Group polynomials over rings
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Aleksandr V. Akishin
Abstract
We consider polynomials over rings such that the polynomials represent Latin squares and define a group operation over the ring. We introduce the notion of a group polynomial, describe a number of properties of these polynomials and the groups generated. For the case of residue rings
Note: Originally published in Diskretnaya Matematika (2019) 31,№2, 3–13 (in Russian).
References
[1] Glukhov M.M., Remizov A.B., Shaposhnikov V.A., The review of k-valued functions theory. Part 1 Reference book, M.: m/u 33965, 1988 (in Russian), 153 pp.Search in Google Scholar
[2] Kurosh A. G., Lectures in General Algebra Chelsea Pub Co, 1963, 335 pp.Search in Google Scholar
[3] Lambek J., Lectures on rings and modules Blaisdell Pub. Co., New York, 1966.Search in Google Scholar
[4] Larin M. V., “Transitive polynomial transformations of residue class rings”, Discrete Math. Appl. 12:2 (2002), 127–140.10.1515/dma-2002-0204Search in Google Scholar
[5] Lidl R., Niederreiter H., Finite Fields Addison-Wesley Publ. Inc., 1983.Search in Google Scholar
[6] Karpov A.V., “Commutative polynomials over prime rings”, Prikladnaya diskretnaya matematika 22:4 (2013), 16–21 (in Russian).10.17223/20710410/22/2Search in Google Scholar
[7] Hazewinkel M., Formal Groups and Applications N.-Y.: Academic Press, 1978.Search in Google Scholar
[8] Lai X., Massey J.A., “Proposal for a new block encryption standard”, EUROCRYPT’90, Lect. Notes Comput. Sci., 473, 389–404.10.1007/3-540-46877-3_35Search in Google Scholar
[9] Lee R.B., Shi Z.J., Rivest R.L., Robshaw V.J.B., “On permutation operations in cipher design”, ITCC’04, 2 (2004), 569–577.10.1109/ITCC.2004.1286714Search in Google Scholar
[10] Li S., Null polynomials modulo n, http://arxiv.org/abs/math/0510217v2Search in Google Scholar
[11] Mullen G.L., “Local polynomials over ℤp”, Fibonacci Quart. 18:2 (1980), 104–107.Search in Google Scholar
[12] Mullen G.L., Stevens H., “Polynomial functions (mod m)”, Acta Math. Hungar. 44 (1984), 237–241.10.1007/BF01950276Search in Google Scholar
[13] Rivest R.L., “Permutation polynomials modulo 2w”, Finite Fields and Their Applications 7 (2001), 287–292.10.1006/ffta.2000.0282Search in Google Scholar
[14] Schnorr C.P., Vaudenay S., “Black box cryptanalysis of hash networks based on multipermutations”, EUROCRYPT’94, Lect. Notes Comput. Sci., 950, 47–57.10.1007/BFb0053423Search in Google Scholar
[15] GOST 28147-89. Information processing systems. Cryptographic protection. Cryptographic transformation algorithm Standart-inform, Moscow, 1990 (in Russian), 26 pp.Search in Google Scholar
[16] Announcing the ADVANCES ENCRYPTION STANDARD (AES), FIPS 197, 2001, https://doi.org/10.6028/NIST.FIPS.19710.6028/NIST.FIPS.197Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Group polynomials over rings
- Maximum subclasses in classes of linear automata over finite fields
- Using binary operations to construct a transitive set of block transformations
- Perfect matchings and K1,p-restricted graphs
- On the waiting times to repeated hits of cells by particles for the polynomial allocation scheme
- Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation
Articles in the same Issue
- Frontmatter
- Group polynomials over rings
- Maximum subclasses in classes of linear automata over finite fields
- Using binary operations to construct a transitive set of block transformations
- Perfect matchings and K1,p-restricted graphs
- On the waiting times to repeated hits of cells by particles for the polynomial allocation scheme
- Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation
