Problem 27:   Primality Testing

1. Implement the Miller-Rabin primality testing algorithm so as to handle numbers at least 256 bits long. As usual, your program should be writtein in C, C++, or Java and should use one of the suggested big number libraries. If your library already contains a probabilistic primality tester, do not use it (except for checking) - implement your own instead. But it's okay to use built-in implementations of modular exponentiation, random number generators, and the other arithmetic functions and predicates.
2. Twin primes are pairs of primes of the form (p, p+2), e.g., (11, 13). (See http://mathworld.wolfram.com/TwinPrimes.html.) Use your program from part (a) to find the smallest p > 2²⁵⁵+100 such that (p, p+2) is a twin prime.

Problem 28:   ElGamal Variants

Problem 29:   Existental Forgery of ElGamal Signatures

Problem 30:   Hash Function Based on Squaring

Problem 31:   Hash Function Based on Matrices