Is n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur


 Dayna Collins
 1 years ago
 Views:
Transcription
1 Is n a Prime Number? Manindra Agrawal IIT Kanpur March 27, 2006, Delft Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 1 / 47
2 Overview 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 2 / 47
3 Outline 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 3 / 47
4 The Problem Given a number n, decide if it is prime. Easy: try dividing by all numbers less than n. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 4 / 47
5 The Problem Given a number n, decide if it is prime. Easy: try dividing by all numbers less than n. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 4 / 47
6 The Problem Given a number n, decide if it is prime efficiently. Not so easy: several nonobvious methods have been found. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 5 / 47
7 The Problem Given a number n, decide if it is prime efficiently. Not so easy: several nonobvious methods have been found. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 5 / 47
8 Efficiently Solving a Problem There should exist an algorithm for solving the problem taking a polynomial in input size number of steps. For our problem, this means an algorithm taking log O(1) n steps. Caveat: An algorithm taking log 12 n steps would be slower than an algorithm taking log log log log n n steps for all practical values of n! Notation: log is logarithm base 2. O (log c n) stands for O(log c n log log O(1) n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 6 / 47
9 Efficiently Solving a Problem There should exist an algorithm for solving the problem taking a polynomial in input size number of steps. For our problem, this means an algorithm taking log O(1) n steps. Caveat: An algorithm taking log 12 n steps would be slower than an algorithm taking log log log log n n steps for all practical values of n! Notation: log is logarithm base 2. O (log c n) stands for O(log c n log log O(1) n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 6 / 47
10 Efficiently Solving a Problem There should exist an algorithm for solving the problem taking a polynomial in input size number of steps. For our problem, this means an algorithm taking log O(1) n steps. Caveat: An algorithm taking log 12 n steps would be slower than an algorithm taking log log log log n n steps for all practical values of n! Notation: log is logarithm base 2. O (log c n) stands for O(log c n log log O(1) n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 6 / 47
11 Efficiently Solving a Problem There should exist an algorithm for solving the problem taking a polynomial in input size number of steps. For our problem, this means an algorithm taking log O(1) n steps. Caveat: An algorithm taking log 12 n steps would be slower than an algorithm taking log log log log n n steps for all practical values of n! Notation: log is logarithm base 2. O (log c n) stands for O(log c n log log O(1) n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 6 / 47
12 Outline 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 7 / 47
13 The Sieve of Eratosthenes Proposed by Eratosthenes (ca. 300 BCE). 1 List all numbers from 2 to n in a sequence. 2 Take the smallest uncrossed number from the sequence and cross out all its multiples. 3 If n is uncrossed when the smallest uncrossed number is greater than n then n is prime otherwise composite. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 8 / 47
14 Time Complexity If n is prime, algorithm crosses out all the first n numbers before giving the answer. So the number of steps needed is Ω( n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 9 / 47
15 Time Complexity If n is prime, algorithm crosses out all the first n numbers before giving the answer. So the number of steps needed is Ω( n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 9 / 47
16 Wilson s Theorem Based on Wilson s theorem (1770). Theorem n is prime iff (n 1)! = 1 (mod n). Computing (n 1)! (mod n) naïvely requires Ω(n) steps. No significantly better method is known! Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 10 / 47
17 Wilson s Theorem Based on Wilson s theorem (1770). Theorem n is prime iff (n 1)! = 1 (mod n). Computing (n 1)! (mod n) naïvely requires Ω(n) steps. No significantly better method is known! Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 10 / 47
18 Outline 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 11 / 47
19 Fundamental Idea Nearly all the efficient algorithms for the problem use the following idea. Identify a finite group G related to number n. Design an efficienty testable property P( ) of the elements G such that P(e) has different values depending on whether n is prime. The element e is either from a small set (in deterministic algorithms) or a random element of G (in randomized algorithms). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 12 / 47
20 Fundamental Idea Nearly all the efficient algorithms for the problem use the following idea. Identify a finite group G related to number n. Design an efficienty testable property P( ) of the elements G such that P(e) has different values depending on whether n is prime. The element e is either from a small set (in deterministic algorithms) or a random element of G (in randomized algorithms). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 12 / 47
21 Fundamental Idea Nearly all the efficient algorithms for the problem use the following idea. Identify a finite group G related to number n. Design an efficienty testable property P( ) of the elements G such that P(e) has different values depending on whether n is prime. The element e is either from a small set (in deterministic algorithms) or a random element of G (in randomized algorithms). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 12 / 47
22 Fundamental Idea Nearly all the efficient algorithms for the problem use the following idea. Identify a finite group G related to number n. Design an efficienty testable property P( ) of the elements G such that P(e) has different values depending on whether n is prime. The element e is either from a small set (in deterministic algorithms) or a random element of G (in randomized algorithms). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 12 / 47
23 Groups and Properties The group G is often: A subgroup of Z n or Z n [ζ] for an extension ring Z n [ζ]. A subgroup of E(Z n ), the set of points on an elliptic curve modulo n. The properties vary, but are from two broad themes. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 13 / 47
24 Groups and Properties The group G is often: A subgroup of Z n or Z n [ζ] for an extension ring Z n [ζ]. A subgroup of E(Z n ), the set of points on an elliptic curve modulo n. The properties vary, but are from two broad themes. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 13 / 47
25 Groups and Properties The group G is often: A subgroup of Z n or Z n [ζ] for an extension ring Z n [ζ]. A subgroup of E(Z n ), the set of points on an elliptic curve modulo n. The properties vary, but are from two broad themes. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 13 / 47
26 Theme I: Factorization of Group Size Compute a complete, or partial, factorization of the size of G assuming that n is prime. Use the knowledge of this factorization to design a suitable property. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 14 / 47
27 Theme I: Factorization of Group Size Compute a complete, or partial, factorization of the size of G assuming that n is prime. Use the knowledge of this factorization to design a suitable property. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 14 / 47
28 Theme II: Fermat s Little Theorem Theorem (Fermat, 1660s) If n is prime then for every e, e n = e (mod n). Group G = Z n and property P is P(e) e n = e in G. This property of Z n is not a sufficient test for primality of n. So try to extend this property to a neccessary and sufficient condition. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 15 / 47
29 Theme II: Fermat s Little Theorem Theorem (Fermat, 1660s) If n is prime then for every e, e n = e (mod n). Group G = Z n and property P is P(e) e n = e in G. This property of Z n is not a sufficient test for primality of n. So try to extend this property to a neccessary and sufficient condition. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 15 / 47
30 Outline 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 16 / 47
31 Lucas Theorem Theorem (E. Lucas, 1891) Let n 1 = t i=1 pd i i where p i s are distinct primes. n is prime iff there is an e Z n such that e n 1 = 1 and gcd(e n 1 p i 1, n) = 1 for every 1 i t. The theorem also holds for a random choice of e. We can choose G = Z n and P to be the property above. The test will be efficient only for numbers n such that n 1 is smooth. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 17 / 47
32 Lucas Theorem Theorem (E. Lucas, 1891) Let n 1 = t i=1 pd i i where p i s are distinct primes. n is prime iff there is an e Z n such that e n 1 = 1 and gcd(e n 1 p i 1, n) = 1 for every 1 i t. The theorem also holds for a random choice of e. We can choose G = Z n and P to be the property above. The test will be efficient only for numbers n such that n 1 is smooth. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 17 / 47
33 Lucas Theorem Theorem (E. Lucas, 1891) Let n 1 = t i=1 pd i i where p i s are distinct primes. n is prime iff there is an e Z n such that e n 1 = 1 and gcd(e n 1 p i 1, n) = 1 for every 1 i t. The theorem also holds for a random choice of e. We can choose G = Z n and P to be the property above. The test will be efficient only for numbers n such that n 1 is smooth. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 17 / 47
34 LucasLehmer Test G is a subgroup of Z n [ 3] containing elements of order n + 1. The property P is: P(e) e n+1 2 = 1 in Z n [ 3]. Works only for special Mersenne primes of the form n = 2 p 1, p prime. For such n s, n + 1 = 2 p. The property needs to be tested only for e = Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 18 / 47
35 LucasLehmer Test G is a subgroup of Z n [ 3] containing elements of order n + 1. The property P is: P(e) e n+1 2 = 1 in Z n [ 3]. Works only for special Mersenne primes of the form n = 2 p 1, p prime. For such n s, n + 1 = 2 p. The property needs to be tested only for e = Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 18 / 47
36 LucasLehmer Test G is a subgroup of Z n [ 3] containing elements of order n + 1. The property P is: P(e) e n+1 2 = 1 in Z n [ 3]. Works only for special Mersenne primes of the form n = 2 p 1, p prime. For such n s, n + 1 = 2 p. The property needs to be tested only for e = Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 18 / 47
37 Time Complexity Raising to n+1 2 th power requires O(log n) multiplication operations in Z n. Overall time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 19 / 47
38 Time Complexity Raising to n+1 2 th power requires O(log n) multiplication operations in Z n. Overall time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 19 / 47
39 PocklingtonLehmer Test Theorem (Pocklington, 1914) If there exists a e such that e n 1 = 1 (mod n) and gcd(e n 1 p j 1, n) = 1 for distinct primes p 1, p 2,..., p t dividing n 1 then every prime factor of n has the form k t j=1 p j + 1. Similar to Lucas s theorem. Let G = Z n and property P precisely as in the theorem. The property is tested for a random e. For the test to work, we need t j=1 n. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 20 / 47
40 PocklingtonLehmer Test Theorem (Pocklington, 1914) If there exists a e such that e n 1 = 1 (mod n) and gcd(e n 1 p j 1, n) = 1 for distinct primes p 1, p 2,..., p t dividing n 1 then every prime factor of n has the form k t j=1 p j + 1. Similar to Lucas s theorem. Let G = Z n and property P precisely as in the theorem. The property is tested for a random e. For the test to work, we need t j=1 n. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 20 / 47
41 PocklingtonLehmer Test Theorem (Pocklington, 1914) If there exists a e such that e n 1 = 1 (mod n) and gcd(e n 1 p j 1, n) = 1 for distinct primes p 1, p 2,..., p t dividing n 1 then every prime factor of n has the form k t j=1 p j + 1. Similar to Lucas s theorem. Let G = Z n and property P precisely as in the theorem. The property is tested for a random e. For the test to work, we need t j=1 n. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 20 / 47
42 Time Complexity Depends on the difficulty of finding prime factorizations of n 1 whose product is at least n. Other operations can be carried out efficiently. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 21 / 47
43 Time Complexity Depends on the difficulty of finding prime factorizations of n 1 whose product is at least n. Other operations can be carried out efficiently. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 21 / 47
44 Elliptic Curves Based Tests Elliptic curves give rise to groups of different sizes associated with the given number. With good probability, some of these groups have sizes that can be easily factored. This motivated primality testing based on elliptic curves. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 22 / 47
45 Elliptic Curves Based Tests Elliptic curves give rise to groups of different sizes associated with the given number. With good probability, some of these groups have sizes that can be easily factored. This motivated primality testing based on elliptic curves. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 22 / 47
46 Elliptic Curves Based Tests Elliptic curves give rise to groups of different sizes associated with the given number. With good probability, some of these groups have sizes that can be easily factored. This motivated primality testing based on elliptic curves. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 22 / 47
47 GoldwasserKilian Test This is a randomized primality proving algorithm. Under a reasonable hypothesis, it is polynomial time on all inputs. Unconditionally, it is polynomial time on all but negligible fraction of numbers. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 23 / 47
48 GoldwasserKilian Test This is a randomized primality proving algorithm. Under a reasonable hypothesis, it is polynomial time on all inputs. Unconditionally, it is polynomial time on all but negligible fraction of numbers. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 23 / 47
49 GoldwasserKilian Test This is a randomized primality proving algorithm. Under a reasonable hypothesis, it is polynomial time on all inputs. Unconditionally, it is polynomial time on all but negligible fraction of numbers. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 23 / 47
50 GoldwasserKilian Test Consider a random elliptic curve over Z n. By a theorem of Lenstra (1987), the number of points of the curve is nearly uniformly distributed in the interval [n n, n n] for prime n. Assuming a conjecture about the density of primes in small intervals, it follows that there are curves with 2q points, for q prime, with reasonable probability. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 24 / 47
51 GoldwasserKilian Test Consider a random elliptic curve over Z n. By a theorem of Lenstra (1987), the number of points of the curve is nearly uniformly distributed in the interval [n n, n n] for prime n. Assuming a conjecture about the density of primes in small intervals, it follows that there are curves with 2q points, for q prime, with reasonable probability. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 24 / 47
52 GoldwasserKilian Test Consider a random elliptic curve over Z n. By a theorem of Lenstra (1987), the number of points of the curve is nearly uniformly distributed in the interval [n n, n n] for prime n. Assuming a conjecture about the density of primes in small intervals, it follows that there are curves with 2q points, for q prime, with reasonable probability. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 24 / 47
53 GoldwasserKilian Test Theorem (GoldwasserKilian) Suppose E(Z n ) is an elliptic curve with 2q points. If q is prime and there exists A E(Z n ) O such that q A = O then either n is provably prime or provably composite. Proof. Let p be a prime factor of n with p n. We have q A = O in E(Z p ) as well. If A = O in E(Z p ) then n can be factored. Otherwise, since q is prime, E(Z p ) q. If 2q < n n then n must be composite. Otherwise, p p > n 2 n which is not possible. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 25 / 47
54 GoldwasserKilian Test Theorem (GoldwasserKilian) Suppose E(Z n ) is an elliptic curve with 2q points. If q is prime and there exists A E(Z n ) O such that q A = O then either n is provably prime or provably composite. Proof. Let p be a prime factor of n with p n. We have q A = O in E(Z p ) as well. If A = O in E(Z p ) then n can be factored. Otherwise, since q is prime, E(Z p ) q. If 2q < n n then n must be composite. Otherwise, p p > n 2 n which is not possible. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 25 / 47
55 GoldwasserKilian Test Theorem (GoldwasserKilian) Suppose E(Z n ) is an elliptic curve with 2q points. If q is prime and there exists A E(Z n ) O such that q A = O then either n is provably prime or provably composite. Proof. Let p be a prime factor of n with p n. We have q A = O in E(Z p ) as well. If A = O in E(Z p ) then n can be factored. Otherwise, since q is prime, E(Z p ) q. If 2q < n n then n must be composite. Otherwise, p p > n 2 n which is not possible. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 25 / 47
56 GoldwasserKilian Test Theorem (GoldwasserKilian) Suppose E(Z n ) is an elliptic curve with 2q points. If q is prime and there exists A E(Z n ) O such that q A = O then either n is provably prime or provably composite. Proof. Let p be a prime factor of n with p n. We have q A = O in E(Z p ) as well. If A = O in E(Z p ) then n can be factored. Otherwise, since q is prime, E(Z p ) q. If 2q < n n then n must be composite. Otherwise, p p > n 2 n which is not possible. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 25 / 47
57 GoldwasserKilian Test 1 Find a random elliptic curve over Z n with 2q points. 2 Prove primality of q recursively. 3 Randomly select an A such that q A = O. 4 Infer n to be prime or composite. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 26 / 47
58 GoldwasserKilian Test 1 Find a random elliptic curve over Z n with 2q points. 2 Prove primality of q recursively. 3 Randomly select an A such that q A = O. 4 Infer n to be prime or composite. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 26 / 47
59 GoldwasserKilian Test 1 Find a random elliptic curve over Z n with 2q points. 2 Prove primality of q recursively. 3 Randomly select an A such that q A = O. 4 Infer n to be prime or composite. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 26 / 47
60 GoldwasserKilian Test 1 Find a random elliptic curve over Z n with 2q points. 2 Prove primality of q recursively. 3 Randomly select an A such that q A = O. 4 Infer n to be prime or composite. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 26 / 47
61 Analysis The algorithm never incorrectly classifies a composite number. With high probability it correctly classifies prime numbers. The running time is O(log 11 n). Improvements by Atkin and others result in a conjectured running time of O (log 4 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 27 / 47
62 Analysis The algorithm never incorrectly classifies a composite number. With high probability it correctly classifies prime numbers. The running time is O(log 11 n). Improvements by Atkin and others result in a conjectured running time of O (log 4 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 27 / 47
63 Analysis The algorithm never incorrectly classifies a composite number. With high probability it correctly classifies prime numbers. The running time is O(log 11 n). Improvements by Atkin and others result in a conjectured running time of O (log 4 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 27 / 47
64 Analysis The algorithm never incorrectly classifies a composite number. With high probability it correctly classifies prime numbers. The running time is O(log 11 n). Improvements by Atkin and others result in a conjectured running time of O (log 4 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 27 / 47
65 AdlemanHuang Test The previous test is not unconditionally polynomial time on a small fraction of numbers. AdlemanHuang (1992) removed this drawback. They first used hyperelliptic curves to reduce the problem of testing for n to that of a nearly random integer of similar size. Then the previous test works with high probability. The time complexity becomes O(log c n) for c > 30! Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 28 / 47
66 AdlemanHuang Test The previous test is not unconditionally polynomial time on a small fraction of numbers. AdlemanHuang (1992) removed this drawback. They first used hyperelliptic curves to reduce the problem of testing for n to that of a nearly random integer of similar size. Then the previous test works with high probability. The time complexity becomes O(log c n) for c > 30! Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 28 / 47
67 AdlemanHuang Test The previous test is not unconditionally polynomial time on a small fraction of numbers. AdlemanHuang (1992) removed this drawback. They first used hyperelliptic curves to reduce the problem of testing for n to that of a nearly random integer of similar size. Then the previous test works with high probability. The time complexity becomes O(log c n) for c > 30! Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 28 / 47
68 Outline 1 The Problem 2 Two Simple, and Slow, Methods 3 Modern Methods 4 Algorithms Based on Factorization of Group Size 5 Algorithms Based on Fermat s Little Theorem 6 An Algorithm Outside the Two Themes Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 29 / 47
69 SolovayStrassen Test A Restatement of FLT If n is odd prime then for every e, 1 e < n, e n 1 2 = ±1 (mod n). When n is prime, e is a quadratic residue in Z n iff e n 1 2 = 1 (mod n). Therefore, if n is prime then ( e n ) = e n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 30 / 47
70 SolovayStrassen Test A Restatement of FLT If n is odd prime then for every e, 1 e < n, e n 1 2 = ±1 (mod n). When n is prime, e is a quadratic residue in Z n iff e n 1 2 = 1 (mod n). Therefore, if n is prime then ( e n ) = e n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 30 / 47
71 SolovayStrassen Test A Restatement of FLT If n is odd prime then for every e, 1 e < n, e n 1 2 = ±1 (mod n). When n is prime, e is a quadratic residue in Z n iff e n 1 2 = 1 (mod n). Therefore, if n is prime then ( e n ) = e n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 30 / 47
72 SolovayStrassen Test Proposed by Solovay and Strassen (1973). A randomized algorithm based on above property. Never incorrectly classifies primes and correctly classifies composites with probability at least 1 2. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 31 / 47
73 SolovayStrassen Test Proposed by Solovay and Strassen (1973). A randomized algorithm based on above property. Never incorrectly classifies primes and correctly classifies composites with probability at least 1 2. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 31 / 47
74 SolovayStrassen Test Proposed by Solovay and Strassen (1973). A randomized algorithm based on above property. Never incorrectly classifies primes and correctly classifies composites with probability at least 1 2. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 31 / 47
75 SolovayStrassen Test 1 If n is an exact power, it is composite. 2 For a random e in Z n, test if ( e n ) = e n 1 2 (mod n). 3 If yes, classify n as prime otherwise it is proven composite. The time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 32 / 47
76 SolovayStrassen Test 1 If n is an exact power, it is composite. 2 For a random e in Z n, test if ( e n ) = e n 1 2 (mod n). 3 If yes, classify n as prime otherwise it is proven composite. The time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 32 / 47
77 SolovayStrassen Test 1 If n is an exact power, it is composite. 2 For a random e in Z n, test if ( e n ) = e n 1 2 (mod n). 3 If yes, classify n as prime otherwise it is proven composite. The time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 32 / 47
78 SolovayStrassen Test 1 If n is an exact power, it is composite. 2 For a random e in Z n, test if ( e n ) = e n 1 2 (mod n). 3 If yes, classify n as prime otherwise it is proven composite. The time complexity is O (log 2 n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 32 / 47
79 Analysis Consider the case when n is a product of two primes p and q. Let a, b Z p, c Z q with a residue and b nonresidue in Z p. Clearly, < a, c > n 1 2 =< b, c > n 1 2 (mod q). If < a, c > n 1 2 < b, c > n 1 2 (mod n) then one of them is not in {1, 1} and so compositeness of n is proven. Otherwise, either ( < a, c > ) < a, c > n 1 2 (mod n), n or ( ) < b, c > n < b, c > n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 33 / 47
80 Analysis Consider the case when n is a product of two primes p and q. Let a, b Z p, c Z q with a residue and b nonresidue in Z p. Clearly, < a, c > n 1 2 =< b, c > n 1 2 (mod q). If < a, c > n 1 2 < b, c > n 1 2 (mod n) then one of them is not in {1, 1} and so compositeness of n is proven. Otherwise, either ( < a, c > ) < a, c > n 1 2 (mod n), n or ( ) < b, c > n < b, c > n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 33 / 47
81 Analysis Consider the case when n is a product of two primes p and q. Let a, b Z p, c Z q with a residue and b nonresidue in Z p. Clearly, < a, c > n 1 2 =< b, c > n 1 2 (mod q). If < a, c > n 1 2 < b, c > n 1 2 (mod n) then one of them is not in {1, 1} and so compositeness of n is proven. Otherwise, either ( < a, c > ) < a, c > n 1 2 (mod n), n or ( ) < b, c > n < b, c > n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 33 / 47
82 Analysis Consider the case when n is a product of two primes p and q. Let a, b Z p, c Z q with a residue and b nonresidue in Z p. Clearly, < a, c > n 1 2 =< b, c > n 1 2 (mod q). If < a, c > n 1 2 < b, c > n 1 2 (mod n) then one of them is not in {1, 1} and so compositeness of n is proven. Otherwise, either ( < a, c > ) < a, c > n 1 2 (mod n), n or ( ) < b, c > n < b, c > n 1 2 (mod n). Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 33 / 47
83 Miller s Test Theorem (Another Restatement of FLT) If n is odd prime and n = s t, t odd, then for every e, 1 e < n, the sequence e 2s 1 t (mod n), e 2s 2 t (mod n),..., e t (mod n) has either all 1 s or a 1 somewhere. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 34 / 47
84 Miller s Test This theorem is the basis for Miller s test (1973). It is a deterministic polynomial time test. It is correct under Extended Riemann Hypothesis. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 35 / 47
85 Miller s Test This theorem is the basis for Miller s test (1973). It is a deterministic polynomial time test. It is correct under Extended Riemann Hypothesis. Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 35 / 47
Primality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationPRIMES is in P. Manindra Agrawal Neeraj Kayal Nitin Saxena
PRIMES is in P Manindra Agrawal Neeraj Kayal Nitin Saxena Department of Computer Science & Engineering Indian Institute of Technology Kanpur Kanpur208016, INDIA Email: {manindra,kayaln,nitinsa}@iitk.ac.in
More informationFaster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers
More informationFactoring integers, Producing primes and the RSA cryptosystem HarishChandra Research Institute
RSA cryptosystem HRI, Allahabad, February, 2005 0 Factoring integers, Producing primes and the RSA cryptosystem HarishChandra Research Institute Allahabad (UP), INDIA February, 2005 RSA cryptosystem HRI,
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationAn Overview of Integer Factoring Algorithms. The Problem
An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm
More informationFactoring integers and Producing primes
Factoring integers,..., RSA Erbil, Kurdistan 0 Lecture in Number Theory College of Sciences Department of Mathematics University of Salahaddin Debember 4, 2014 Factoring integers and Producing primes Francesco
More informationFactoring integers, Producing primes and the RSA cryptosystem
Factoring integers,..., RSA Erbil, Kurdistan 0 Lecture in Number Theory College of Sciences Department of Mathematics University of Salahaddin Debember 1, 2014 Factoring integers, Producing primes and
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationRecent Breakthrough in Primality Testing
Nonlinear Analysis: Modelling and Control, 2004, Vol. 9, No. 2, 171 184 Recent Breakthrough in Primality Testing R. Šleževičienė, J. Steuding, S. Turskienė Department of Computer Science, Faculty of Physics
More informationCryptography and Network Security Number Theory
Cryptography and Network Security Number Theory XiangYang Li Introduction to Number Theory Divisors b a if a=mb for an integer m b a and c b then c a b g and b h then b (mg+nh) for any int. m,n Prime
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More information3. Computational Complexity.
3. Computational Complexity. (A) Introduction. As we will see, most cryptographic systems derive their supposed security from the presumed inability of any adversary to crack certain (number theoretic)
More informationOn Generalized Fermat Numbers 3 2n +1
Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationStudy of algorithms for factoring integers and computing discrete logarithms
Study of algorithms for factoring integers and computing discrete logarithms First IndoFrench Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department
More informationCryptography and Network Security Chapter 8
Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:
More information2 Primality and Compositeness Tests
Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 16351642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationTHE MATHEMATICS OF PUBLIC KEY CRYPTOGRAPHY.
THE MATHEMATICS OF PUBLIC KEY CRYPTOGRAPHY. IAN KIMING 1. Forbemærkning. Det kan forekomme idiotisk, at jeg som dansktalende og skrivende i et danskbaseret tidsskrift med en (formentlig) primært dansktalende
More informationArithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJPRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJPRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More informationDiscrete Mathematics, Chapter 4: Number Theory and Cryptography
Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility
More informationSmooth numbers and the quadratic sieve
Algorithmic Number Theory MSRI Publications Volume 44, 2008 Smooth numbers and the quadratic sieve CARL POMERANCE ABSTRACT. This article gives a gentle introduction to factoring large integers via the
More informationRSA and Primality Testing
and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2
More informationNumber Theory Learning Module 2 Prime Numbers and Integer Factorization in Sage 1
Learning Module 2 Prime Numbers and Integer Factorization in Sage 1 1 Objectives. Learn how to test for primality and generate prime numbers and in Sage. Learn how to work with factorizations in Sage.
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationComputational Number Theory
Computational Number Theory C. Pomerance 1 Introduction Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented
More informationCIS 5371 Cryptography. 8. Encryption 
CIS 5371 Cryptography p y 8. Encryption  Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: Allornothing secrecy.
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationPrimality Testing and Factorization Methods
Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,
More informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationElements of Applied Cryptography Public key encryption
Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationLecture 13: Factoring Integers
CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method
More informationHYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS. 1. Thoery and Algorithm
HYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS WENHAN WANG 1. Thoery and Algorithm The idea of the method using hyperelliptic curves to factor integers is similar to the elliptic curve factoring method.
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationComputer and Network Security
MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest
More information2.1 Complexity Classes
15859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look
More informationFACTORING. n = 2 25 + 1. fall in the arithmetic sequence
FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard,
More informationPRIME PYTHAGOREAN TRIANGLES
PRIME PYTHAGOREAN TRIANGLES HARVEY DUBNER AND TONY FORBES Abstract. A prime Pythagorean triangle has three integer sides of which the hypotenuse and one leg are primes. In this article we investigate their
More informationFactorization Methods: Very Quick Overview
Factorization Methods: Very Quick Overview Yuval Filmus October 17, 2012 1 Introduction In this lecture we introduce modern factorization methods. We will assume several facts from analytic number theory.
More informationRecent progress in additive prime number theory
Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns in
More informationThe Quadratic Sieve Factoring Algorithm
The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 Introduction Mathematicians have been attempting to find better and faster ways to factor composite
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationDetermining the Optimal Combination of Trial Division and Fermat s Factorization Method
Determining the Optimal Combination of Trial Division and Fermat s Factorization Method Joseph C. Woodson Home School P. O. Box 55005 Tulsa, OK 74155 Abstract The process of finding the prime factorization
More informationI. Introduction. MPRI Cours 2122. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.
F. Morain École polytechnique MPRI cours 2122 20132014 3/22 F. Morain École polytechnique MPRI cours 2122 20132014 4/22 MPRI Cours 2122 I. Introduction Input: an integer N; logox F. Morain logocnrs
More informationInteger factoring and modular square roots
Integer factoring and modular square roots Emil Jeřábek Institute of Mathematics of the Academy of Sciences Žitná 25, 115 67 Praha 1, Czech Republic, email: jerabek@math.cas.cz July 28, 2015 Abstract BureshOppenheim
More informationRuntime and Implementation of Factoring Algorithms: A Comparison
Runtime and Implementation of Factoring Algorithms: A Comparison Justin Moore CSC290 Cryptology December 20, 2003 Abstract Factoring composite numbers is not an easy task. It is classified as a hard algorithm,
More information15 Prime and Composite Numbers
15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationEMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS. Carl Pomerance
ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS Carl Pomerance Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of computing an integer l with g l
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More informationOn the numbertheoretic functions ν(n) and Ω(n)
ACTA ARITHMETICA LXXVIII.1 (1996) On the numbertheoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationA new probabilistic public key algorithm based on elliptic logarithms
A new probabilistic public key algorithm based on elliptic logarithms Afonso Comba de Araujo Neto, Raul Fernando Weber 1 Instituto de Informática Universidade Federal do Rio Grande do Sul (UFRGS) Caixa
More informationCryptography: RSA and the discrete logarithm problem
Cryptography: and the discrete logarithm problem R. Hayden Advanced Maths Lectures Department of Computing Imperial College London February 2010 Public key cryptography Assymmetric cryptography two keys:
More informationOn primeorder elliptic curves with embedding degrees k = 3, 4 and 6
On primeorder elliptic curves with embedding degrees k = 3, 4 and 6 Koray Karabina and Edlyn Teske University of Waterloo ANTS VIII, Banff, May 20, 2008 K. Karabina and E. Teske (UW) Primeorder elliptic
More informationThe Prime Facts: From Euclid to AKS
The Prime Facts: From Euclid to AKS c 2003 Scott Aaronson 1 Introduction My idea for this talk was to tell you only the simplest, most basic things about prime numbers the things you need to know to call
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationRSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p1)(q1) = φ(n). Is this true?
RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p1)(q1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e 1
More informationA number field is a field of finite degree over Q. By the Primitive Element Theorem, any number
Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and
More informationLecture 10: Distinct Degree Factoring
CS681 Computational Number Theory Lecture 10: Distinct Degree Factoring Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview Last class we left of with a glimpse into distant degree factorization.
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More information54 Prime and Composite Numbers
54 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee
More informationConsequently, for the remainder of this discussion we will assume that a is a quadratic residue mod p.
Computing square roots mod p We now have very effective ways to determine whether the quadratic congruence x a (mod p), p an odd prime, is solvable. What we need to complete this discussion is an effective
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More informationNumber Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures
Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationIntegers and division
CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.
More informationOn Factoring Integers and Evaluating Discrete Logarithms
On Factoring Integers and Evaluating Discrete Logarithms A thesis presented by JOHN AARON GREGG to the departments of Mathematics and Computer Science in partial fulfillment of the honors requirements
More informationMATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction
MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationMath 453: Elementary Number Theory Definitions and Theorems
Math 453: Elementary Number Theory Definitions and Theorems (Class Notes, Spring 2011 A.J. Hildebrand) Version 542011 Contents About these notes 3 1 Divisibility and Factorization 4 1.1 Divisibility.......................................
More informationA Comparison Of Integer Factoring Algorithms. Keyur Anilkumar Kanabar
A Comparison Of Integer Factoring Algorithms Keyur Anilkumar Kanabar Batchelor of Science in Computer Science with Honours The University of Bath May 2007 This dissertation may be made available for consultation
More informationSTUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION. Takayuki Yato. A Senior Thesis. Submitted to
STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION by Takayuki Yato A Senior Thesis Submitted to Department of Information Science Faculty of Science The University of Tokyo on
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationUSING LUCAS SEQUENCES TO FACTOR LARGE INTEGERS NEAR GROUP ORDERS
USING LUCAS SEQUENCES TO FACTOR LARGE INTEGERS NEAR GROUP ORDERS Zhenxiang Zhang* Dept. of Math., Anhui Normal University, 241000 Wuhu, Anhui, P.R. China email: zhangzhx@mail.ahwhptt.net.cn (Submitted
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationCRC Press has granted the following specific permissions for the electronic version of this book:
This is a Chapter from the Handbook of Applied Cryptography, by A. Menezes, P. van Oorschot, and S. Vanstone, CRC Press, 1996. For further information, see www.cacr.math.uwaterloo.ca/hac CRC Press has
More informationElementary Number Theory
Elementary Number Theory A revision by Jim Hefferon, St Michael s College, 2003Dec of notes by W. Edwin Clark, University of South Florida, 2002Dec L A TEX source compiled on January 5, 2004 by Jim Hefferon,
More informationOn primeorder elliptic curves with embedding degrees k = 3, 4 and 6
On primeorder elliptic curves with embedding degrees k = 3, 4 and 6 Koray Karabina and Edlyn Teske Dept. of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada NL 3G1, kkarabina@uwaterloo.ca,
More informationc 2007 Society for Industrial and Applied Mathematics ÉRIC SCHOST
SIAM J. COMPUT. Vol. 36, No. 6, pp. 1777 1806 c 2007 Society for Industrial and Applied Mathematics LINEAR RECURRENCES WITH POLYNOMIAL COEFFICIENTS AND APPLICATION TO INTEGER FACTORIZATION AND CARTIER
More informationGeneric attacks and index calculus. D. J. Bernstein University of Illinois at Chicago
Generic attacks and index calculus D. J. Bernstein University of Illinois at Chicago The discretelogarithm problem Define Ô = 1000003. Easy to prove: Ô is prime. Can we find an integer Ò ¾ 1 2 3 Ô 1 such
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationFactoring Algorithms
Institutionen för Informationsteknologi Lunds Tekniska Högskola Department of Information Technology Lund University Cryptology  Project 1 Factoring Algorithms The purpose of this project is to understand
More informationA New Generic Digital Signature Algorithm
Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study
More informationA 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction StarkHeegner Theorem. Let d > 0 be a squarefree integer then Q( d) has
A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency
More informationDIVISORS IN A DEDEKIND DOMAIN. Javier Cilleruelo and Jorge JiménezUrroz. 1 Introduction
DIVISORS IN A DEDEKIND DOMAIN Javier Cilleruelo and Jorge JiménezUrroz 1 Introduction Let A be a Dedekind domain in which we can define a notion of distance. We are interested in the number of divisors
More informationThe application of prime numbers to RSA encryption
The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered
More informationFactorizations of a n ± 1, 13 a < 100
Factorizations of a n ± 1, 13 a < 100 Richard P. Brent Computer Sciences Laboratory, Australian National University GPO Box 4, Canberra, ACT 2601, Australia email: rpb@cslab.anu.edu.au and Herman J. J.
More information