International Association
for Cryptologic Research

We revisit the matrix model for non-interference (NI) probing security of masking gadgets introduced by Belaïd et al. at CRYPTO 2017. This leads to two main results. 1) We generalise the theorems on which this model is based, so as to be able to apply them to masking schemes over any finite field --- in particular GF(2)--- and to be able to analyse the *strong* non-interference (SNI) security notion. We also follow Faust et al. (TCHES 2018) to additionally consider a *robust* probing model that takes hardware defects such as glitches into account. 2) We exploit this improved model to implement a very efficient verification algorithm that improves the performance of state-of-the-art software by three orders of magnitude. We show applications to variants of NI and SNI multiplication gadgets from Barthe et al. (EUROCRYPT~2017) which we verify to be secure up to order 11 after a significant parallel computation effort, whereas the previous largest proven order was 7; SNI refreshing gadgets (ibid.); and NI multiplication gadgets from Gross et al. (TIS@CCS 2016) secure in presence of glitches. We also reduce the randomness cost of some existing gadgets, notably for the implementation-friendly case of 8 shares, improving here the previous best results by 17% (resp. 19%) for SNI multiplication (resp. refreshing).

Designing a block cipher or cryptographic permutation can be approached in many different ways. One such approach, popularized by AES, consists in grouping the bits along the S-box boundaries, e.g., in bytes, and in consistently processing them in these groups. This aligned approach leads to hierarchical structures like superboxes that make it possible to reason about the differential and linear propagation properties using combinatorial arguments. In contrast, an unaligned approach avoids any such grouping in the design of transformations. However, without hierarchical structure, sophisticated computer programs are required to investigate the differential and linear propagation properties of the primitive. In this paper, we formalize this notion of alignment and study four primitives that are exponents of different design strategies. We propose a way to analyze the interactions between the linear and the nonlinear layers w.r.t. the differential and linear propagation, and we use it to systematically compare the four primitives using non-trivial computer experiments. We show that alignment naturally leads to different forms of clustering, e.g., of active bits in boxes, of two-round trails in activity patterns, and of trails in differentials and linear approximations.