Figure 1.
Round function of an SPN and of a Feistel network
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November
2020, 14(4): 727-743.
doi: 10.3934/amc.2020093
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 |
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.
Keywords:
Symmetric Cryptography,
Block Cipher,
Trapdoor,
Group Generated by Round Functions,
Partitions.
Mathematics Subject Classification:
Primary: 94A60, 20B05; Secondary: 20B35.
Citation:
Riccardo Aragona, Marco Calderini, Roberto Civino. Some group-theoretical results on Feistel Networks in a long-key scenario. Advances in Mathematics of Communications,
2020, 14
(4)
: 727-743.
doi: 10.3934/amc.2020093
![]()
References:
| [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. |
show all references
References:
| [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. |
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