Towards a General Construction of Recursive MDS Diffusion Layers

Abstract : MDS matrices are of great importance in the design of block ciphers and hash functions. MDS matrices are in general not sparse and have a large description and thus induces costly implementation in soft-ware/hardware. To overcome this problem, in particular for applications in lightweight cryptography, it was proposed by Guo et. al. to use recur-sive MDS matrices. Such matrices can be computed as a power of companion matrices. Following this, some ad-hoc techniques are proposed to find recursive MDS matrices which are suitable for hardware/software implementation. In another direction, coding theoretic techniques are used to directly construct recursive MDS matrices: Berger technique uses Gabidulin codes and Augot et. al. technique uses shortened BCH codes. In this paper, we provide a necessary and sufficient condition to construct recursive MDS matrices from non-singular diagonable companion matrices. Then we provide three methods to construct recursive MDS matrices. Moreover, recursive MDS matrices obtained through our first method are same as those obtained using shortened BCH codes. The other two methods provide those companion matrices which can be obtained from Gabidulin codes. However, our formulation of necessary and sufficient condition provides many more possibilities to explore to get recursive MDS matrices.
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Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉
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Kishan Chand Gupta, Sumit Kumar Pandey, Ayineedi Venkateswarlu. Towards a General Construction of Recursive MDS Diffusion Layers. Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉. 〈hal-01276436〉

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