Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications
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- 作者：Kishan Chand Gupta ; Indranil Ghosh Ray
- 关键词：Diffusion ; InvMixColumn operation ; Involutory matrix ; MDS matrix ; MixColumn operation ; Orthogonal matrix ; 94A60
- 刊名：Cryptography and Communications
- 出版年：2015
- 出版时间：June 2015
- 年：2015
- 卷：7
- 期：2
- 页码：257-287
- 全文大小：592 KB
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- 刊物主题：Coding and Information Theory
Mathematics of Computing - 出版者：Springer New York
- ISSN：1936-2455
文摘
MDS matrices incorporate diffusion layers in block ciphers and hash functions. MDS matrices are in general not sparse and have a large description and thus induce costly implementations both in hardware and software. It is also nontrivial to find MDS matrices which could be used in lightweight cryptography. In the AES MixColumn operation, a circulant MDS matrix is used which is efficient as its elements are of low hamming weights, but no general constructions and study of MDS matrices from d×d circulant matrices for arbitrary d is available in the literature. In a SAC 2004 paper, Junod et al. constructed a new class of efficient matrices whose submatrices were circulant matrices and they coined the term circulating-like matrices for these new class of matrices. We call these matrices as Type-I circulant-like matrices. In this paper we introduce a new type of circulant-like matrices which are involutory by construction and we call them Type-II circulant-like matrices. We study the MDS properties of d×d circulant, Type-I and Type-II circulant-like matrices and construct new and efficient MDS matrices which are suitable for lightweight cryptography for d up to 8. We also consider orthogonal and involutory properties of such matrices and study the construction of efficient MDS matrices whose inverses are also efficient. We explore some interesting and useful properties of circulant, Type-I and Type-II circulant-like matrices which are prevalent in many parts of mathematics and computer science.