International Association
for Cryptologic Research

S-boxes are the most popular nonlinear building blocks used in symmetrickey primitives. Both cryptographic properties and implementation cost of an S-box are crucial for a good cipher design, especially for lightweight ones. This paper aims to determine the exact minimum area of optimal 4-bit S-boxes (whose differential uniform and linearity are both 4) under certain standard cell library. Firstly, we evaluate the upper and lower bounds upon the minimum area of S-boxes, by proposing a Prim-like greedy algorithm and utilizing properties of balanced Boolean functions to construct bijective S-boxes. Secondly, an SAT-aided automatic search tool is proposed that can simultaneously consider multiple cryptographic properties such as the uniform, linearity, algebraic degree, and the implementation costs such as area, and gate depth complexity. Thirdly, thanks to our tool, we manage to find the exact minimum area for different types of 4-bit S-boxes.The measurement in this paper uses the gate equivalent (GE) as standard unit under UMC 180 nm library, all 2/3/4-input logic gates are taken into consideration. Our results show that the minimum area of optimal 4-bit S-box is 11 GE and the depth is 3. If we do not use the 4-input gates, this minimum area increases to 12 GE and the depth in this case is 4, which is the same if we only use 2-input gates. If we further require that the S-boxes should not have fixed points, the minimum area continue increasing a bit to 12.33 GE while keeping the depth. Interestingly, the same results are also obtained for non-optimal 4-bit bijective S-boxes as long as their differential uniform U(S) < 16 and linearity L(S) < 8 (i.e., there is no non-trivial linear structures) if only 2-input and 3-input gates are used. But the minimum area reduce to 9 GE if 4-input gates are involved. More strictly, if we require the algebraic degree of all coordinate functions of optimal S-boxes be 3, the minimum area is 14 GE with fixed point and 14.33 GE without fixed point, and the depth increases sharply to 8.Besides determining the exact minimum area, our tool is also useful to search for a better implementation of existing S-boxes. As a result, we find out an implementation of Keccak’s 5-bit S-box with 17 GE. As a contrast, the designer’s original circuit has an area of 23.33 GE, while the optimized result by Lu et al. achieves an area of 17.66 GE. Also, we find out the first optimized implementation of SKINNY’s 8-bit S-box with 26.67 GE.

Maximum Distance Separable (MDS) matrices have been widely used in symmetric cryptographic primitives because of their excellent cryptographic properties. However, due to the heavy area cost, larger-scale MDS matrices than 4 x 4 ones are limited in ciphers, although they have a larger branch number. In this paper, we propose a general method for constructing MDS matrices with low implementation area cost, using matrix decomposition, automatic search, and symbolic computation techniques. According to matrix decomposition theory, every invertible matrix can be decomposed into a sequence of elementary matrices, including Type-1 (row switching), Type-2 (row multiplication) and Type-3 (row addition) elementary matrices. So, we first propose a greedy algorithm to construct MDS matrix patterns with as few Type-3 elementary matrices as possible. Then, we build an automatic search model to minimize the implementation area of multiplication coefficients used in Type-3 elementary matrices. Lastly, another greedy strategy is raised to further reduce the implementation area of MDS matrix patterns by introducing a few Type-2 elementary matrices. In comparison to previous methods, our approach is more general and effective for constructing lower-area MDS matrices. To demonstrate the efficiency of our method, we apply the framework on constructing m x m MDS matrices over F2n or GL(n, F2), where m ∈ {4, 5, 6, 7, 8} and n ∈ {4, 8, 16, 32, 64}. The 4 x 4 MDS matrices constructed by our method can also reach the minimum area. The 5 x 5, 6 x 6 and 7 x 7 MDS matrices constructed by our method have lower area compared to previous ones. While the 8 x 8 MDS matrices with n ∈ {16, 32, 64} constructed by our method also have lower area compared to previous ones.