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2015 | OriginalPaper | Chapter

Direct Construction of Recursive MDS Diffusion Layers Using Shortened BCH Codes

Authors : Daniel Augot, Matthieu Finiasz

Published in: Fast Software Encryption

Publisher: Springer Berlin Heidelberg

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Abstract

MDS matrices allow to build optimal linear diffusion layers in block ciphers. However, MDS matrices cannot be sparse and usually have a large description, inducing costly software/hardware implementations. Recursive MDS matrices allow to solve this problem by focusing on MDS matrices that can be computed as a power of a simple companion matrix, thus having a compact description suitable even for constrained environments. However, up to now, finding recursive MDS matrices required to perform an exhaustive search on families of companion matrices, thus limiting the size of MDS matrices one could look for. In this article we propose a new direct construction based on shortened BCH codes, allowing to efficiently construct such matrices for whatever parameters. Unfortunately, not all recursive MDS matrices can be obtained from BCH codes, and our algorithm is not always guaranteed to find the best matrices for a given set of parameters.

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Appendix
Available only for authorised users
Footnotes
1
If the non-systematic part \(M\) of an MDS generator matrix contained a 0, then the line of \(G_M\) containing this zero would have Hamming weight \(\le k\), which is in contradiction with the minimal distance of the code. More generally, for an MDS code \(\varGamma _M\), for any \(i\le k\) all the \(i\times i\) minors of \(M\) must be non-zero.
 
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Metadata
Title
Direct Construction of Recursive MDS Diffusion Layers Using Shortened BCH Codes
Authors
Daniel Augot
Matthieu Finiasz
Copyright Year
2015
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-46706-0_1

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