Geodesic
Lifting of Solutions of an Exponential Congruence
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 76-86.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : Polynomial algorithm, Fermat quotients
Keywords: discrete logarithm, ring of algebraic integers, $\mathfrak{p}$-adic logarithm. 
@article{MZM_2006_80_1_a9,
     author = {I. A. Popovyan},
     title = {Lifting of {Solutions} of an {Exponential} {Congruence}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {76--86},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a9/}
}
TY  - JOUR
AU  - I. A. Popovyan
TI  - Lifting of Solutions of an Exponential Congruence
JO  - Matematičeskie zametki
PY  - 2006
SP  - 76
EP  - 86
VL  - 80
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a9/
LA  - ru
ID  - MZM_2006_80_1_a9
ER  - 
%0 Journal Article
%A I. A. Popovyan
%T Lifting of Solutions of an Exponential Congruence
%J Matematičeskie zametki
%D 2006
%P 76-86
%V 80
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a9/
%G ru
%F MZM_2006_80_1_a9
I. A. Popovyan. Lifting of Solutions of an Exponential Congruence. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 76-86. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a9/

[1] H. Riesel, “Some soluble cases of the discrete logarithm problem”, BIT, 28:4 (1988) | DOI | MR

[2] M. A. Cherepnëv, “O nekotorom svoistve diskretnogo logarifma”, Tez. dokl. XII mezhd. konf. “Problemy teoreticheskoi kibernetiki”, N. Novgorod, 1999

[3] Yu. V. Nesterenko, “Chastnye Ferma i $p$-adicheskie logarifmy”, Trudy po diskretnoi matematike, 5, Fizmatlit, M., 2002, 173–188

[4] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, Nauka, M., 1964 | MR

[5] E. Artin, J. Tate, Class Field Theory, Benjamin, N.Y.–Amsterdam, 1967 | MR | Zbl