Abstract
This paper studies the problem of constructing lightweight involutory maximal distance separable (MDS) matrices. The authors find the exact lower bound of the XOR counts for 4 × 4 involutory MDS matrices over \(\mathbb{F}_{2^{4}}\). Further, some new structures of 4 × 4 involutory MDS matrices over \(\mathbb{F}_{2^{m}}\) are provided to construct involutory MDS matrices and the authors constructed the lightest 4 × 4 involutory MDS matrices over \(\mathbb{F}_{2^{8}}\) so far by using these structures.
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References
Shannon C E, Communication theory of secrecy systems, The Bell System Technical Journal, 1949, 28(4): 656–715.
Sajadieh M, Dakhilalian M, Mala H, et al., On construction of involutory MDS matrices from Vandermonde matrices in GF(2q), Des. Codes Cryptography, 2012, 64(3): 287–308.
Gupta K C and Ray I G, On constructions of involutory MDS matrices, Progress in Cryptology — AFRICACRYPT 2013, Eds. by Youssef M, Nitaj A, and Hassanien A E, Cairo, 2013.
Nakahara J and Abrahão E, A new involutory MDS matrix for the AES, International Journal of Network Security, 2009, 9(2): 109–116.
Gupta K C and Ray I G, On constructions of circulant MDS matrices for lightweight cryptography, ISPEC 2014, Eds. by Huang X and Zhou J, Fuzhou, China, 2014.
Liu M and Sim S M, Lightweight MDS generalized circulant matrices, FSE 2016, Eds. by Peyrin T, Bochum, 2016.
Beierle C, Kranz T, and Leander G, Lightweight multiplication in GF(2n) with applications to MDS matrices, FSE 2016, Ed. by Peyrin T, Bochum, 2016.
Jean J, Peyrin T, Sim S M, et al., Optimizing implementations of lightweight building blocks, IACR Transactions on Symmetric Cryptology, 2017, 2017(4): 130–168.
Kranz T, Leander G, Stoffelen K, et al., Shorter linear straight-line programs for MDS matrices, IACR Transactions on Symmetric Cryptology, 2017, 2017(4): 188–211.
Khoo K, Peyrin P, Poschmann A, et al., Foam: Searching for hardware-optimal SPN structures and components with a fair comparison, Cryptographic Hardware and Embedded Systems — CHES 2014, Eds. by Batina L and Robshwa M, Busan, South Korea, 2014.
Li Y and Wang M, On the construction of lightweight circulant involutory MDS matrices, FSE 2016, Ed. by Peyrin T, Bochum, 2016.
Sarkar S and Syed H, Lightweight diffusion layer: Importance of Toeplitz matrices, IACR Transactions on Symmetric Cryptology, 2016, 2016(1): 95–113.
Bai J, Li T, Sun Y, et al., The lightest 4 × 4 MDS matrices over GL(4, \(\mathbb{F}_2\)), Cryptology ePrint Archive, Report 2016/686, 2016, https://eprint.iacr.org/2016/686.
Sim S M, Khoo K, Oggier F, et al., Lightweight MDS involution matrices, FSE 2015, Ed. by Leander G, Istanbul, 2015.
Cui T, Jin C, and Kong Z, On compact Cauchy matrices for substitution-permutation networks, IEEE Transactions on Computers, 2015, 64(7): 2098–2102.
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This research was supported in part by the National Natural Science Foundation of China under Grant No. 11371356 & 61877058, CAS Project QYZDJ-SSW-SYS022 and the Strategy Cooperation Project AQ-1701.
This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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Bai, J., Sun, Y. & Wang, D. On the Construction of Involutory MDS Matrices over \(\mathbb{F}_{2^{m}}\). J Syst Sci Complex 33, 836–848 (2020). https://doi.org/10.1007/s11424-019-8125-0
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DOI: https://doi.org/10.1007/s11424-019-8125-0