On Deterministic Reduction of Factoring Integers to Computing the Exponents of Elements in Modular Group

Article type: Research Article

Authors: Pomykała, Jacek^*

Affiliations: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland. [email protected]

Correspondence: [*] Address for correspondence: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland.

Abstract: In the paper we prove that all but at most x/A(x) positive integers n ≤ x can be completely factored in deterministic polynomial time C(x), querying the prime decomposition exponent oracle at most D(x) times. The functions A(x), C(x) and D(x) have the polynomial growth (of log x) at infinity.

Keywords: Brun-Titchmarsh inequality, factoring algorithms, Z_n^*-generating sets, Dirichlet characters, smooth numbers, Discrete Logarithm Problem for Composite Numbers

DOI: 10.3233/FI-2017-1521

Journal: Fundamenta Informaticae, vol. 152, no. 3, pp. 289-295, 2017