## Cryptology ePrint Archive: Report 2016/831

Reducing the Number of Non-linear Multiplications in Masking Schemes

Jürgen Pulkus and Srinivas Vivek

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 $\mathbb{F}_{2^d}$, 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 $\mathbb{F}_{2^{16}}$.

Category / Keywords: implementation / side-channel countermeasure, masking, probing security, block cipher, software implementation, polynomial evaluation

Original Publication (in the same form): IACR-CHES-2016
DOI:
10.1007/978-3-662-53140-2_23