# Public Key Cryptography: RSA and Lots of Number Theory

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1 Public Key Cryptography: RSA and Lots of Number Theory

2 Public vs. Private-Key Cryptography We have just discussed traditional symmetric cryptography: Uses a single key shared between sender and receiver Asymmetric (public key) cryptography was introduced by Diffie & Hellman and is a dramatically different approach to cryptography: Two keys are used: a public and a private key Alice generates a public and a private key The Public Key is given to anyone who would like to securely communicate with Alice Alice keeps the Private key private and she may decrypt messages encrypted with the public key Asymmetric cryptography is NOT a replacement of symmetric cryptography There are many scenarios where symmetric is much better than asymmetric (such as bulk data encryption)

3 Public-Key Cryptography

4 Pros and Cons of Public Key Cryptography There are several advantages to public key cryptography: Asymmetry allows for easier key distribution: We do not need a trusted third party (KDC) to distribute keys Asymmetry provides proof of origin: The secret key is something that only one entity knows. Can be extended to form chains of trust: Public Key Certificate frameworks provide a natural way to model trust. There are several disadvantages to public key cryptography: Computation Burden: By its very nature, public key cryptography is not as fast or computationally efficient as symmetric cryptography. Communication Overhead: PKIs typically require significant communication overhead (frequent and large messages).

5 Public-Key Characteristics Public-Key algorithms rely on two keys with the characteristics that it is: computationally infeasible to find decryption key knowing only algorithm & encryption key computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)

6 Public-Key Cryptosystems From Stalling s Book

7 RSA The RSA algorithm is the most popular public key scheme and was invented by Rivest, Shamir & Adleman of MIT in 1977 Based on exponentiation in a finite field over integers modulo a prime Exponentiation takes O((log n) 3 ) operations (easy) Exponentiation is accomplished through repeated squaring Uses large integer operations Requires finding large primes The security of RSA is based on the (believed) intractability of the factoring of the product of two large primes Difficulty of factoring is based upon the size of the factors Factorization of RSA composite takes O(e log n log log n ) operations (hard)

8 Setup of RSA Alice wishes to generate a public key and a private key 1. She first generates two large random primes p and q 2. She computes the composite n=pq, and the Euler Phi function ϕ( n) = ( p 1)( q 1) 3. She chooses a random encryption exponent e such that gcd( e, ϕ( n)) = 1 4. She finds the decryption exponent by: ed = 1 modϕ( n) Public key is {e,n} Private key is {d,p,q}

9 Encryption with RSA Suppose Bob wishes to encrypt a message M and send it to Alice. He acquires her public key {e,n} He computes the ciphertext: c = m e mod Alice can decrypt by using her private key {d,p,q} via m Requirement: We are doing operations modulo n, so message must be smaller than m n = c d mod n

10 RSA Example 1. Select primes: p= & q= Compute n = pq = Compute phi(n)=(p 1)(q-1) 4. Select e : gcd(e,phi(n))=1; choose e= Determine d: de=1 mod phi(n) and d < phi(n). Value is d= Example encryption 1. Suppose m= Ciphertext c=m e mod n = (30120) 9007 = Reconstruct plaintext: m=c d mod n = = 30120

11 Why RSA Works OK so we ve covered what RSA is now lets look at why it works, and how to make it work in practice. There are several key observations, most built from number theory, that we will need. We will cover the following: Euler s Theorem How to Calculate Inverses Modular Exponentiation Finding Primes We will briefly discuss the security of RSA today. More on the security of RSA will come in a lecture on attacks and cryptanalysis.

12 Euler s Theorem Fermat s Little Theorem: If p is a prime and p does not divide a, then a p 1 1 (mod p). Euler s Theorem is a generalization of Fermat s Little Theorem Euler s Theorem: If gcd(a, n) = 1, then a φ(n) 1 (mod n). What is φ(n)? It s the amount of numbers between 1 and n that relatively prime to n. Example: φ(pq) = (p-1)(q-1) Euler/Fermat Example: Compute (mod 101). Solution: From Fermat s theorem, we know that (mod 101). Therefore, (2 100 ) (mod 101).

13 Why Does Euler Make RSA Work? Basic Principle: Let a, n, x, y be integers with n 1 and gcd(a, n) = 1. If x y (mod φ(n)), then a x a y (mod n). In other words, if you want to work mod n, you should work mod φ(n) in the exponent. In RSA, we choose ed 1 (mod φ(n)), so m ed m 1 = m (mod n). Or, more explicitly ed 1 (mod φ(n)) means So ed = 1 + k φ(n) m ed m 1+kφ(n) = m (mod n)

14 How to Calculate Inverses for RSA? We need to calculate e and d such that ed 1 (mod φ(n)). How do we do this? Step 1: Choose a random e such that gcd(e, φ(n)) =1. Why? How? Step 2: Now, find d. The wrong way: e 1 e 2 e 3 3e and e so mod ϕ 2e mod mod on... ( n) ϕ( n) ϕ( n) The Correct Way: Use Extended Euclidean Algorithm!

15 The Plain Euclidean Algorithm The (plain) Euclidean Algorithm finds the gcd(a,b): Example: gcd(1180, 482) 1180 = = = = = Last non-zero remainder is the gcd.

16 Plain Euclidean Algorithm, pg 2. Formally, the Euclidean algorithm for calculating gcd(a,b): Suppose that a>b Divide b into a: a = q 1 b + r 1 If r 1 =0 then b a and gcd(a,b)=b else represent b by b=q 2 r 1 +r 2 3. Continue in this way until remainder is zero. The gcd is last non-zero remainder. a = q 1 b + r 1 b = q 2 r 1 + r 2 r 1 = q 3 r 2 + r 3... r k-2 = q k r k-1 + r k r k-1 = q k+1 r k +0 gcd(a,b) = r k

17 Getting closer to inverses We can prove the following result using the Euclidean Algorithm: Theorem: Let a and b be two integers, with at least one of a, b nonzero, and let d = gcd(a, b). Then there exist integers x, y such that ax + by = d. In particular, if a and b are relatively prime, then there exist integers x, y with ax + by = 1. How do we use this? Suppose we know a, b, x, y as above. Then the inverse of a (mod b) is x. Why? So, we need to find these x and y! Euclidean algorithm will give us x and y, if we do bookkeeping!

18 Showing ax+by=gcd(a,b gcd(a,b) The proof of the previous theorem just involves substitution. r 1 =a-q 1 b let x 1 =1, y 1 = -q 1, so r 1 = x 1 a + y 1 b Next step r 2 = b q 2 r 1 plug in earlier result r 2 = b q 2 (x 1 a + y 1 b) = -x 1 q 2 a + (b) (1-y 1 q 2 ) let x 2 = - x 1 q 2, y 2 = (1-y 1 q 2 ) So r 2 = x 2 a + y 2 b Follow this process repeatedly: If r i = x i a + y i b Then r i+1 = r i-1 q i+1 r i = x i-1 a + y i-1 b - q i+1 (x i a + y i b) = a (x i-1 -q i+1 x i ) + b (y i-1 q i+1 y i ) = a x i+1 + b y i+1 Since this holds for any r i, it holds for the last one r k =gcd(a,b).

19 Extended Euclidean Algorithm The idea in the proof leads to the Extended Euclidean Algorithm Input: a, b non-negative with a> b Output: d=gcd(a,b) and x and y such that ax+by=d If b=0 { d=a; x=1; y=0; return(d,x,y); } x2=1; x1=0; y2=0; y1=1; While b>0 { q=floor(a/b); r=a-q*b; x=x2-q*x1; y=y2-q*y1; a=b; b=r; x2=x1; x1=x; y2=y1; y1=y; } d=a; x=x2; y=y2; Return(d,x,y)

20 Implementation Detail: How to multiply fast! RSA needs calculations like m e mod n. How do can we do this quickly? If we just do sequential multiplication, it will take forever! (Remember, n is on the order of 1000 bits!!! And so is e!!!) To do it effectively, we use Repeated Squaring: Example: Let s do (mod 789) Since 1234 = (1234 = in binary), thus (mod 789).

21 Making Primes, Principles pg. 1 Basic Principle: Let n be an integer and suppose there exist integers x and y with x 2 y 2 (mod n), but x! ±y (mod n). Then n is composite. Moreover, gcd(x y, n) gives a nontrivial factor of n. Proof. Let d = gcd(x y, n). If d = n then x y (mod n), which is assumed not to happen. Suppose d = 1. We know that if a bc and gcd(a, b) = 1, then a c. In our case, since n divides x 2 y 2 = (x y)(x+y) and d = 1, we must have that n divides x + y, which contradicts the assumption that x! y (mod n). Therefore, d is not = 1, n, so d is a nontrivial factor of n. Example: Since (mod 35), but 12! ±2 (mod 35), we know that 35 is composite. Moreover, gcd(12 2, 35) = 5 is a nontrivial factor of 35.

22 Making Primes, Principles pg. 2 We may use Fermat s Little Theorem to prove numbers are not prime. Here s the way: Suppose you have a number n and want to show it is not prime. Choose a number a, and calculate a n-1 (mod n) If this does not equal 1, then n cannot be prime. Why? Example: Show 35 is not prime. Hence 35 is not prime = = 11 * 4 = 9!= 1 (mod 35) But, what if a n-1 (mod n) = 1? This does not mean n is prime. Numbers n such that a n-1 (mod n)=1 for a particular a are said to be pseudoprimes base a. a is said to be a liar for n.

23 Making Primes, Miller-Rabin pg. 1 s Fact: Let n be an odd prime and let n 1 = 2 r, where r is odd. Let a be any integer such that gcd(a,n)=1. Then either a r 1( mod n) or a 2jr 1( mod n) for some 0 j s 1. s Definition: Let n be an odd composite with n 1 = 2 r. Let a [ 1,n 1]. If either a r 1( mod n) or a 2jr 1( mod n), for some 0 j s 1 then n is a strong pseudoprime base a, and a is a strong liar for n. Fact: If n is an odd composite integer, then at most 1/4 of the numbers a are strong liars for n. We can use this in a Monte-Carlo algorithm to produce primes : Test t different a s. Probability of falsely identifying a prime is ( 1) t 4

24 Miller-Rabin Primality Test, pg. 2 Generate a random (odd) integer n such that n-1 = 2 s r For k=1 to t do Choose a random integer 2 a n 2 Calculate y=a r (mod n) If ( (y!= 1) & (y!= n-1) ) then j=1; While ( ( j <= s-1) & (y!= n-1) ) do y=y 2 (mod n) If y=1 then Return( Composite ); j++; enddo If (y!= n-1) then Return( Composite ); endif endfor Return( Probably Prime );

25 OK, we know how to make primes Now what? Not all primes are good There are some things we should check for when choosing primes Make certain (p-1) or (q-1) do not have many small factors! Why? Else, the (p-1)-factorization Method will make n easy to factor Make p and q of different lengths Why? The following result applies Theorem: Suppose p and q are primes with q < p < 2q. Let n=pq, and choose e and d as in the RSA algorithm. If d < (1/3)n 1/4, then d can be calculated quickly. Make certain adversary doesn t know many of the digits of p or q. Why? The following result applies Theorem: Let n=pq have m digits. If we know the first m/4 or the last m/4 digits of p then we can efficiently factor n.

26 A Little on the Security of RSA The security of RSA is based upon the assumption that factoring the product of two large primes is hard. What if we assume factoring is impossible, then what are the logical implications? Most arguments go like this: If factoring is hard, and XYZ is directly related to factoring, then XYZ is hard. Or, say it another way Assume XYZ is easy, then show XYZ is equivalent to factoring, which contradicts the fact that factoring is impossible!

27 An Example of this Principle Suppose Eve sees n and e (they re public!!!). We claim she can t figure out φ(n). Proof: We show that knowing n and φ(n) is equivalent to factoring, i.e. finding p and q! p and q are the roots of (x-p)(x-q) = x 2 -(p+q)x + pq. Note that n-φ(n)+1 = pq - (p-1)(q-1) +1 = p+q So (x-p)(x-q) = x 2 (n-φ(n)+1)x + n We can solve this using quadratic formula p,q = ( n ϕ( n) + 1) ± ( n ϕ( n) + 1) So, if we could find φ(n) we would be able to factor n!!! This removes the easy way to find d by finding φ(n) n

28 Factorization and Fermat Factorization Modern factorization methods involve significant mathematical machinery. However, we may use a simple factoring method to see what not to do when setting up RSA Fermat Factorization: Start with n=pq. We try to write n=x 2 y 2 = (x+y)(x-y) Can we even do this? Yes, always! Let p=x+y and q = x-y x+y = p 2x = p+q x = (p+q)/2 x-y = q -(x+y = p) -2y = q-p y=(p-q)/2 x y = q So this is always possible. Now, try n+1 2, n+2 2, n+3 2, until we find a square If n+y 2 = x 2 then we are done! n=x 2 y 2 This method only works well when (x+y) and (x-y) are close! That is, when p and q are close! So, we must not choose p and q too close.

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