Reducing the number of non-linear multiplications in masking schemes

Abstract

In recent years, methods to securely mask S-boxes against side-channel attacks by representing them as polynomials over finite binary fields have become quite efficient. A good cost model for this is to count how many non-linear multiplications are needed. In this work we improve on the current state-of-the-art generic method published by Coron–Roy–Vivek at CHES 2014 by working over slightly larger fields than strictly needed. This leads us, for example, to evaluate DES S-boxes with only 3 non-linear multiplications and, as a result, obtain 25% improvement in the running time for secure software implementations of DES when using three or more shares. On the theoretical side, we prove a logarithmic upper bound on the number of non-linear multiplications required to evaluate any d-bit S-box, when ignoring the cost of working in unreasonably large fields. This upper bound is lower than the previous lower bounds proved under the assumption of working over the field F 2d, and we show this bound to be sharp. We also achieve a way to evaluate the AES S-box using only 3 non-linear multiplications over F216.