Codes over L(GF(2)~m,GF(2)~m), MDS Diffusion Matrices and Cryptographic Applications

摘要

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 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.