Optimizing Implementations of Linear Layers

Abstract

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.