Geodesic
On cosets in the direct product of groups whose images by bijective mappings from factors to groups are cosets
Diskretnaya Matematika, Tome 35 (2023) no. 4, pp. 18-45
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This paper is concerned with images of cosets in the direct product of groups by bijective mappings from factors to groups. We prove necessary and sufficient conditions on bijective mappings for existence a coset in the direct product of two groups whose image is a coset. Under some constraints on bijective mappings, we describe the cosets in the direct product of groups, whose images by bijective mappings from factors to groups are cosets, We described the cosets in the direct product of elementary abelian 2-groups whose images by the inversion permutation of nonzero elements of a finite field on factors are cosets. We also describe the similar cosets for the permutation used an an $s$-box of Kuznyechik algorithm, Under some constraints on bijective mappings, we describe the automorphisms of the direct product groups which commute with bijective mappings from factors to groups.
Keywords: invariant coset attack, self-similiarity attack, $s$-box layer, Kuznyechik algorithm.
Mots-clés : inversion permutation of nonzero elements of a finite field 
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D. A. Burov. On cosets in the direct product of groups whose images by bijective mappings from factors to groups are cosets. Diskretnaya Matematika, Tome 35 (2023) no. 4, pp. 18-45. http://geodesic.mathdoc.fr/item/DM_2023_35_4_a1/

[1] Burov D. A., “Podgruppy pryamogo proizvedeniya grupp, invariantnye otnositelno deistviya podstanovok na somnozhitelyakh”, Diskretnaya matematika, 31:4 (2019), 3–19 | DOI

[2] Burov D. A., “O svyazi lineinoi i raznostnoi kharakteristik otobrazhenii dvoichnykh vektornykh prostranstv s kharakteristikami rasseivaniya po blokam sistem imprimitivnosti gruppy sdvigov dvoichnogo vektornogo prostranstva”, Diskretnaya matematika, 35:1 (2023), 3–34 | DOI

[3] Glukhov M. M., “O 2-tranzitivnykh proizvedeniyakh regulyarnykh grupp podstanovok”, Trudy po diskretnoi matematike, 3 (2000), 37–52

[4] Kholl M., Teoriya grupp, IL, M., 1962, 468 pp.

[5] Barkan E., Biham E., “In how many ways can you write Rijndael”, Asiacrypt 2002, Lect. Notes Comput. Sci., 2501, 2002, 160–175 | DOI | MR | Zbl

[6] Beyne T., “Block cipher invariants as eigenvectors of correlation matrices”, J. Cryptology, 33 (2020), 1156–1183 | DOI | MR | Zbl

[7] Bidwell J. N. S., “Automorphisms of direct products of finite groups II”, Arch. Math., 91:2 (2008), 111–121 | DOI | MR | Zbl

[8] Bouillaguet C., Dunkelman O., Leurent G., Fouque P.-A., “Another look at complementation properties”, FSE 2010, Lect. Notes Comput. Sci., 6147, 2010, 347–364 | DOI | Zbl

[9] Bulygin S., Walter M., Buchmann J., “Full analysis of Printcipher with respect to invariant subspace attack: efficient key recovery and countermeasures”, Des. Codes Cryptogr., 73 (2014), 997–1022 | DOI | MR | Zbl

[10] Burov D. A., Pogorelov B. A., “An attack on 6 rounds of Khazad”, Matematicheskie voprosy kriptografii, 7:2 (2016), 35–46 | DOI | MR | Zbl

[11] Fomin D. B., “On the impossibility of an invariant attack on Kuznyechik”, J. Computer Virology and Hacking Techniques, 18:1 (2022), 61–67 | DOI

[12] Guo J., Jean J., Nicolic I., Qiao K., Sasaki Y., Sim S. M., “Invariant subspace attack against Midori64 and the resistant criteria for S-box designs”, IACR Trans. Symm. Cryptology, 2016:1 (2016), 33–56 | DOI | MR

[13] Kolomeec N., Bykov D., On the image of an affine subspace under the inverse function within a finite field, 2022, arXiv: 2206.14980

[14] Leander G., Minaud B., Ronjom S., “A generic approach to invariant subspace attacks: cryptanalysis of Robin, iScream and Zorro”, Eurocrypt 2015, Lect. Notes Comput. Sci., 9056, 2015, 254–283 | DOI | MR | Zbl

[15] Todo Y., Leander G., Sasaki Y., “Nonlinear invariant attack practical attack on full SCREAM, iSCREAM, and Midori64”, Asiacrypt 2016, Lect. Notes Comput. Sci., 10032, 2016, 3–33 | DOI | MR | Zbl

[16] Courtois N., “The inverse S-box, non-linear polynomial relations and cryptanalysis of block ciphers”, AES 2004, Lect. Notes Comput. Sci., 3373, 2004, 170–188 | DOI | MR

[17] Leander G., Abdelraheem M., Alkhzaimi H., Zenner E., “A cryptanalysis of PRINT cipher: the invariant subspace attack”, Crypto 2011, Lect. Notes Comput. Sci., 6841, 2011, 206–221 | DOI | MR | Zbl

[18] Ranea A., Preneel B., “On self-equivalence encodings in white-box implementations”, SAC, Lect. Notes Comput. Sci., 12804, 2020, 639–669 | DOI | MR

[19] Remak R., “Uber die darstellung der endlichen gruppen als untergruppen direct produkte”, J. Reine Angew. Math., 1 (1930), 1–44 | MR

[20] Ronjom S., Invariant subspaces in Simpira, IACR Cryptology Archive, Report 2016/248, 2016