Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions

B. A. Pogorelov ; M. A. Pudovkina

Diskretnaya Matematika, Tome 34 (2022) no. 1, pp. 103-125.

Résumé

The class of nonabelian 2-groups $H$ with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group $H$, and put forward criteria of their bijectivity. For the generalized group of quaternions of order $2^m$, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.

Détail

Keywords: orthomorphism, complete transformation, dihedral group, generalized quaternion group, semidihedral group, modular maximal cyclic group.

Bibliographie

[1] Mann H. B., “On orthogonal Latin squares”, Bull. Amer. Math. Soc., 50 (1944), 249–257 | DOI | MR | Zbl

[2] Bose R. C., Chakravarti I. M., Knuth D. E., “On methods of constructing sets of mutually orthogonal Latin squares using a computer. I”, Technometrics, 2:4 (1960), 507–516 | DOI | MR | Zbl

[3] Johnson D. M., Dulmage A. L., Mendelsohn N .S., “Orthomorphisms of groups and orthogonal Latin squares. I”, Canad. J. Math., 13 (1961), 356–372 | DOI | MR | Zbl

[4] Dénes J., Keedwell A. D., Latin Squares and Their Applications, Akadémiai Kiadó, Budapest, 1974, 547 pp. | MR | Zbl

[5] Evans A. B., “Generating orthomorphisms of $GF(q)+$”, Discrete Mathematics, 63:1 (1987), 21–26 | DOI | MR | Zbl

[6] Evans A. B., “Orthomorphisms of ${Z_p}$”, Discrete Mathematics, 64:2-3 (1987), 147–156 | DOI | MR | Zbl

[7] Bowler A., “Orthomorphisms of dihedral groups”, Discrete Mathematics, 167–168 (1997), 141–144 | DOI | MR | Zbl

[8] Evans A. B., Orthomorphism Graphs of Groups, Springer-Verlag, Berlin, 1992, 114 pp. | MR | Zbl

[9] Evans A. B., “Mutually orthogonal Latin squares based on general linear groups”, Designs, Codes and Cryptography, 71:3 (2014), 479–492 | DOI | MR | Zbl

[10] Evans A. B., “On orthogonal orthomorphisms of cyclic and non-abelian groups. II”, J. Comb. Designs, 15:3 (2007), 195–209 | DOI | MR | Zbl

[11] Evans A. B., “Applications of complete mappings and orthomorphisms of finite groups”, Quasigroups and Related Systems, 23:1 (2015), 5–30 | MR | Zbl

[12] Glukhov M. M., “O primeneniyakh kvazigrupp v kriptografii”, Prikladnaya diskretnaya matematika, 2:2 (2008), 28–32 | Zbl

[13] Mittenthal L., “Block substitutions using orthomorphic mappings”, Adv. Appl. Math., 16:1 (1995), 59–71 | DOI | MR | Zbl

[14] Junod P., Vaudenay S., “FOX: a new family of block ciphers”, SAC'04, Lect. Notes Comput. Sci., 3357, 2004, 114–129 | DOI | MR

[15] Yun A., Park J. H., Lee J., “On Lai–Massey and quasi-Feistel ciphers”, Designs, Codes and Cryptography, 58 (2011), 45–72 | DOI | MR | Zbl

[16] Gilboa S., Gueron S., Balanced permutations Even-Mansour ciphers, Cryptology ePrint Archive. 2014. Report 2014/642, 2014, 21 pp.

[17] Vaudenay S., “On the Lai-Massey Scheme”, ASIACRYPT'1999, Lect. Notes Comput. Sci., 1716, 1999, 9–19 | MR

[18] Massey J. L., “SAFER K-64: a byte-oriented block-ciphering algorithm”, FSE 94, Lect. Notes Comput. Sci., 1267, 1994, 1–17

[19] Glukhov M. M., “K analizu nekotorykh sistem otkrytogo raspredeleniya klyuchei, osnovannykh na neabelevykh gruppakh”, Matematicheskie voprosy kriptografii, 1:4 (2010), 5–22 | Zbl

[20] Glukhov M. M., “O matritsakh perekhodov raznostei pri ispolzovanii nekotorykh modulyarnykh grupp”, Matematicheskie voprosy kriptografii, 4:4 (2013), 27–47 | Zbl

[21] Biham E., Shamir A., Differential Cryptanalysis of the Data Encryption Standard, Springer, New York, 1993, 188 pp. | MR | Zbl

[22] Carter G., Dawson E., Nielsen L., “DESV: A Latin square variation of DES”, Proc. Workshop Selected Areas in Cryptography (SAC'95), 1997, 158–172

[23] Hawkes P., O'Connor L., “XOR and Non-XOR differential probabilities”, EUROCRYPT'99, Lect. Notes Comput. Sci., 1592, 1999, 141–144 | MR

[24] Hall M., Paige L. J., “Complete mappings of finite groups”, Pacific J. Math., 5:4 (1955), 541–549 | DOI | MR | Zbl

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

[26] Alagic G., Russell A., “Quantum-secure symmetric-key cryptography based on hidden shifts”, EUROCRYPT 2017, Lect. Notes Comput. Sci., 10212, no. 3, 2017, 65–93 | DOI | MR | Zbl

[27] Anan M. V., Targhi E. E., Tabia G. N., Unruh D., “Post-quantum security of the CBC, CFB, OFB, nCTR, and XTS modes of operation”, PQCrypto 2016, Lect. Notes Comput. Sci., 9606, 2016, 44–63 | DOI | MR | Zbl

[28] Berkovich Y., Groups of Prime Power Order, v. 1, de Gruyter Expos. in Math., 46, W. de Gruyter GmbH Co., Berlin, 2008, 532 pp. | MR | Zbl

[29] Berkovich Y., Groups of Prime Power Order, v. 3, de Gruyter Expos. in Math., 56, W. de Gruyter GmbH Co., Berlin, 2011, 666 pp. | DOI | MR | Zbl

[30] Pogorelov B. A., Pudovkina M. A., “Svoistva regulyarnykh predstavlenii neabelevykh $2$-grupp s tsiklicheskoi podgruppoi indeksa $2$”, Matematicheskie voprosy kriptografii, 12:4 (2021), 65–85

[31] Kholl M., Teoriya grupp, Izd-vo inostrannoi literatury, Moskva, 1962, 468 pp.

[32] Glukhov M. M., “O chislovykh parametrakh, svyazannykh s zadaniem konechnykh grupp sistemami obrazuyuschikh elementov”, Trudy po diskretnoi matematike, 1, TVP, M., 1997, 43–66 | MR

[33] Trishin A. E., “O pokazatele nelineinosti kusochno-lineinykh podstanovok additivnoi gruppy polya $\mathbb F_{2^n}$”, Prikladnaya diskretnaya matematika, 30:4 (2015), 32–42 | Zbl