No abstract

We give a new algebraic proof of the non-existence of circulant involutory MDS matrices with coefficients in fields of characteristic 2. In odd characteristics we give parameters for the potential existence. If we relax circulancy to θ-circulancy, then there is no restriction to the existence of θ-circulant involutory MDS matrices even for fields of characteristic 2. Finally, we relax further the involutory definition and propose a new direct construction of almost involutory θ-circulant MDS matrices. We show that they can be interesting in hardware implementations.

We consider highly structured truncated differential paths to mount a new rebound attack on Grøstl-512, a hash functions based on two AES-like permutations, P1024 and Q1024, with non-square input and output registers. We explain how such differential paths can be computed using a Mixed-Integer Linear Programming approach. Together with a SuperSBox description, this allows us to build a rebound attack with a 6-round inbound phase whereas classical rebound attacks have 4-round inbound phases. This yields the first distinguishing attack on a 11-round version of P1024 and Q1024 with about 272 computations and a memory complexity of about 256 bytes, to be compared with the 296 computations required by the corresponding generic attack. Previous best results on this permutation reached 10 rounds with a computational complexity of about 2392 operations, to be compared with the 2448 computations required by the corresponding generic attack.

A good linear diffusion layer is a prerequisite in the design of block ciphers. Usually it is obtained by combining matrices with optimal diffusion property over the Sbox alphabet. These matrices are constructed either directly using some algebraic properties or by enumerating a search space, testing the optimal diffusion property for every element. For implementation purposes, two types of structures are considered: Structures where all the rows derive from the first row and recursive structures built from powers of companion matrices. In this paper, we propose a direct construction for new recursive-like MDS matrices. We show they are quasi-involutory in the sense that the matrix-vector product with the matrix or with its inverse can be implemented by clocking a same LFSR-like architecture. As a direct construction, performances do not outperform the best constructions found with exhaustive search. However, as a new type of construction, it offers alternatives for MDS matrices design.