Algebraic construction of cryptographically good binary linear transformations

Abstract

Maximum Distance Separable (MDS) and Maximum Distance Binary Linear (MDBL) codes are used as diffusion layers in the design of the well-known block ciphers like the Advanced Encryption Standard, Khazad, Camellia, and ARIA. The reason for the use of these codes in the design of block ciphers is that they provide optimal diffusion effect to meet security of a round function of a block cipher. On the other hand, the constructions of these diffusion layers are various. For example, whereas the Advanced Encryption Standard uses a 4 Â 4 MDS matrix over GF(2 8 ), ARIA uses a 16 Â 16 involutory binary matrix over GF(2 8 ). The most important cryptographic property of a diffusion layer is the branch number of that diffusion layer, which represents the diffusion rate and measures security against linear and differential cryptanalysis. Therefore, MDS and Maximum Distance Binary Linear codes, which provide maximum branch number for a diffusion layer, are preferred in the design of block ciphers as diffusion layers. In this paper, we present a new algebraic construction method based on MDS codes for 8 Â 8 and 16 Â 16 involutory and non-involutory binary matrices of branch numbers 5 and 8, respectively. By using this construction method, we also show some examples of these diffusion layers.

Efficient methods to generate cryptographically significant binary diffusion layers

“…In [16] and [17], an algebraic method is presented to construct 8 × 8, 16 × 16 and 32 × 32 binary matrices with a maximum or maximum achievable branch number by taking into consideration the number of fixed points at the same time. These studies focus on achieving good implementation properties.…”

Hardware accelerated search for resource-efficient and secure permutation matrices

“…A limited search (focusing only on cyclic matrices of this form) results already in rather competitive non-involutory matrices with HW of only 76, whereas the best reported involutory matrices have HW of 112 [12]. We therefore focus our search on HW below 76 and 112 for non-involutory and involutory matrices, respectively.…”

Generating binary diffusion layers with maximum/high branch numbers and low search complexity

“…constructed 16 × 16 and 32 × 32 binary matrices with branch numbers of 8 and 10, respectively, giving good implementation properties on 8‐bit and 32‐bit processors. In , 8 × 8 and 16 × 16 binary matrices with maximum branch numbers and good implementation properties were proposed with an algebraic approach, where 2 × 2 and 4 × 4 MDS or almost MDS matrices (Hadamard and circulant matrices with branch numbers of 4 or 5) over were used. In , 32 × 32 involutory binary matrices with the maximum branch number of 12 were constructed by 8 × 8 matrices over (Hadamard matrices with branch numbers of 7 or 8).…”

“…In , 32 × 32 involutory binary matrices with the maximum branch number of 12 were constructed by 8 × 8 matrices over (Hadamard matrices with branch numbers of 7 or 8). In , 20 × 20 and 24 × 24 binary matrices with branch numbers of 8 and 10, respectively, were presented by a different interpretation of the method given in and . In , Albrecht et al .…”

“…In this study, we combine and extend the strategies of , , and with a different point of view. We first find cyclic binary matrix groups having algebraically suitable properties, and then we construct binary matrices by using the elements of these cyclic groups in 2 × 2, 4 × 4, and 8 × 8 Hadamard matrices and 4 × 4, 12 × 12, and 16 × 16 circulant matrices.…”