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About permutations on the sets of tuples from elements of the finite field
V. S. Kugurakov, A. F. Gainutdinova, V. T. Dubrovin Kazan Federal University, Kazan, 420008
Russia
Abstract:
The following problem was considered: let $S=S_1\times S_2\times
\dots \times S_m$ be the Cartesian product of subsets $S_i$ that are
subgroups of the multiplicative group of a finite field ${\mathbb
F}_q$ of $q$ elements or their extensions by adding a zero element;
a map $f: S\rightarrow S$ of $S$ into itself can be specified by a
system of polynomials $f_1, \dots, f_m\in {\mathbb F}_q [x_1, \dots,
x_m ]$. Necessary and sufficient conditions, for which the map
$f=\langle f_1, \dots ,f_m\rangle$ is bijective, were obtained. Then
this problem was generalized to the case when the subsets $S_i$ are
any subsets of ${\mathbb F}_q$. The obtained results can be used to
construct $S$-boxes and $P$-boxes in block ciphers and to calculate
automorphism groups of error-correcting codes.
Keywords:
cryptography, error-correcting codes, finite fields, permutation polynomials.
Received: 11.03.2019
Citation:
V. S. Kugurakov, A. F. Gainutdinova, V. T. Dubrovin, “About permutations on the sets of tuples from elements of the finite field”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 161, no. 2, Kazan University, Kazan, 2019, 292–300
Linking options:
https://www.mathnet.ru/eng/uzku1518 https://www.mathnet.ru/eng/uzku/v161/i2/p292
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Abstract page: | 252 | Full-text PDF : | 116 | References: | 26 |
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