Abstract
Many block ciphers and hash functions require the diffusion property of Maximum Distance Separable (MDS) matrices. Serial matrices with the MDS property obtain a trade-off between area requirement and clock cycle performance to meet the needs of lightweight cryptography. In this paper, we propose a new class of serial-type matrices called Diagonal-Serial Invertible (DSI) matrices with the sparse property. These matrices have a fixed XOR count (contributed by the connecting XORs) which is half that of existing matrices. We prove that for matrices of order 4, our construction gives the matrix with the lowest possible fixed XOR cost. We also introduce the Reversible Implementation (RI) property, which allows the inverse matrix to be implemented using the similar hardware resource as the forward matrix, even when the two matrices have different finite field entries. This allows us to search for serial-type matrices which are lightweight in both directions by just focusing on the forward direction. We obtain MDS matrices which outperform existing lightweight (involutory) matrices.
S. M. Sim—Supported by the Singapore National Research Foundation Fellowship 2012 (NRF-NRFF2012-06).
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Notes
- 1.
The indices starts from 1 for the convenience of latter discussions.
- 2.
This observation has also been pointed out in [8].
- 3.
We can multiply the XOR counts of all matrices in Table 2 by 2 to get matrices over \(\mathrm {GF}(2^8)\) but we do not include most of them in Table 3 to prevent congestion. But we can easily see that the best (sparse DSI) matrices we get directly from \(\mathrm {GF}(2^8)/\mathsf {0x1c3}\) do outperform 2 copies of the best matrices over \(\mathrm {GF}(2^4)\) for \(5 \le k \le 7\).
- 4.
- 5.
Given that sparse DSI matrices of order 4 can be 4-MDS, having \(q>8\) would be a bad trade-off between area and clock cycle.
- 6.
\((1\ 2\ 3\ 4)\) is a cycle permutation expression, where the component in the 1st position goes to 2nd position, 2nd to 3rd, 3rd to 4th, and finally the component in the last position goes to the 1st position.
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Toh, D., Teo, J., Khoo, K., Sim, S.M. (2018). Lightweight MDS Serial-Type Matrices with Minimal Fixed XOR Count. In: Joux, A., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2018. AFRICACRYPT 2018. Lecture Notes in Computer Science(), vol 10831. Springer, Cham. https://doi.org/10.1007/978-3-319-89339-6_4
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