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

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### 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.

Outline

- Primitive Element Theorem
- Diffie Hellman Key Distribution
- ElGamal Encryption
- ElGamal Digital Signatures

Goldwasser

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)

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 = qe+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).

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.

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.

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.

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.

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.

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

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

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

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