Cryptography and computer security
Master Program, first year,
Web programming and Internet technologies;
Mathematical and software support for mobile devices;
Computer mathematics and system analysis.
Tuition students mathematical methods underlying the construction of modern public key cryptosystems; introduction to methods of providing computer and network security, mathematical foundation of public key cryptography algorithms; acquaintance with methods of analysis of cryptosystems.
As a result of the study of the discipline the student should be able to:
Algebra and number theory
Divisibility in the ring of integers. GCD, LCM, solvability of linear Diophantine equations.
The law of distribution of Prime numbers, the evaluation of distances between neighboring primes.
Euclidean algorithm,extended Euclidean algorithm, computational complexity of the Euclidean algorithm.
Residues, their properties, a solution of a linear residue, the Chinese remainder theorem, multiplicative functions, Euler’s function, Euler’s theorem, Fermat’s little theorem.
Binary algorithms of exponentiation. Sliding window method.
Computational complexity of algorithms: polynomial, subexponential, exponential algorithms.
Quadratic residues. The Legendre Symbol. Euler’s theorem for quadratic residues, Quadratic reciprocity law. Jacobi Symbols. The calculation of the Legendre symbol.
An algorithm for solving quadratic residues. A general algorithm of solutions of polynomial residues.
The Karatsuba method for multiplying integers. Multiplication of integers using the Chinese remainder theorem. An operation of Montgomery and Barrett reduction.
Deterministic primality test, the Miller-Rabin primality test. The algorithm for constructing large prime numbers.
Algorithm for finding the element of a cyclic group with a given order.
Elliptic curves, a group of elliptic curve points. Obtaining of the addition formulae. The algorithm for computing multiples of a point.
Efficient methods for computing the addition operation on elliptic curve points. Curves in the form of Weierstrass, Montgomery, Edwards.
The RSA cryptosystem.
Diffie-Hellman algorithm for key distribution.
Signature scheme of El Gamal.
Algorithms for digital signature on elliptic curves ECDSA, GOST R 34.10-2012, STB 34.101.45-2013.
Verbal, visual, problem-based, practical, dialog-based and heuristic.
Examinations marks are given taking into account: