Università Roma Tre

The concept of bitwise operation. Estimation of the number of bitwise operations (machine time) to execute fundamental operatons.
Polynomial and exponential algorithms. Divisibility. Euclid algorithm and bezout identity executed through bitwise operations. Congruences in Z. Chinese
remainder and successive squaring algorithm.

RSA
The Adleman, Shamir e Rivest algorithm . Formulation and analysis of the algorithm. RSA attacks.
Recalls on squaring congruences; Legendre and Jacobi symbols. A plynomial algorithm to calculate Jacobi symbol. Rabin cryptosystem: description, security analysis and examples. Prime numbers distribution. Primality tests: Fermat, Pocklington algorithms. Euler and strong pseudo-primality condition. Carmichael numbers. Montecarlo algorithms. Solovay-Strassen and Miller-Rabin tests. Pollard factorization methods: p-1and $\rho$.

FINITE FIELDS

HANDBOOK OF APPLIED CRYPTOGRAPHY (Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone)

N. Koblitz, A Course in Number Theory and Cryptography, GTM-Springer.