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

Autores: Muharrem Tolga Sakalli, Bora Aslan
Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 259, Nº 2, 2014, págs. 485-494
Idioma: inglés
DOI: 10.1016/j.cam.2013.05.008
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Resumen
- 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.