International Association
for Cryptologic Research

In this paper, we compare several non-tight degree evaluation methods i.e., Boura and Canteaut’s formula, Carlet’s formula as well as Liu’s numeric mapping and division property proposed by Todo, and hope to find the best one from these methodsfor practical applications. Specifically, for the substitution-permutation-network (SPN) ciphers, we first deeply explore the relationships between division property of an Sbox and its algebraic properties (e.g., the algebraic degree of its inverse). Based on these findings, we can prove theoretically that division property is never worse than Boura and Canteaut’s and Carlet’s formulas, and we also experimentally verified that the division property can indeed give a better bound than the latter two methods. In addition, for the nonlinear feedback shift registers (NFSR) based ciphers, according to the propagation of division property and the core idea of numeric mapping, we give a strict proof that the estimated degree using division property is never greater than that of numeric mapping. Moreover, our experimental results on Trivium and Kreyvium indicate the division property actually derives a much better bound than the numeric mapping. To the best of our knowledge, this is the first time to give a formal discussion on the relationships between division property and other degree evaluation methods, and we present the first theoretical proof and give the experimental verification to illustrate that division property is the optimal one among these methods in terms of the accuracy of the upper bounds on algebraic degree.

In this paper, we propose a new heuristic algorithm to search efficient implementations (in terms of Xor count) of linear layers used in symmetric-key cryptography. It is observed that the implementation cost of an invertible matrix is related to its matrix decomposition if sequential-Xor (s-Xor) metric is considered, thus reducing the implementation cost is equivalent to constructing an optimized matrix decomposition. The basic idea of this work is to find various matrix decompositions for a given matrix and optimize those decompositions to pick the best implementation. In order to optimize matrix decompositions, we present several matrix multiplication rules over F2, which are proved to be very powerful in reducing the implementation cost. We illustrate this heuristic by searching implementations of several matrices proposed recently and matrices already used in block ciphers and Hash functions, and the results show that our heuristic performs equally good or outperforms Paar’s and Boyar-Peralta’s heuristics in most cases.