Abstract
In the present paper, a polynomial algorithm is suggested for reducing the problem of taking the discrete logarithm in the ring of algebraic integers modulo a power of a prime ideal to a similar problem with the power equal to one. Explicit formulas are obtained; instead of the Fermat quotients, in the case of residues in the ring of rational integers, these formulas use other polynomially computable logarithmic functions, like the \(\mathfrak{p}\)-adic logarithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
H. Riesel, “Some soluble cases of the discrete logarithm problem,” BIT, 28 (1988), no. 4, 839–851.
M. A. Cherepnev, “On a property of the discrete logarithm,” in: Abstracts of Reports. XII Internat. Conference “Problems in Theoretical Cybernetics” [in Russian], Nizhnii Novgorod, 1999.
Yu. V. Nesterenko, “Fermat quotients and p-adic logarithms,” in: Proceedings in Discrete Mathematics [in Russian], vol. 5, Fizmatlit, Moscow, 2002, pp. 173–188.
A. I. Borevich and I. R. Shafarevich, Number Theory, Nauka, Moscow, 1964; English transl.: Academic Press, New York-London, 1966.
E. Artin and J. Tate, Class Field Theory, Benjamin, New York-Amsterdam, 1967.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 80, no. 1, 2006, pp. 76–86.
Original Russian Text Copyright © 2006 by I. A. Popovyan.
Rights and permissions
About this article
Cite this article
Popovyan, I.A. Lifting of solutions of an exponential congruence. Math Notes 80, 72–82 (2006). https://doi.org/10.1007/s11006-006-0110-y
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0110-y