1 / 12

ElGamal Public Key Cryptography

ElGamal Public Key Cryptography. CS 303 Alg. Number Theory & Cryptography Jeremy Johnson.

maya-glass
Download Presentation

ElGamal Public Key Cryptography

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ElGamal Public Key Cryptography CS 303 Alg. Number Theory & Cryptography Jeremy Johnson Taher ElGamal, "A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms", IEEE Transactions on Information Theory, v. IT-31, n. 4, 1985, pp469–472 or CRYPTO 84, pp10–18, Springer-Verlag.

  2. Outline • Primitive Element Theorem • Diffie Hellman Key Distribution • ElGamal Encryption • ElGamal Digital Signatures Goldwasser

  3. Public Key Cryptography • Let M be a message and let C be the encrypted message (ciphertext). A public key cryptosystem has a separate method E() for encrypting and D() decrypting. • D(E(M)) = M • Both E() and D() are easy to compute • Publicly revealing E() does not make it easy to determine D() • E(D(M)) = M - needed for signatures • The collection of E()’s are made publicly available but the D()’s remain secret. Called a one-way trap-door function (hard to invert, but easy if you have the secret information)

  4. Order • Definition. Let b Zn* The order of b is the smallest positive integer satisfying be 1 (mod n). • Theorem 9.1. If b has order e modulo n and if j is a positive integer such that bj 1 (mod n), then e|j. Proof. j = qe+r, 0  r < e. bj 1  (be)qbr  br(mod n). This implies that r = 0, since e is the smallest power of b equivalent to 1 mod n. • Corollary 9.2. Let b Zn*. ord(b)|(n).

  5. Primitive Element Theorem • Zp* = <>, i.e. ord() = p-1. • Example • Z7* = <3> 31=3, 32=2, 33=6, 34=4, 35=5, 36=1 • Z13* = <2> 21=2, 22=4, 23=8, 24=3, 25=6, 26=12, 27=11, 28=9, 29=5, 210=10, 211=7, 212=1 • Note. ord() = p-1  {1,, 2,…, p-1} distinct.

  6. Discrete Logarithms • Discrete log problem • Given Zp* = <> • log(y) = x, if y = x. • Example • Z13* = <2> 21=2, 22=4, 23=8, 24=3, 25=6, 26=12, 27=11, 28=9, 29=5, 210=10, 211=7, 212=1 • Log2(5) = 9.

  7. Properties of Primitive Elements • Theorem 9.4. If b has order e modulo n, then ord(bi) = e/gcd(e,i). • Theorem 9.7. Let p be a prime and d a divisor of p-1, then the number of positive integers less than p with order d is (d). • Corollary. The number of primitive elements mod p is equal to (p-1) > 1.

  8. Some Lemmas • Lemma 9.6. Let P(x) be a polynomial of degree t and let p be a prime. If p does not divide the coefficient of xt in P(x), then P(x)  0 (mod p), has at most t solutions mod p. Proof. By induction on the degree of P(x)=t. P(x1) = 0  P(x) = P1(x)(x - x1), and the degree of P1(x) = t-1. • Lemma 9.8. The sum of (d) over the divisors of n = n. • Example: n=12. (1)+ (2)+ (3)+ (4)+ (6)+ (12)=1+1+2+2+2+4 = 12.

  9. Proof of Theorem 9.7 • Theorem 9.7. Let p be a prime and d a divisor of p-1, then the number of positive integers less than p with order d is (d). Proof. If there is an element a of order d, then by Theorem 9.4, ai, gcd(i,d)=1 is also of order d. By Lemma 9.6, 1, a, a2,…,ad-1 are the roots of P(x)=xd-1, and there (d) elements of order d. Since every elements is of order d|p-1 and p-1 = d|p-1 (d), there must be an element of order d for every d|p-1 and hence exactly (d) of them.

  10. Public Key Distribution • The goal is for two users to securely exchange a key over an insecure channel. The key is then used in a normal cryptosystem • Diffie-Hellman Key Exchange • Y = X mod q (q prime,  primitive – all elements are powers of ) • X = log Y mod q [discrete log] • Yi = Xi mod q [for each user] • Kij = Xi*Xj mod q [shared key] • Kij = YiXj mod q = YjXi mod q

  11. ElGamal Encryption • Zp* = <>, m  Zp message • B encrypts a message to A. • A: x random, h = x, public key = (p, ,h) • B: y random, k = y, shared key K = hy = xy • EA(m) = (c1,c2), c1 = k, c2=mK mod p. • DA((c1,c2)) = c2*(1/K) mod p, K = c1 x = xy • Security depends on Computational Diffie-Hellman (CDH) assumption: given (, x,y) it is hard to compute xy • Do not use same k twice

  12. ElGamal Digital Signature • Zp* = <>, m  Zp message • A signs message m. • A: h = x, public key = (p, ,h), secret key = x. • A: k random with gcd(k,p-1)=1 • r = k (mod p) • s = (m – xr)(1/k) mod p-1 [m = sk + xr (mod p-1)] • Signature = (r,s) • Verify m=rshr

More Related