Discrete Mathematics 0800-MDYS

Integer division, relation of divisibily
The Euclid algorithm for fingind the GCD of integers, proof of its validity and estimationg the time complexity
The extended Euclid algorithm
Prime numbers, composite numbers and relatively prime numbers
Factorisation and the Fundamental theorem of arithmetic
The density of primes
The sieve of Eratostenes
Congruence relation, residue rings
The Euler function and the small Fermat's theorem
Rings of pulynomials, residue rings of polynomials, construction of finite fields
Linear congruences, Chinese remainder theorem
Error correcting codes, redundant coding, Hamming distance
The Hamming and the Singleton bound
Cyclic codes
Reed-Solomon codes
Groups, subgroups, group homomorphisms, the kernel and the image
Action of a group on a set. Orbits and stabilisers
Cosets, normal divisor, quotient group, Lagrange theorem
Classification of finite groups, the Sylow theorem
Cayley theorem
The multiplicative group of a ring, discrete logarithm
Fermat algorithm, Pollards p-1 method, exponentation by squaring, the Polladrs. $rho$ method
Leibnitz's test, Miller's-Rabin's test, pseudoprime and strong pseudoprime numbers, Lucas test
Gauss method of finding the rank of an element
Methods of finding the discrete logarithm - Shanks, $rho$ Pollard's and Pohling's-Hellmans
El-Gammal cryptosystem - encryption and subscription
RSA cryptosystem - encryption and subscription
Projective space, non-singular ellptic curves and their group structure. The Hasse's theorem
El-Gammal cryptosystem over an elliptic curve
Simple and directed graphs, graph isomorphisms, thr group of a graph automorphisms
Trees
The Euler problem and the Euler theorem, Fleury's theorem
Hamiltom graphs, Ore theorem