the area of a right triangle with rational side lengths? translate naturally into questions about ellitic curves. Other questions, such as the famous Birch and SwinnertonDyer conjecture, describe mysterious structure that mathematicians exect ellitic curves to have. One can also associate finite abelian grous to ellitic curves, and in many cases these grous are well suited to the construction of crytosystems. In articular, ellitic curves are widely believed to rovide good security with smaller key sizes, something that is useful in many alications, for examle, if we are going to rint an encrytion key on a ostage stam, it is helful if the key is short! Morover, there is a way to use ellitic curves to factor integers, which lays a crucial role in sohisticated attacks on the RSA ublickey crytosystem of Section 3.3. This chater is a brief introduction to ellitic curves that builds on the ideas of Chaters 1 3 and introduces several dee theorems and ideas that we will not rove. In Section 6.1, we define ellitic curves and draw some ictures of them, and then in Section 6.2 we describe how to ut a grou structure on the set of oints on an ellitic curve. Sections 6.3 and 6.4 are about how to aly ellitic curves to two crytograhic roblems constructing ublickey crytosystems and factoring integers. Finally, in Section 6.5, we consider ellitic curves over the rational numbers, and exlain a dee connection between ellitic curves and a 1,000year old unsolved roblem.
131 Ellitic Curves FIGURE 6.1. The ellitic curve y 2 = x 3 5x + 4 over R 6.1 The Definition Definition (Ellitic Curve). An ellitic curve over a field K is a curve defined by an equation of the form y 2 = x 3 + ax + b, where a, b K and 16(4a b 2 ) 0. The condition that 16(4a b 2 ) 0 imlies that the curve has no singular oints, which will be essential for the alications we have in mind (see Exercise 6.1). SAGE Examle We use the ElliticCurve command to create an ellitic curve over the rational field Q and draw the lot in Figure 6.1. sage: E = ElliticCurve([5, 4]) sage: E Ellitic Curve defined by y^2 = x^35*x + 4 over Rational Field sage: P = E.lot(thickness=4,rgbcolor=(0.1,0.7,0.1)) sage: P.show(figsize=[4,6]) We will use ellitic curves over finite fields to factor integers in Section 6.3 and to construct crytosystems in Section 6.4. The following Sage code creates an ellitic curve over the finite field of order 37 and lots it, as illustrated in Figure 6.2.
132 6.2 The Grou Structure on an Ellitic Curve 125 sage: E = ElliticCurve(GF(37), [1,0]) sage: E Ellitic Curve defined by y^2 = x^3 + x over Finite Field of size 37 sage: E.lot(ointsize=45) FIGURE 6.2. The ellitic curve y 2 = x 3 + x over Z/37Z In Section 6.2, we will ut a natural abelian grou structure on the set E(K) = {(x, y) K K : y 2 = x 3 + ax + b} {O} of Krational oints on an ellitic curve E over K. Here, O may be thought of as a oint on E at infinity. Figure 6.2 contains a lot of the oints of y 2 = x 3 + x over the finite field Z/37Z, though note that we do not exlicitly draw the oint at O at infinity. Remark If K has characteristic 2 (i.e., we have = 0 in K), then for any choice of a, b, the quantity 16(4a b 2 ) K is 0, so according to Definition there are no ellitic curves over K. There is a similar roblem in characteristic 3. If we instead consider equations of the form y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, we obtain a more general definition of ellitic curves, which correctly allows for ellitic curves in characteristics 2 and 3; these ellitic curves are oular in crytograhy because arithmetic on them is often easier to efficiently imlement on a comuter. 6.2 The Grou Structure on an Ellitic Curve Let E be an ellitic curve over a field K, given by an equation y 2 = x 3 + ax + b. We begin by defining a binary oeration + on E(K).
133 Ellitic Curves Algorithm (Ellitic Curve Grou Law). Given P 1, P 2 E(K), this algorithm comutes a third oint R = P 1 + P 2 E(K). 1. [Is P i = O?] If P 1 = O set R = P 2 or if P 2 = O set R = P 1 and terminate. Otherwise write (x i, y i ) = P i. 2. [Negatives] If x 1 = x 2 and y 1 = y 2, set R = O and terminate. { (3x a)/(2y 1 ) if P 1 = P 2, 3. [Comute λ] Set λ = (y 1 y 2 )/(x 1 x 2 ) otherwise. 4. [Comute Sum] Then R = ( λ 2 x 1 x 2, λx 3 ν ), where ν = y 1 λx 1 and x 3 = λ 2 x 1 x 2 is the xcoordinate of R. Note that in Ste 3, if P 1 = P 2, then y 1 0; otherwise, we would have terminated in the revious ste. Theorem The binary oeration + defined in Algorithm endows the set E(K) with an abelian grou structure, with identity O. Before discussing why the theorem is true, we reinterret + geometrically, so that it will be easier for us to visualize. We obtain the sum P 1 +P 2 by finding the third oint P 3 of intersection between E and the line L determined by P 1 and P 2, then reflecting P 3 about the xaxis. (This descrition requires suitable interretation in cases 1 and 2, and when P 1 = P 2.) This is illustrated in Figure 6.3, in which (0, 2)+(1, 0) = (3, 4) on y 2 = x 3 5x+4. SAGE Examle We create the ellitic curve y 2 = x 3 5x+4 in Sage, then add together P = (1, 0) and Q = (0, 2). We also comute P +P, which is the oint O at infinity, which is reresented in Sage by (0 : 1 : 0), and comute the sum P + Q + Q + Q + Q, which is surrisingly large. sage: E = ElliticCurve([5,4]) sage: P = E([1,0]); Q = E([0,2]) sage: P + Q (3 : 4 : 1) sage: P + P (0 : 1 : 0) sage: P + Q + Q + Q + Q (350497/ : / : 1) To further clarify the above geometric interretation of the grou law, we rove the following roosition. Proosition (Geometric Grou Law). Suose P i = (x i, y i ), i = 1, 2 are distinct oints on an ellitic curve y 2 = x 3 + ax + b, and that x 1 x 2. Let L be the unique line through P 1 and P 2. Then L intersects the grah of E at exactly one other oint Q = ( λ 2 x 1 x 2, λx 3 + ν ), where λ = (y 1 y 2 )/(x 1 x 2 ) and ν = y 1 λx 1.
134 6.2 The Grou Structure on an Ellitic Curve FIGURE 6.3. The Grou Law: (1, 0) + (0, 2) = (3, 4) on y 2 = x 3 5x + 4 Proof. The line L through P 1, P 2 is y = y 1 + (x x 1 )λ. Substituting this into y 2 = x 3 + ax + b, we get (y 1 + (x x 1 )λ) 2 = x 3 + ax + b. Simlifying, we get f(x) = x 3 λ 2 x 2 + = 0, where we omit the coefficients of x and the constant term since they will not be needed. Since P 1 and P 2 are in L E, the olynomial f has x 1 and x 2 as roots. By Proosition 2.5.3, the olynomial f can have at most three roots. Writing f = (x x i ) and equating terms, we see that x 1 + x 2 + x 3 = λ 2. Thus, x 3 = λ 2 x 1 x 2, as claimed. Also, from the equation for L we see that y 3 = y 1 + (x 3 x 1 )λ = λx 3 + ν, which comletes the roof. To rove Theorem means to show that + satisfies the three axioms of an abelian grou with O as identity element: existence of inverses, commutativity, and associativity. The existence of inverses follows immediately from the definition, since (x, y) + (x, y) = O. Commutativity is also clear from the definition of grou law, since in Parts 1 3, the recie is unchanged if we swa P 1 and P 2 ; in Part 4 swaing P 1 and P 2 does not change the line determined by P 1 and P 2, so by Proosition it does not change the sum P 1 + P 2. It is more difficult to rove that + satisfies the associative axiom, i.e., that (P 1 + P 2 ) + P 3 = P 1 + (P 2 + P 3 ). This fact can be understood from at least three oints of view. One is to reinterret the grou law geometrically (extending Proosition to all cases), and thus transfer the roblem to a question in lane geometry. This aroach is beautifully exlained
135 Ellitic Curves with exactly the right level of detail in [ST92, I.2]. Another aroach is to use the formulas that define + to reduce associativity to checking secific algebraic identities; this is something that would be extremely tedious to do by hand, but can be done using a comuter (also tedious). A third aroach (see [Sil86] or [Har77]) is to develo a general theory of divisors on algebraic curves, from which associativity of the grou law falls out as a natural corollary. The third aroach is the best, because it oens u many new vistas; however, we will not ursue it further because it is beyond the scoe of this book. SAGE Examle In the following Sage session, we use the formula from Algorithm to verify that the grou law holds for any choice of oints P 1, P 2, P 3 on any ellitic curve over Q such that the oints P 1, P 2, P 3, P 1 + P 2, P 2 + P 3 are all distinct and nonzero. We define a olynomial ring R in 8 variables. sage: R.<x1,y1,x2,y2,x3,y3,a,b> = QQ[] We define the relations the x i will satisfy, and a quotient ring Q in which those relations are satisfied. (Quotients of olynomial rings are a generalization of the construction Z/nZ that may be viewed as the quotient of the ring Z of integers by the relation that sets n to equal 0.) sage: rels = [y1^2  (x1^3 + a*x1 + b),... y2^2  (x2^3 + a*x2 + b),... y3^2  (x3^3 + a*x3 + b)]... sage: Q = R.quotient(rels) We define the grou oeration, which assumes the oints are distinct. sage: def o(p1,p2):... x1,y1 = P1; x2,y2 = P2... lam = (y1  y2)/(x1  x2); nu = y1  lam*x1... x3 = lam^2  x1  x2; y3 = lam*x3  nu... return (x3, y3) We define three oints, add them together via P 1 + (P 2 + P 3 ) and (P 1 + (P 2 + P 3 )), and observe that the results are the same modulo the relations. sage: P1 = (x1,y1); P2 = (x2,y2); P3 = (x3,y3) sage: Z = o(p1, o(p2,p3)); W = o(o(p1,p2),p3) sage: (Q(Z[0].numerator()*W[0].denominator() ... Z[0].denominator()*W[0].numerator())) == 0 True sage: (Q(Z[1].numerator()*W[1].denominator() ... Z[1].denominator()*W[1].numerator())) == 0 True
136 6.3 Integer Factorization Using Ellitic Curves Integer Factorization Using Ellitic Curves In 1987, Hendrik Lenstra ublished the landmark aer [Len87] that introduces and analyzes the Ellitic Curve Method (ECM), which is a owerful algorithm for factoring integers using ellitic curves. Lenstra s method is also described in [ST92, IV.4], [Dav99, VIII.5], and [Coh93, 10.3]. Lenstra s algorithm is well suited for finding mediumsized factors of an integer N, which today means between 10 to 40 decimal digits. The ECM method is not directly used for factoring RSA challenge numbers (see Section 1.1.3), but it is used on auxiliary numbers as a crucial ste in the number field sieve, which is the best known algorithm for hunting for such factorizations. Also, imlementation of ECM tyically requires little memory. H. Lenstra Pollard s ( 1)Method Lenstra s discovery of ECM was insired by Pollard s ( 1)method, which we describe in this section. Definition (Power Smooth). Let B be a ositive integer. If n is a ositive integer with rime factorization n = ei i, then n is Bower smooth if ei i B for all i. For examle, 30 = is B ower smooth for B = 5, 7, but 150 = is not 5ower smooth (it is B = 25ower smooth). We will use the following algorithm in both the Pollard 1 and ellitic curve factorization methods. Algorithm (Least Common Multile of First B Integers). Given a ositive integer B, this algorithm comutes the least common multile of the ositive integers u to B. 1. [Sieve] Using, for examle, the rime sieve (Algorithm 1.2.3), comute a list P of all rimes B. 2. [Multily] Comute and outut the roduct P log (B). Proof. Set m = lcm(1, 2,..., B). Then, ord (m) = max({ord (n) : 1 n B}) = ord ( r ), where r is the largest ower of that satisfies r B. Since r B < r+1, we have r = log (B).
137 Ellitic Curves SAGE Examle We imlement Algorithm in Sage and comute the least common multile for B = 100 using both the above algorithm and a naive algorithm. We use math.log below so that log (B) is comuted quickly using double recision numbers. sage: def lcm_uto(b):... return rod([^int(math.log(b)/math.log())... for in rime_range(b+1)]) sage: lcm_uto(10^2) sage: LCM([1..10^2]) Algorithm as imlemented above in Sage takes about a second for B = Let N be a ositive integer that we wish to factor. We use the Pollard ( 1)method to look for a nontrivial factor of N as follows. First, we choose a ositive integer B, usually with at most six digits. Suose that there is a rime divisor of N such that 1 is Bower smooth. We try to find using the following strategy. If a > 1 is an integer not divisible by, then by Theorem , a 1 1 (mod ). Let m = lcm(1, 2, 3,..., B), and observe that our assumtion that 1 is Bower smooth imlies that 1 m, so a m 1 (mod ). Thus gcd(a m 1, N) > 1. If gcd(a m 1, N) < N also then gcd(a m 1, N) is a nontrivial factor of N. If gcd(a m 1, N) = N, then a m 1 (mod q r ) for every rime ower divisor q r of N. In this case, reeat the above stes but with a smaller choice of B or ossibly a different choice of a. Also, it is a good idea to check from the start whether or not N is not a erfect ower M r and, if so, relace N by M. We formalize the algorithm as follows: Algorithm (Pollard 1 Method). Given a ositive integer N and a bound B, this algorithm attemts to find a nontrivial factor g of N. (Each rime g is likely to have the roerty that 1 is Bower smooth.) 1. [Comute lcm] Use Algorithm to comute m = lcm(1, 2,..., B). 2. [Initialize] Set a = [Power and gcd] Comute x = a m 1 (mod N) and g = gcd(x, N). 4. [Finished?] If g 1 or N, outut g and terminate.
138 6.3 Integer Factorization Using Ellitic Curves [Try Again?] If a < 10 (say), relace a by a + 1 and go to ste 3. Otherwise, terminate. For fixed B, Algorithm often slits N when N is divisible by a rime such that 1 is Bower smooth. Aroximately 15 ercent of rimes in the interval from and are such that 1 is 10 6 ower smooth, so the Pollard method with B = 10 6 already fails nearly 85 ercent of the time at finding 15digit rimes in this range (see also Exercise 6.10). We will not analyze Pollard s method further, since it was mentioned here only to set the stage for the ellitic curve factorization method. The following examles illustrate the Pollard ( 1)method. Examle In this examle, Pollard works erfectly. Let N = We try to use the Pollard 1 method with B = 5 to slit N. We have m = lcm(1, 2, 3, 4, 5) = 60; taking a = 2, we have and so 61 is a factor of (mod 5917) gcd(2 60 1, 5917) = gcd(3416, 5917) = 61, Examle In this examle, we relace B with a larger integer. Let N = With B = 5 and a = 2, we have (mod ), and gcd(2 60 1, ) = 1. With B = 15, we have and m = lcm(1, 2,..., 15) = , (mod ), gcd( , N) = 2003, so 2003 is a nontrivial factor of Examle In this examle, we relace B by a smaller integer. Let N = Suose B = 7, so m = lcm(1, 2,..., 7) = 420, (mod 4331), and gcd( , 4331) = 4331, so we do not obtain a factor of If we relace B by 5, Pollard s method works: (mod 4331), and gcd(2 60 1, 4331) = 61, so we slit 4331.
139 Ellitic Curves Examle In this examle, a = 2 does not work, but a = 3 does. Let N = 187. Suose B = 15, so m = lcm(1, 2,..., 15) = , (mod 187), and gcd( , 187) = 187, so we do not obtain a factor of 187. If we relace a = 2 by a = 3, then Pollard s method works: (mod 187), and gcd( , 187) = 11. Thus 187 = SAGE Examle We imlement the Pollard ( 1)method in Sage and use our imlementation to do all of the above examles. sage: def ollard(n, B=10^5, sto=10):... m = rod([^int(math.log(b)/math.log())... for in rime_range(b+1)])... for a in [2..sto]:... x = (Mod(a,N)^m  1).lift()... if x == 0: continue... g = gcd(x, N)... if g!= 1 or g!= N: return g... return 1 sage: ollard(5917,5) 61 sage: ollard(779167,5) 1 sage: ollard(779167,15) 2003 sage: ollard(4331,7) 1 sage: ollard(4331,5) 61 sage: ollard(187, 15, 2) 1 sage: ollard(187, 15) Motivation for the Ellitic Curve Method Fix a ositive integer B. If N = q with and q rime, and we assume that 1 and q 1 are not Bower smooth, then the Pollard ( 1) method is unlikely to work. For examle, let B = 20 and suose that N = = Note that neither 59 1 = 2 29 nor = 4 25 is Bower smooth. With m = lcm(1, 2, 3,..., 20) = , we have 2 m (mod N),
140 6.3 Integer Factorization Using Ellitic Curves 133 and gcd(2 m 1, N) = 1, so we do not find a factor of N. As remarked above, the roblem is that 1 is not 20ower smooth for either = 59 or = 101. However, notice that 2 = 3 19 is 20ower smooth. Lenstra s ECM relaces (Z/Z), which has order 1, by the grou of oints on an ellitic curve E over Z/Z. It is a theorem that #E(Z/Z) = + 1 ± s for some nonnegative integer s < 2 (see [Sil86, V.1] for a roof). Also, every value of s subject to this bound occurs, as one can see using comlex multilication theory. For examle, if E is the ellitic curve y 2 = x 3 + x + 54 over Z/59Z, then by enumerating oints one sees that E(Z/59Z) is cyclic of order 57. The set of numbers ± s for s 15 contains 14 numbers that are Bower smooth for B = 20, which illustrates that working with an ellitic curve gives us more flexibility. For examle, 60 = is 5ower smooth and 70 = is 7ower smooth Lenstra s Ellitic Curve Factorization Method Algorithm (Ellitic Curve Factorization Method). Given a ositive integer N and a bound B, this algorithm attemts to find a nontrivial factor g of N or oututs Fail. 1. [Comute lcm] Use Algorithm to comute m = lcm(1, 2,..., B). 2. [Choose Random Ellitic Curve] Choose a random a Z/NZ such that 4a (Z/NZ). Then P = (0, 1) is a oint on the ellitic curve y 2 = x 3 + ax + 1 over Z/NZ. 3. [Comute Multile] Attemt to comute mp using an ellitic curve analog of Algorithm If at some oint we cannot comute a sum of oints because some denominator in Ste 3 of Algorithm is not corime to N, we comute the greatest common divisor g of this denominator with N. If g is a nontrivial divisor, outut it. If every denominator is corime to N, outut Fail. If Algorithm fails for one random ellitic curve, there is an otion that is unavailable with Pollard s ( 1)method we may reeat the above algorithm with a different ellitic curve. With Pollard s method we always work with the grou (Z/NZ), but here we can try many grous E(Z/NZ) for many curves E. As mentioned above, the number of oints on E over Z/Z is of the form + 1 t for some t with t < 2 ; Algorithm thus has a chance if + 1 t is Bower smooth for some t with t < 2.
141 Ellitic Curves Examles For simlicity, we use an ellitic curve of the form y 2 = x 3 + ax + 1, which has the oint P = (0, 1) already on it. We factor N = 5959 using the ellitic curve method. Let m = lcm(1, 2,..., 20) = = , where x 2 means x is written in binary. First, we choose a = 1201 at random and consider y 2 = x x + 1 over Z/5959Z. Using the formula for P + P from Algorithm we comute 2 i P = 2 i (0, 1) for i B = {4, 5, 6, 7, 8, 13, 21, 22, 23, 24, 26, 27}. Then i B 2i P = mp. It turns out that during no ste of this comutation does a number not corime to 5959 aear in any denominator, so we do not slit N using a = Next, we try a = 389 and at some stage in the comutation we add P = (2051, 5273) and Q = (637, 1292). When comuting the grou law exlicitly, we try to comute λ = (y 1 y 2 )/(x 1 x 2 ) in (Z/5959Z), but we fail since x 1 x 2 = 1414 and gcd(1414, 5959) = 101. We thus find a nontrivial factor 101 of SAGE Examle We imlement ellitic curve factorization in Sage, then use it to do the above examle and some other examles. sage: def ecm(n, B=10^3, trials=10):... m = rod([^int(math.log(b)/math.log())... for in rime_range(b+1)])... R = Integers(N)... # Make Sage think that R is a field:... R.is_field = lambda : True... for _ in range(trials):... while True:... a = R.random_element()... if gcd(4*a.lift()^3 + 27, N) == 1: break... try:... m * ElliticCurve([a, 1])([0,1])... excet ZeroDivisionError, msg:... # msg: "Inverse of <int> does not exist"... return gcd(integer(str(msg).slit()[2]), N)... return 1 sage: set_random_seed(2) sage: ecm(5959, B=20) 101 sage: ecm(next_rime(10^20)*next_rime(10^7), B=10^3)
142 6.3.5 A Heuristic Exlanation 6.4 Ellitic Curve Crytograhy 135 Let N be a ositive integer and, for simlicity of exosition, assume that N = 1 r with the i distinct rimes. It follows from Lemma that there is a natural isomorhism f : (Z/NZ) (Z/ 1 Z) (Z/ r Z). When using Pollard s method, we choose an a (Z/NZ), comute a m, then comute gcd(a m 1, N). This gcd is divisible exactly by the rimes i such that a m 1 (mod i ). To reinterret Pollard s method using the above isomorhism, let (a 1,..., a r ) = f(a). Then (a m 1,..., a m r ) = f(a m ), and the i that divide gcd(a m 1, N) are exactly the i such that a m i = 1. By Theorem , these i include the rimes j such that j 1 is Bower smooth, where m = lcm(1,..., m). We will not define E(Z/NZ) when N is comosite, since this is not needed for the algorithm (where we assume that N is rime and hoe for a contradiction). However, for the remainder of this aragrah, we retend that E(Z/N Z) is meaningful and describe a heuristic connection between Lenstra and Pollard s methods. The significant difference between Pollard s method and the ellitic curve method is that the isomorhism f is relaced by an isomorhism (in quotes) g : E(Z/NZ) E(Z/ 1 Z) E(Z/ r Z) where E is y 2 = x 3 + ax + 1, and the a of Pollard s method is relaced by P = (0, 1). We ut the isomorhism in quotes to emhasize that we have not defined E(Z/N Z). When carrying out the ellitic curve factorization algorithm, we attemt to comute mp, and if some comonents of f(q) are O, for some oint Q that aears during the comutation, but others are nonzero, we find a nontrivial factor of N. 6.4 Ellitic Curve Crytograhy The idea to use ellitic curves in crytograhy was indeendently roosed by Neil Koblitz and Victor Miller in the mid 1980s. In this section, we discuss an analog of DiffieHellman that uses an ellitic curve instead of (Z/Z). We then discuss the ElGamal ellitic curve crytosystem Ellitic Curve Analogs of DiffieHellman The DiffieHellman key exchange from Section 3.2 works well on an ellitic curve with no serious modification. Michael and Nikita agree on a secret key as follows:
143 Ellitic Curves 1. Michael and Nikita agree on a rime, an ellitic curve E over Z/Z, and a oint P E(Z/Z). 2. Michael secretly chooses a random m and sends mp. 3. Nikita secretly chooses a random n and sends np. 4. The secret key is nmp, which both Michael and Nikita can comute. Presumably, an adversary can not comute nmp without solving the discrete logarithm roblem (see Problem and Section below) in E(Z/Z). For wellchosen E, P, and, exerience suggests that the discrete logarithm roblem in E(Z/Z) is much more difficult than the discrete logarithm roblem in (Z/Z) (see Section for more on the ellitic curve discrete log roblem) The ElGamal Crytosystem and Digital Rights Management This section is about the ElGamal crytosystem, which works well on an ellitic curve. This section draws on a aer by a comuter hacker named Beale Screamer who cracked a Digital Rights Management (DRM) system. The ellitic curve used in the DRM is an ellitic curve over the finite field k = Z/Z, where = The number in base 16 is 89ABCDEF F7, which includes counting in hexadecimal, and digits of e, π, and 2. The ellitic curve E is y 2 = x x We have #E(k) = , and the grou E(k) is cyclic with generator B = ( , ).
144 6.4 Ellitic Curve Crytograhy 137 Our heroes Nikita and Michael share digital music when they are not out fighting terrorists. When Nikita installed the DRM software on her comuter, it generated a rivate key n = , which it hides in bits and ieces of files. In order for Nikita to lay Juno Reactor s latest hit juno.wma, her web browser contacts a website that sells music. After Nikita sends her credit card number, that website allows Nikita to download a license file that allows her audio layer to unlock and lay juno.wma. As we will see below, the license file was created using the ElGamal ublickey crytosystem in the grou E(k). Nikita can now use her license file to unlock juno.wma. However, when she shares both juno.wma and the license file with Michael, he is frustrated because even with the license, his comuter still does not lay juno.wma. This is because Michael s comuter does not know Nikita s comuter s rivate key (the integer n above), so Michael s comuter can not decryt the license file. We now describe the ElGamal crytosystem, which lends itself well to imlementation in the grou E(Z/Z). To illustrate ElGamal, we describe how Nikita would set u an ElGamal crytosystem that anyone could use to encryt messages for her. Nikita chooses a rime, an ellitic curve E over Z/Z, and a oint B E(Z/Z), and ublishes, E, and B. She also chooses a random integer n, which she kees secret, and ublishes nb. Her ublic key is the fourtule (, E, B, nb). Suose Michael wishes to encryt a message for Nikita. If the message is encoded as an element P E(Z/Z), Michael comutes a random integer r and the oints rb and P + r(nb) on E(Z/Z). Then P is encryted as the air (rb, P + r(nb)). To decryt the encryted message, Nikita multilies rb by her secret key n to find n(rb) = r(nb), then subtracts this from P + r(nb) to obtain P = P + r(nb) r(nb). Remark It also make sense to construct an ElGamal crytosystem in the grou (Z/Z). Returning to our story, Nikita s license file is an encryted message to her. It contains the air of oints (rb, P + r(nb)), where and rb = ( , ) P + r(nb) = ( , ).
145 Ellitic Curves When Nikita s comuter lays juno.wma, it loads the secret key n = into memory and comutes n(rb) = ( , ). It then subtracts this from P + r(nb) to obtain P = ( , ). The xcoordinate is the key that unlocks juno.wma. If Nikita knew the rivate key n that her comuter generated, she could comute P herself and unlock juno.wma and share her music with Michael. Beale Screamer found a weakness in the imlementation of this system that allows Nikita to detetermine n, which is not a huge surrise since n is stored on her comuter after all. SAGE Examle We do the above examles in Sage: sage: = sage: E = ElliticCurve(GF(), \... [ , ]) sage: E.cardinality() sage: E.cardinality().is_rime() True sage: B = E([ , ]) sage: n= sage: r= sage: P = E([ , ]) sage: encryt = (r*b, P + r*(n*b)) sage: encryt[1]  n*encryt[0] == P # decryting works True The Ellitic Curve Discrete Logarithm Problem Problem (Ellitic Curve Discrete Log Problem). Suose E is an ellitic curve over Z/Z and P E(Z/Z). Given a multile Q of P, the ellitic curve discrete log roblem is to find n Z such that np = Q.
146 6.4 Ellitic Curve Crytograhy 139 For examle, let E be the ellitic curve given by y 2 = x 3 + x + 1 over the field Z/7Z. We have E(Z/7Z) = {O, (2, 2), (0, 1), (0, 6), (2, 5)}. If P = (2, 2) and Q = (0, 6), then 3P = Q, so n = 3 is a solution to the discrete logarithm roblem. If E(Z/Z) has order or ± 1, or is a roduct of reasonably small rimes, then there are some methods for attacking the discrete log roblem on E, which are beyond the scoe of this book. It is therefore imortant to be able to comute #E(Z/Z) efficiently, in order to verify that the ellitic curve one wishes to use for a crytosystem doesn t have any obvious vulnerabilities. The naive algorithm to comute #E(Z/Z) is to try each value of x Z/Z and count how often x 3 +ax+b is a erfect square mod, but this is of no use when is large enough to be useful for crytograhy. Fortunately, there is an algorithm due to Schoof, Elkies, and Atkin for comuting #E(Z/Z) efficiently (olynomial time in the number of digits of ), but this algorithm is beyond the scoe of this book. In Section 3.2.1, we discussed the discrete log roblem in (Z/Z). There are general attacks called index calculus attacks on the discrete log roblem in (Z/Z) that are slow, but still faster than the known algorithms for solving the discrete log in a general grou (one with no extra structure). For most ellitic curves, there is no known analog of index calculus attacks on the discrete log roblem. At resent, it aears that given, the discrete log roblem in E(Z/Z) is much harder than the discrete log roblem in the multilicative grou (Z/Z). This suggests that by using an ellitic curvebased crytosystem instead of one based on (Z/Z), one gets equivalent security with much smaller numbers, which is one reason why building crytosystems using ellitic curves is attractive to some crytograhers. For examle, Certicom, a comany that strongly suorts ellitic curve crytograhy, claims: [Ellitic curve cryto] devices require less storage, less ower, less memory, and less bandwidth than other systems. This allows you to imlement crytograhy in latforms that are constrained, such as wireless devices, handheld comuters, smart cards, and thinclients. It also rovides a big win in situations where efficiency is imortant. For an utodate list of ellitic curve discrete log challenge roblems that Certicom sonsors, see [Cer]. For examle, in Aril 2004, a secific crytosystem was cracked that was based on an ellitic curve over Z/Z, where has 109 bits. The first unsolved challenge roblem involves an ellitic curve over Z/Z, where has 131 bits, and the next challenge after that is one in which has 163 bits. Certicom claims at [Cer] that the 163bit challenge roblem is comutationally infeasible.
147 Ellitic Curves FIGURE 6.4. Louis J. Mordell 6.5 Ellitic Curves Over the Rational Numbers Let E be an ellitic curve defined over Q. The following is a dee theorem about the grou E(Q). Theorem (Mordell). The grou E(Q) is finitely generated. That is, there are oints P 1,..., P s E(Q) such that every element of E(Q) is of the form n 1 P n s P s for integers n 1,... n s Z. Mordell s theorem imlies that it makes sense to ask whether or not we can comute E(Q), where by comute we mean find a finite set P 1,..., P s of oints on E that generate E(Q) as an abelian grou. There is a systematic aroach to comuting E(Q) called descent (see, for examle, [Cre97, Cre, Sil86]). It is widely believed that the method of descent will always succeed, but nobody has yet roved that it will. Proving that descent works for all curves is one of the central oen roblems in number theory, and is closely related to the Birch and SwinnertonDyer conjecture (one of the Clay Math Institute s million dollar rize roblems). The crucial difficulty amounts to deciding whether or not certain exlicitly given curves have any rational oints on them or not (these are curves that have oints over R and modulo n for all n). The details of using descent to comute E(Q) are beyond the scoe of this book. In several laces below, we will simly assert that E(Q) has a certain structure or is generated by certain elements. In each case, we comuted E(Q) using a comuter imlementation of this method The Torsion Subgrou of E(Q) For any abelian grou G, let G tor be the subgrou of elements of finite order. If E is an ellitic curve over Q, then E(Q) tor is a subgrou of E(Q), which must be finite because of Theorem (see Exercise 6.6).
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