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