Elementary Abelian regular subgroups of vector space affine group related to cryptanalysis.~II

M. A. Goltvanitsa

Matematičeskie voprosy kriptografii, Tome 15 (2024), pp. 9-47.

Résumé

Let $p$ be a prime number, $(V,+)$ be a finite-dimensional vector space over finite field $\mathbb{F}_p$ of cardinality $p$. We investigate elementary Abelian regular subgroups $\mathcal{T}$ of affine group $\mathrm{AGL}(V)$. Each such subgroup defines new binary operation $\circ$ on the set $V$ and can be utilized in cryptanalysis, especially in cryptanalysis of block ciphers constructed as SP-networks. In the previous paper we propose the first practical algorithm for testing whether given s-box preserving zero belong to the normalizer of some group $\mathcal{T}$ in $\mathrm{Sym}(V)$. In this paper we generalize this algorithm for an arbitrary s-box. We find some arithmetic properties of linear groups associated with groups $\mathcal{T}$. Basing on utilizing automorphisms of direct sums of commutative algebras we suggest the first practical method for construction of $\circ$-affine SP-networks with an arbitrary block size.

Bibliographie

[1] Glukhov M.M., Elizarov V.P., Nechaev A.A., Algebra, Lan, M., 2015, 606 pp.

[2] Hegedus P., “Regular subgroups of the affine group”, J. Algebra, 225:2 (2000), 740–742 | DOI | MR | Zbl

[3] Caranti A., Francesca D., Sala M., “Abelian regular subgroups of the affine group and radical rings”, Publ. Math. (Debr.), 69:3 (2006), 297–308 | DOI | MR | Zbl

[4] Catino F., Colazzo P., “On regular subgroups of the affine group”, Bull. Aust. Math. Soc., 91:1 (2015), 76–85 | DOI | MR | Zbl

[5] Pellegrini M., Tamburini M., “More on regular subgroups of the affine group”, Linear Algebra Appl., 505 (2016), 126–151 | DOI | MR | Zbl

[6] Pellegrini M., Tamburini M., “Regular subgroups of the affine group with no translations”, J. Algebra, 478 (2017), 410–418 | DOI | MR | Zbl

[7] Calderini M., Sala M., “On differential uniformity of maps that may hide an algebraic trapdoor”, Int. Conf. Algebraic Informatics, Lect. Notes Comput. Sci., 9270, Springer, 2015, 70–78 | DOI | MR | Zbl

[8] Brunetta C., Calderini M., Sala M., “On hidden sums compatible with a given block cipher diffusion layer”, Discr. Math., 342:2 (2019), 373–386 | DOI | MR | Zbl

[9] Civino R., Blondeau C., Sala M., “Differential attacks: using alternative operations”, Des. Codes and Cryptogr., 87:2-3 (2019), 225–247 | DOI | MR | Zbl

[10] Schafer R.D., “Alternative algebras over an arbitrary field”, Bull. Amer. Math Soc., 49 (1943), 549–555 | DOI | MR | Zbl

[11] Dixon J., “Maximal Abelian subgroups of the symmetric groups”, Can. J. Math., 23:3 (1971), 426–438 | DOI | MR | Zbl

[12] Calderini M., Civino R., Sala M., “On properties of translation groups in the affine general linear group with applications to cryptography”, J. Algebra, 569 (2021), 658–680 | DOI | MR | Zbl

[13] Goltvanitsa M.A., “Elementarnye regulyarnye abelevy podgruppy affinnoi gruppy vektornogo prostranstva v svyazi s analizom kriptograficheskikh primitivov”, Matematicheskie voprosy kriptografii, 14:4 (2023), 25–53 | DOI | MR | Zbl

[14] Kargapolov M.I., Merzlyakov Yu.I., Osnovy teorii grupp, Lan, M., 2009, 288 pp. | MR

[15] Isaacs M., Finite group theory, Amer. Math. Soc., 2008, 350 pp. | MR | Zbl

[16] Daemen J., Rijmen V., The Design of Rijndael: AES-the Advanced Encryption Standard, Springer Science Business Media, 2013 | MR

[17] Kash F., Moduli i koltsa, Mir, M., 1981, 368 pp.

[18] Aragona R., Calderini M., Sala M., The role of Boolean functions in hiding sums as trapdoors for some block ciphers | DOI

[19] Biryukov A., Cannier C. De, Braeken A., Preneel B., “A toolbox for cryptanalysis: linear and affine equivalence algorithms”, EUROCRYPT 2003, Lect. Notes Comput. Sci., 2656, 33–50 | DOI | MR | Zbl

[20] Kormen T., Laizerson Ch., Rivest R., Shtain K., Algoritmy: postroenie i analiz, Dialektika-Vilyams, M., 2020, 1328 pp.

[21] Calderini M., On Boolean functions, symmetric cryptography and algebraic coding theory, Ph.D, University of Trento, 2015