Abstract
The aim of this paper is to provide a general framework in the study of binary block codes. The main objective is to present a general approach in order to explore MDS diffusion matrices used for example in the design of block ciphers with a Substitution Permutation Network design (the so-called SPN block-ciphers).
In order to analyze these codes, we consider additive block codes over binary m-tuples. We are interested in the distance properties related to the block structure. To do this, we introduce a notion of \(\mathcal{L}\)-codes that are codes over the non-commutative ring of linear endomorphisms of GF(2)m. We study the main properties of these codes, especially the notion of duality in this context. We show how most of the known families of block codes can be interpreted in this context. Finally, we conclude by practical examples that allow to derive MDS diffusion matrices over GF(2)m from MDS matrices constructed over smaller blocks.
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References
Augot, D., Finiasz, M.: Exhaustive search for small dimension recursive MDS diffusion layers for block ciphers and hash functions. In: Proceedings of the 2013 IEEE International Symposium on Information Theory, Istanbul, Turkey, July 7-12, pp. 1551–1555. IEEE (2013)
Berger, T.P., El Amrani, N.: Codes over finite quotients of polynomial rings. Finite Fields and Their Applications 25, 165–181 (2014)
Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Information Security and Cryptography. Springer (2002)
Guo, J., Peyrin, T., Poschmann, A.: The PHOTON family of lightweight hash functions. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 222–239. Springer, Heidelberg (2011)
Huffman, W.C.: Codes and groups. In: Huffman, W.C., Pless, V. (eds.) Handbook of Coding Theory II, ch.17. Elsevier Science Inc., New York (1998)
Lally, K., Fitzpatrick, P.: Algebraic structure of quasicyclic codes. Discrete Applied Mathematics 111(1-2), 157–175 (2001)
Ling, S., Niederreiter, H., Solé, P.: On the algebraic structure of quasi-cyclic codes IV: repeated roots. Des. Codes Cryptography 38(3), 337–361 (2006)
MacWilliams, F.J., Sloane, N.J.A.: The theory of Error Correcting Codes. North-Holland, Amsterdam (1986)
Silvester, J.R.: Determinants of block matrices. The Mathematical Gazette 84(3), 460–467 (2000)
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Berger, T.P., El Amrani, N. (2015). Codes over \(\mathcal{L}(GF(2)^m,GF(2)^m)\), MDS Diffusion Matrices and Cryptographic Applications. In: El Hajji, S., Nitaj, A., Carlet, C., Souidi, E. (eds) Codes, Cryptology, and Information Security. C2SI 2015. Lecture Notes in Computer Science(), vol 9084. Springer, Cham. https://doi.org/10.1007/978-3-319-18681-8_16
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DOI: https://doi.org/10.1007/978-3-319-18681-8_16
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18680-1
Online ISBN: 978-3-319-18681-8
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