Theoretical algorithms in cryptography

This book is devoted to algorithmic number theory, to develop intensively the last thirty years in the direction of the theory of numbers, which has important applications in cryptography. The relevance of this trend has increased immeasurably in the 70-ies of XX century with the advent of cryptographic Diffie-Hellman and RSA. At the present time, according to some estimates, almost the entire world fleet means of asymmetric cryptography in mathematical terms based on the number-theoretic problems.
For the purposes of cryptography (for practical implementation and validation of cryptographic resistance, and to develop methods for their opening).
• algorithms are easy integers;
• factorization methods (ie. E. Search methods factoring integers into factors);
• calculations using elliptic curves over finite fields;
• algorithms for computing discrete logarithms;
• Methods of decomposition of polynomials over finite fields and over the field of rational numbers;
• methods of solving systems of linear equations over finite fields;
• algorithms for arithmetic operations with large integers;
• algorithms for polynomial arithmetic.