Number theoretic algorithms in cryptography (Translations of mathematical monographs, 232)

Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important: algorithms for primality testing, factorization algorithms for integers and for polynomials in one variable, applications of the theory of elliptic curves, algorithms for computation of discrete logarithms, algorithms for solving linear equations over finite fields, algorithms for performing arithmetic operations on large integers. The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.

Primality testing and construction of large primes. Factorization of integers with exponential complexity. Factorization of integers with subexponential complexity. Application of elliptic curves to primality testing and factorization of integers. Algorithms for computing discrete logarithm. Factorization of polynomials over finite fields. Reduced lattice bases and their applications. Factorization of polynomials over the field of rational numbers with polynomial complexity. Discrete Fourier transform and its applications. High-precision integer arithmetic. Solving systems of linear equations over finite fields. Facts from number theory. Bibliography. Index.