Abstract

Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heuristic expected running time Lp[1/3; 3 2/3]. For numbers of a special form, there is an asymptotically slower but more practical version of the algorithm.

Citations

553	Factoring Polynomials with Rational Coefficients - Lenstra, Lovàsz - 1982
240	Approximate formulas for some functions of prime numbers - Rosser, Schoenfeld - 1962
199	Factoring integers with elliptic curves - Lenstra - 1987
177	On Lovász’ lattice reduction and the nearest lattice point problem - Babai - 1986
143	Solving sparse linear equations over finite fields - Wiedemann - 1986
131	Factoring polynomials over large finite fields - Berlekamp - 1970
107	Transcendental number theory - Baker - 1975
73	Discrete logarithms in finite fields and their cryptographic significance - ODLYZKO - 1985
65	Saunders: “On Wiedemann’s method of solving sparse linear systems - Kaltofen, David - 1991
59	On a problem of Oppenheim concerning “Factorisatio Numerorum - Canfield, Erdős, et al. - 1983
59	The number field sieve - Lenstra, Lenstra, et al. - 1990
33	Odlyzko, Computation of discrete logarithms in prime fields - LaMacchia, M
30	Searching for primitive roots in finite fields - Shoup - 1992
22	The discrete logarithm problem, in Cryptology and Computational - McCurley - 1990
20	Factoring numbers using singular integers - Adleman
18	Fast, rigorous factorization and discrete logarithm algorithms, Discrete Algorithms and Complexity, edited by - Pomerance - 1987
16	rigorous factorization and discrete logarithm algorithms - Fast
15	On an irreducibility theorem - Brillhart, Filaseta, et al. - 1981
15	Discrete logarithms in GF(p - Coppersmith, Odlyzko, et al. - 1986
10	On the distribution in short intervals of integers having no large prime factor - Friedlander, Lagarias - 1987
5	Private communication - Pomerance - 1996
4	Factoring integers with the number field sieve, preprint - Pomerance
4	On the maximal modulus of conjugates of an algebraic integer - Dobrowolski - 1978
1	Modifications to the number field sieve, preprint - Coppersmith
1	A rigorous time bound for factoring integers, preprint - Pomerance
1	Class numbers and units - Zantema - 1982