On the algebraic construction of cryptographically good 32×32 binary linear transformations

- A new algebraic method to construct cryptographically good 32×32 binary matrices.
- How to construct 32×32 involutory binary matrices of branch number 12.
- To construct non-involutory binary matrices of branch number 11 with a fixed point.

Binary linear transformations (also called binary matrices) have matrix representations over GF(2). Binary matrices are used as diffusion layers in block ciphers such as Camellia and ARIA. Also, the 8×8 and 16×16 binary matrices used in Camellia and ARIA, respectively, have the maximum branch number and therefore are called Maximum Distance Binary Linear (MDBL) codes. In the present study, a new algebraic method to construct cryptographically good 32×32 binary linear transformations, which can be used to transform a 256-bit input block to a 256-bit output block, is proposed. When constructing these binary matrices, the two cryptographic properties; the branch number and the number of fixed points are considered. The method proposed is based on 8×8 involutory and non-involutory Finite Field Hadamard (FFHadamard) matrices with the elements of GF(24). How to construct 32×32 involutory binary matrices of branch number 12, and non-involutory binary matrices of branch number 11 with one fixed point, are described.