Number Theory and Cryptography

Finite Fields. Modular arithmetic. Number Theory. The sieve of Eratosthenes and other factoring methods. The extended Eulidean algorithm. The Euler function. Linear Diophantine equations and congruences. The fundamental theorems: The fundamental theorem of Arithmetic, Euler, Fermat, Wilson, the Chinese remainder theorem the prime number theorem. Quadratic congruences, Legendre and Jacobi symbols. The Quadratic reciprocity law. Numbers: perfect, Mersenne, Fermat and amicable numbers. Cryptology. Historical overview. Classical cryptosystems: encryption, decryption and cryptanalysis of the cryptosystems: additive, multiplicative, affine, Vigenere, Playfair and Hill. The Discrete logarithm problem (D.L.P.). Public Key cryptosystems: The R.S.A., Merkle, Hellman- the Diffie-Hellman problem, Elgamal, Massey – Omura. Digital Signatures. Elliptic Curves. Combinatorial Designs and Cryptography.