Abstract
MDS matrices are used as building blocks of diffusion layers in block ciphers, and XOR count is a metric that estimates the hardware implementation cost. In this paper we report the minimum value of XOR counts of 4 × 4 MDS matrices over F24 and F28 , respectively. We give theoretical constructions of Toeplitz MDS matrices and show that they achieve the minimum XOR count. We also prove that Toeplitz matrices cannot be both MDS and involutory. Further we give theoretical constructions of 4 × 4 involutory MDS matrices over F24 and F28 that have the best known XOR counts so far: for F24 our construction gives an involutory MDS matrix that actually improves the existing lower bound of XOR count, whereas for F28 , it meets the known lower bound.
Highlights
Lightweight cryptography is about cryptosystems that require low implementation costs, and this topic has drawn huge attention over the last few years
In this paper we have obtained the minimum values of XOR counts of 4 × 4 maximum distance separable (MDS) matrices over F24 and F28
We have considered the polynomial basis as this is a conventional choice in practice
Summary
Lightweight cryptography is about cryptosystems that require low implementation costs, and this topic has drawn huge attention over the last few years. In 2014, [14] introduced the metric XOR count that measured the cost of hardware implementation of a diffusion matrix. [20] made a huge search effort to find lightweight diffusion matrices, and they observed that XOR count distribution varies with different irreducible polynomial that generate the field. For F24 our construction gives an involutory MDS matrix (Example 3) with XOR count 16 + 4 · 3 · 4 which improves the existing lower bound 24 + 4 · 3 · 4. On the other hand for F28 , our construction gives an involutory MDS matrix (Example 2) with XOR count 64 + 4 · 3 · 8 that matches with the existing known lower bound
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