RSA Preliminaries

Division Algorithm Given integers a and b≠0, there exist unique Integers q and r such that a = bq +r where 0 ≤ r < |b| a is the dividend, b is the divisor, and r is the remainder. b divides a or b is a factor of a if r=0.

Greatest Common Divisor The greatest common divisor of two nonzero integers a and b is defined to be the largest positive integer that divides (i.e., is a factor of) both a and b. The greatest common divisor of a and b is denoted by gcd(a,b). We say that a and b are relatively prime if gcd(a,b)=1.

The Euclidean Algorithm EUCLID(a,b) if b=0 then return a else return EUCLID(b, a mod b)

A useful relation Fact: If d = gcd(a,b), then there exist unique integers x and y such that d = ax + by x and y can be found using the “extended Euclidean algorithm”

Euler’s phi function For any positive integer n, φ(n) (Euler’s phi function) is defined to be the number of positive integers less than n that are relatively prime to n. φ(n) = nπp|n(1-1/p), (where p runs through all prime factors of n) Note that if p is prime, then φ(p) = p-1.

The Theorems of Euler and Fermat Euler’s Theorem aφ(n) = 1 mod n for all a in Zn* , where Zn* is the set of all a in Zn such that gcd(a,n)=1 Fermat’s Theorem If p is prime, then ap = a mod p for all a in Zp and if gcd(a,p)=1, ap-1 = 1 mod p for all a in Zp*

The Chinese Remainder Theorem Suppose n1, n2, …, nk are positive integers which are pairwise relatively prime. Then, for any given integers a1,a2, …, ak, there exists an integer x solving the system of simultaneous congruences x = a1 mod n1 x = a2 mod n2 All solutions are … congruent modulo x = ak mod nk N=n1n2 … nk

Public-key Cryptosystems Each participant has a public key and a secret key. Every public and secret key is a one-to-one function from the set D to D, where D is the set of permissible messages. Alice: PA, SA Bob: PB,SB

Sending a Message Bob uses Alice’s public key to send an encrypted message M to Alice. C = PA(M) M -> PA -----------------------------> SA -> M communication channel Alice uses her secret key to decrypt M

Public and Secret keys are Matched Pairs E.g., M=SA(PA(M)) = PA(SA(M))

Digital Signatures Suppose Alice wished to send Bob a digitally signed response M’ 1. Alice computes her digital signature for the message M’: σ = SA(M’) 2. Alice sends the pair (M’, σ) to Bob. 3. When Bob receives (M’, σ) , he can verify that it originated from Alice by verifying that M’ =PA (σ )

The RSA Cryptosystem A participant creates his public and secret keys as follows: Select an random two large primes, p and q Compute n = pq. Compute e that is relatively prime to φ(n)=(p-1)(q-1) Compute the modulo φ(n) inverse d of e. Publish the pair P=(e,n) as RSA public key. Keep secret the pair S=(d,n) as RSA secret key. The domain D of messages is Zn The transformation of a message M associated with a public key P=(e,n) is P(M)=Me and the transformation of a “ciphertext” C associated with a secret key S=(d,n) is S(C)=Cd

The Correctness of RSA SA(PA(M)) = Med=Mde=PA(SA(M)) We have ed = 1 mod (p-1)(q-1). This means that when ed is divided by (p-1)(q-1), the remainder is 1 and so by the division algorithm, ed = k(p-1)(q-1)+1 for some k. Thus, Med = M1+k(p-1)(q-1)=M M(p-1)k(q-1) =M(1)k(q-1) mod p (by Fermat’s theorem) =M mod p if gcd(M,p)=1.

The Correctness of RSA 2 Also, Med = M1+k(p-1)(q-1)=M M(q-1)k(p-1) =M(1)k(p-1) mod q (by Fermat’s theorem) =M mod q if gcd(M,q)=1 Thus, Med =M mod p Med =M mod q These two congruences hold even when either p or q is not relatively prime to M and so Med =M mod n (by the Chinese Remainder Theorem)

Example (for an extremely simple, breakable code) Suppose that Alice chooses p=7, q=11 and e=13. Then PA = (13,77) and d=37 since 13*37=1 mod 60 and so her secret key is SA = (37,n). Suppose that Bob wants to send 52 to Alice. What is the encrypted message?

Encrypted message Bob uses Alice’s public key to encrypt his message: 5213 = 17 mod 77

Decrypted message Alice receives the encrypted message 17. Alice uses her secret key (37,77) to decrypt Bob’s message 1737 = 52 mod 77

Verification Suppose that Alice wants to verify to Bob that she received his message. She sends encodes the message she received with her secret key and sends it to Bob: 5237 = 24 mod 77 Bob uses Alices’s public key to obtain 2413 = 52 mod 77

Breaking the code Messages encrypted with RSA can be decrypted by determining primes p and q such that n=pq since in that case a d can be determined such that de = 1 mod φ(n)

Asignment Write an openMP program such that given an integer n, the program determines two primes p and q such that n = pq, if such primes exist.