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

• Published in: Journal of computational and applied mathematics Vol. 259; pp. 485 - 494
• Main Authors: Sakallı, Muharrem Tolga, Aslan, Bora
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.03.2014
• Subjects: Binary linear transformation; Block cipher; Branch number; Cryptography; Finite fields; Fixed points; Finite fields; Fixed points; Binary linear transformation; Cryptography; Branch number; Block cipher
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2013.05.008

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. •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.