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

M. A. Goltvanitsa

Matematičeskie voprosy kriptografii, Tome 14 (2023), pp. 25-53.

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 $\mathrm{AGL}(V)$. Every such subgroup determines new binary operation $\circ$ on the set $V$ and can be used in cryptanalysis. We investigate the structure properties of the group of linear maps associated with the group $\mathcal{T}$. The membership criterion for the right regular representation of group $(V, +)$ to belong to the normalizer of $\mathcal{T}$ in symmetric group $\mathrm{Sym}\,(V)$ is obtained. A practically realizable algorithm for testing whether given $\mathrm{s}$-box belongs to the normalizer of some group $\mathcal{T}$ in $\mathrm{Sym}\,(V)$ is proposed and investigated.

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

[10] Civino R., Blondeau C., Sala M., “Differential attacks: using alternative operations”, Des. Codes and Cryptogr., 87:2-3 (2019), 225–247 | 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] Kargapolov M.I., Merzlyakov Yu.I., Osnovy teorii grupp, Lan, M., 2009, 288 pp. | MR

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

[15] Daemen J., Rijmen V., The Design of Rijndael: AES-the Advanced Encryption Standard, Springer Science Business Media, 2013, 238 pp. | MR

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

[17] Aragona R., Calderini M., Sala M., The role of Boolean functions in hiding sums as trapdoors for some block ciphers, arXiv: 1411.7681

[18] Lam T.Y., A First Course in Noncommutative Rings, Springer, New York, 2012, 397 pp. | MR

[19] Dzhekobson N., Stroenie kolets, IL, M., 1961, 392 pp.

[20] 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, 2003, 33–50 | DOI | MR | Zbl

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