Some group-theoretical results on Feistel Networks in a long-key scenario

Riccardo Aragona; Marco Calderini; Roberto Civino

doi:10.3934/amc.2020093

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2020, Volume 14, Issue 4: 727-743. Doi: 10.3934/amc.2020093

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Some group-theoretical results on Feistel Networks in a long-key scenario

1.
DISIM, University of L'Aquila, Via Vetoio, 67100 Coppito (AQ), Italy
2.
Department of Informatics, University of Bergen, Postboks 7803, N-5020 Bergen, Norway

^* Corresponding author: Riccardo Aragona
^* Corresponding author: Riccardo Aragona

Received: December 1, 2019

Revised: May 1, 2020

Early access: July 2, 2020

Published online: November 1, 2020

All the authors are members of INdAM-GNSAGA (Italy). Part of this work was carried out during Marco Calderini's visit at University of L'Aquila, supported by the Meltzer Research Fund and Trond Mohn Foundation.

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Abstract

The study of the trapdoors that can be hidden in a block cipher is and has always been a high-interest topic in symmetric cryptography. In this paper we focus on Feistel-network-like ciphers in a classical long-key scenario and we investigate some conditions which make such a construction immune to the partition-based attack introduced recently by Bannier et al.

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Mathematics Subject Classification: Primary: 94A60, 20B05; Secondary: 20B35.

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Figure 1. Round function of an SPN and of a Feistel network

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[1]	R. Aragona, M. Calderini, R. Civino, M. Sala and I. Zappatore, Wave-shaped round functions and primitive groups, Advances in Mathematics of Communications, 13 (2019), 67-88. doi: 10.3934/amc.2019004.
[2]	R. Aragona, M. Calderini, A. Tortora and M. Tota, On the primitivity of PRESENT and other lightweight ciphers, J. Algebra Appl., 17 (2018), 1850115 (16 pages). doi: 10.1142/S0219498818501153.
[3]	R. Aragona, A. Caranti and M. Sala, The group generated by the round functions of a GOST-like cipher, Ann. Mat. Pura Appl., 196 (2017), 1-17. doi: 10.1007/s10231-016-0559-6.
[4]	R. Aragona, A. Caranti, F. Dalla Volta and M. Sala, On the group generated by the round functions of translation based ciphers over arbitrary fields, Finite Fields Appl., 25 (2014), 293-305. doi: 10.1016/j.ffa.2013.10.005.
[5]	A. Bannier, N. Bodin and E. Filiol, Partition-Based Trapdoor Ciphers, Cryptology ePrint Archive, Report 2016/493, 2016.
[6]	A. Bannier and E. Filiol, Partition-based Trapdoor Ciphers, IntechOpen, London, 2017.
[7]	E. Biham and A. Shamir, Differential cryptanalysis of {DES-like} cryptosystems, Journal of Cryptology, 4 (1991), 3-72. doi: 10.1007/BF00630563.
[8]	A. Bogdanov et al., PRESENT: An ultra-lightweight block cipher, CHES '07, Lecture Notes in Comput. Sci., 4727 (2007), 450–466.
[9]	M. Calderini, A note on some algebraic trapdoors for block ciphers, Advances in Mathematics of Communications, 12 (2018), 515-524. doi: 10.3934/amc.2018030.
[10]	M. Calderini, R. Civino and M. Sala, On properties of translation groups in the affine general linear group with applications to cryptography, preprint, arXiv: math.GR/1702.00581, 2017.
[11]	A. Caranti, F. Dalla Volta and M. Sala, On some block ciphers and imprimitive groups, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 339-350. doi: 10.1007/s00200-009-0100-x.
[12]	A. Caranti, F. Dalla Volta and M. Sala, An application of the O'Nan-Scott theorem to the group generated by the round functions of an AES-like cipher, Des. Codes Cryptogr., 52 (2009), 293-301. doi: 10.1007/s10623-009-9283-1.
[13]	D. Coppersmith and E. Grossman, Generators for certain alternating groups with applications to cryptography, SIAM J. Appl. Math., 29 (1975), 624-627. doi: 10.1137/0129051.
[14]	J. Daemen and V. Rijmen, The design of Rijndael: AES – the Advanced Encryption Standard, Information Security and Cryptography, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4.
[15]	Federal information processing standards publication, Data Encryption Standard and Others, National Bureau of Standards, US Department of Commerce, 1977.
[16]	E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Ann. Sci. École Norm. Sup., 6 (1889), 9-102. doi: 10.24033/asens.317.
[17]	C. Harpes and J. L. Massey, Partitioning cryptanalysis,, Fast Software Encryption, Lecture Notes in Comput. Sci., 1267 (1997), 13–27. doi: 10.1007/BFb0052331.
[18]	Jr. B. S. Kaliski, R. L. Rivest and A. T. Sherman, Is the Data Encryption Standard a group? (Results of cycling experiments on DES), J. Cryptology, 1 (1988), 3-36. doi: 10.1007/BF00206323.
[19]	K. G. Paterson, Imprimitive permutation groups and trapdoors in iterated block ciphers,, Fast Software Encryption, Lecture Notes in Comput. Sci., 1636 (1999), 201–214. doi: 10.1007/3-540-48519-8_15.
[20]	J. Petrillo, Goursat's other theorem, The College Mathematics Journal, 40 (2009), 119-124.
[21]	C. E. Shannon, Communication theory of secrecy systems, Bell System Tech., 28 (1949), 656-715. doi: 10.1002/j.1538-7305.1949.tb00928.x.
[22]	R. Sparr and R. Wernsdorf, Group theoretic properties of Rijndael-like ciphers, Discrete Appl. Math., 156 (2008), 3139-3149. doi: 10.1016/j.dam.2007.12.011.
[23]	R. Wernsdorf, The round functions of RIJNDAEL generate the alternating group, Fast Software Encryption, Lecture Notes in Comput. Sci., 2365 (2002), 143–148. doi: 10.1007/3-540-45661-9_11.
[24]	R. Wernsdorf, The round functions of SERPENT generate the alternating group, 2000. Available from: http://csrc.nist.gov/archive/aes/round2/comments/20000512-rwernsdorf.pdf.
[25]	R. Wernsdorf, The one-round functions of the DES generate the alternating group, Advances in Cryptology-EUROCRYPT '92, Lecture Notes in Comput. Sci., 658 (1993), 99–112. doi: 10.1007/3-540-47555-9_9.