Geodesic
Discrete logarithm in an arbitrary quotient ring of polynomials of one variable over a finite field
Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 120-132
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the question of solvability and solution of the congruence relation $a^n(x)\equiv b(x)\pmod{F(x)}$ over a finite field for an arbitrary polynomial $F(x)$. In the case where $F(x)$ is a power of an irreducible polynomial, we give an algorithm of lifting of the solution, that is, the solution of the congruence $$ a^n(x)\equiv b(x)\pmod{f^\alpha(x)} $$ reduces to the solution of the congruence $a^n(x)\equiv b(x)\pmod{f(x)}$. For this case we obtain necessary and sufficient conditions for solvability of exponential congruences. If $F(x)$ is not a power of an irreducible polynomial, then the solution, as before, reduces to the solution of congruences of the form $a^n(x)\equiv b(x)\pmod{f_i(x)}$, but the question of solvability reduces to checking the solvability of congruences of the form $$ a^n(x)\equiv b(x)\pmod{f_i(x)f_j(x)}, $$ where $f_i(x)$ and $f_j(x)$ are irreducible divisors of $F(x)$. For the moduli of the form $f_i(x)f_j(x)$ the result is obtained for some special cases. In addition, we describe a constructive isomorphism of the quotient ring of polynomials $R=GF(p^m)[x]/(f^\alpha(x))$ and a chain ring represented in the form $\overline R=GF(p^r)[x]/(x^t)$, so that the results obtained for polynomials are extended to finite chain rings of prime characteristics. In particular, for the chain rings represented in the form $GF(p^r)[x]/(x^t)$ we give necessary and sufficient conditions for solvability of exponential congruences. 
@article{DM_2010_22_2_a8,
     author = {A. V. Markelova},
     title = {Discrete logarithm in an arbitrary quotient ring of polynomials of one variable over a~finite field},
     journal = {Diskretnaya Matematika},
     pages = {120--132},
     year = {2010},
     volume = {22},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2010_22_2_a8/}
}
TY  - JOUR
AU  - A. V. Markelova
TI  - Discrete logarithm in an arbitrary quotient ring of polynomials of one variable over a finite field
JO  - Diskretnaya Matematika
PY  - 2010
SP  - 120
EP  - 132
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DM_2010_22_2_a8/
LA  - ru
ID  - DM_2010_22_2_a8
ER  - 
%0 Journal Article
%A A. V. Markelova
%T Discrete logarithm in an arbitrary quotient ring of polynomials of one variable over a finite field
%J Diskretnaya Matematika
%D 2010
%P 120-132
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/DM_2010_22_2_a8/
%G ru
%F DM_2010_22_2_a8
A. V. Markelova. Discrete logarithm in an arbitrary quotient ring of polynomials of one variable over a finite field. Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 120-132. http://geodesic.mathdoc.fr/item/DM_2010_22_2_a8/

[1] Buchmann J., Jacobson M. J., Teske E., “On some computational problems in finite abelian groups”, Math. Comput., 66:220 (1997), 1663–1687 | DOI | MR | Zbl

[2] Vasilenko O. N., Teoretiko-chislovye algoritmy v kriptografii, MTsNMO, Moskva, 2003 | MR

[3] Vasilenko O. N., “O diskretnom logarifmirovanii v nekotorykh gruppakh”, Vestnik MGU. Cer. 1. Matem., mekhan., 2000, no. 5, 53–55 | MR | Zbl

[4] Cohen H., A course in computational algebraic number theory, Springer, Berlin, 1993 | MR

[5] Bach E., Shallit J., Algorithmic number theory, v. 1, MIT, Massachusetts, 1996 | MR | Zbl

[6] Vinogradov I. M., Osnovy teorii chisel, Nauka, Moskva, 1972 | MR

[7] Popovyan I. A., “Pod'em reshenii pokazatelnogo sravneniya”, Matematicheskie zametki, 80:1 (2006), 76–86 | MR | Zbl

[8] Van der Varden B. L., Algebra, Nauka, Moskva, 1976 | MR

[9] Elizarov V. P., Konechnye koltsa, Gelios ARV, Moskva, 2006