# Public Key Cryptography and RSA. Review: Number Theory Basics

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1 Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and n. Definition Any integer number n > 1 that is not prime is called a composite number. Theorem (Fundamental Theorem of Arithmetic) e 1 e2 2 1 n = p p... p ek k Definition The greatest common divisor of a and b, denoted by gcd(a, b), is the largest number that divides both a and b. Definition Two integers a > 0 and b > 0 are relatively prime if gcd(a, b) =

2 Review: Extended Euclidian Algorithm Input: a, b Output: (d,x,y) s.t. d=gcd(a,b) and ax + by = d d=a; t=b; x=1; y=0; r=0; s=1; while (t>0) { q = d/t u=x-qr; v=y-qs; w=d-qt x=r; y=s; d=t r=u; s=v; t=w } return (d, x, y) How many times before this loop stops? Invariants: gcd(a,b)=gcd(d,t) ax + by = d ar + bs = t Euclidian Algorithm 3 Review: Chinese Reminder Theorem (CRT) Let n 1, n 2,,,, n k be integers s.t. gcd(n i, n j ) = 1, i j. x a x... x a 1 2 mod mod n n 1 2 a k mod n k There exists a unique solution modulo n = n 1 n 2 n k The solution is given by ρ(a1,a2,,ak) = (Σ a i m i y i ) mod n, where m i = n / n i, and y i = m i -1 mod n i 4 2

3 Review: Euler Phi Function Definition: A reduced set of residues (RSR) modulo m is a set of integers R each relatively prime to m, so that every integer relatively prime to m is congruent to exactly one integer in R. Definition: Given n, Z n * ={a 0<a<n and gcd(a,n)=1} is the standard RSR modulo n. Definition Given an integer n, Φ(n) = Z n * is the size of RSR modulo n. Theorem: Fact: If gcd(m,n) = 1, Φ(mn) = Φ(m) Φ(n) Φ(p)=p-1 for prime p 5 Review: Euler s Theorem Euler s Theorem Given integer n > 1, such that gcd(a, n) = 1 then a Φ(n) 1 (mod n) Corollary: Given integer n > 1, such that gcd(a, n) = 1 then a Φ(n)-1 mod n is a multiplicative inverse of a mod n. Corollary: Given integer n > 1, x, y, and a positive integers with gcd(a, n) = 1. If x y (mod Φ(n)), then a x a y (mod n). Corollary (Fermat s Little Theorem): a p-1 1 (mod n) 6 3

4 Why public key cryptography? Overview of Public Key Cryptography RSA Lecture Outline square & multiply algorithm RSA implementation Pohlig-Hellman Fall 2004/Lecture 13 7 Limitation of Secret Key (Symmetric) Cryptography Secret key cryptography symmetric encryption confidentiality (privacy) MAC (keyed hash) authentication (integrity) Sender and receiver must share the same key needs secure channel for key distribution impossible for two parties having no prior relationship Other limitation of authentication scheme cannot authenticate to multiple receivers does not have non-repudiation 8 4

5 Public Key Cryptography Overview Proposed in Diffie and Hellman (1976) New Directions in Cryptography public-key encryption schemes public key distribution systems Diffie-Hellman key agreement protocol digital signature Public-key encryption was proposed in 1970 by James Ellis in a classified paper made public in 1997 by the British Governmental Communications Headquarters Diffie-Hellman key agreement and concept of digital signature are still due to Diffie & Hellman 9 Public Key Encryption Public-key encryption each party has a PAIR (K, K -1 ) of keys: K is the public key and K -1 is the secret key, such that D K -1[E K [M]] = M Knowing the public-key and the cipher, it is computationally infeasible to compute the private key Public-key crypto system is thus known to be asymmetric crypto systems The public-key K may be made publicly available, e.g., in a publicly available directory Many can encrypt, only one can decrypt 10 5

6 Public Key Cryptography Overview Public key distribution systems two parties who do not share any private information through communications arrive at some secret not known to any eavesdroppers Authentication with public keys: Digital Signature the authentication tag of a message can only be computed by one user, but can be verified by many called one-way message authentication in [Diffie & Hellman, 1976] 11 Public-Key Encryption Needs One-way Trapdoor Functions Given a public-key crypto system, Alice has public key K E K must be a one-way function, knowing y= E K [x], it should be difficult to find x However, E K must not be one-way from Alice s perspective. The function E K must have a trapdoor such that knowledge of the trapdoor enables one to invert it 12 6

7 Trapdoor One-way Functions Definition: A function f: {0,1}* {0,1}* is a trapdoor one-way function iff f(x) is a one-way function; however, given some extra information it becomes feasible to compute f -1 : given y, find x s.t. y = f(x) 13 RSA Algorithm Invented in 1978 by Ron Rivest, Adi Shamir and Leonard Adleman Published as R L Rivest, A Shamir, L Adleman, "On Digital Signatures and Public Key Cryptosystems", Communications of the ACM, vol 21 no 2, pp , Feb 1978 Security relies on the difficulty of factoring large composite numbers Essentially the same algorithm was discovered in 1973 by Clifford Cocks, who works for the British intelligence 14 7

8 The Multiplicative Group Z pq * Let p and q be two large primes Denote their product n=pq. The multiplicative group Z n *= Z pq * contains all integers in the range [1,pq-1] that are relatively prime to both p and q The size of the group is Φ(pq) = (p-1)(q-1)=n-(p+q)+1 For every x Z pq *, x (p-1)(q-1) 1 15 Exponentiation in Z pq * Motivation: We want to use exponentiation for encryption Let e be an integer, 1<e<(p-1)(q-1) When is the function f(x)=x e, a one-to-one function in Z pq *? If x e is one-to-one, then it is a permutation in Z pq *. 16 8

9 Exponentiation in Z pq * Claim: If e is relatively prime to (p-1)(q-1) then f(x)=x e is a one-to-one function in Z pq * Proof by constructing the inverse function of f. As gcd(e,(p-1)(q-1))=1, then there exists d and k s.t. ed=1+k(p-1)(q-1) Let y=x e, then y d =(x e ) d =x 1+k(p-1)(q-1) =x (mod pq), i.e., g(y)=y d is the inverse of f(x)=x e. 17 RSA Public Key Crypto System Key generation: Select 2 large prime numbers of about the same size, p and q Compute n = pq, and Φ(n) = (q-1)(p-1) Select a random integer e, 1 < e < Φ(n), s.t. gcd(e, Φ(n)) = 1 Compute d, 1< d< Φ(n) s.t. ed 1 mod Φ(n) Public key: (e, n) Secret key: d 18 9

10 RSA Description (cont.) Encryption Given a message M, 0 < M < n M Z n {0} use public key (e, n) compute C = M e mod n C Z n {0} Decryption Given a ciphertext C, use private key (d) Compute C d mod n = (M e mod n) d mod n = M ed mod n = M 19 RSA Example p = 11, q = 7, n = 77, Φ(n) = 60 d = 13, e = 37 (ed = 481; ed mod 60 = 1) Let M = 15. Then C M e mod n C (mod 77) = 71 M C d mod n M (mod 77) =

11 Why does RSA work? Need to show that (M e ) d (mod n) = M, n = pq We have shown that when M Z pq *, i.e., gcd(m, n) = 1, then M ed M (mod n) What if M Z pq {0} Z pq *, e.g., gcd(m, n) = p. ed 1 (mod Φ(n)), so ed = kφ(n) + 1, for some integer k. M ed mod p = (M mod p) ed mod p = 0 so M ed M mod p M ed mod q = (M k*φ(n) mod q) (M mod q) = M mod q so M ed M mod q As p and q are distinct primes, it follows from the CRT that M ed M mod pq 21 Square and Multiply Algorithm for Exponentiation Computing (x) c mod n Example: suppose that c=53= x 53 =(x 26 ) 2 x=(((x 3 ) 2 ) 2 x) 2 ) 2 x =(((x 2 x) 2 ) 2 x) 2 ) 2 x mod n Alg: Square-and-multiply (x, n, c = c k-1 c k-2 c 1 c 0 ) z=1 for i k-1 downto 0 { z z 2 mod n if c i = 1 then z (z x) mod n } return z 22 11

12 Efficiency of computation modulo n Suppose that n is a k-bit number, and 0 x,y n computing (x+y) mod n takes time O(k) computing (x-y) mod n takes time O(k) computing (xy) mod n takes time O(k 2 ) computing (x -1 ) mod n takes time O(k 3 ) computing (x) c mod n takes time O((log c) k 2 ) 23 RSA Implementation n, p, q The security of RSA depends on how large n is, which is often measured in the number of bits for n. Current recommendation is 1024 bits for n. p and q should have the same bit length, so for 1024 bits RSA, p and q should be about 512 bits. P or q should not be small! 24 12

13 RSA Implementation Select p and q prime numbers In general, select numbers, then test for primality Many implementations use the Rabin-Miller test, (probabilistic test) 25 Finding large prime numbers Attacks on RSA Factoring Next 26 13

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