Optimal First-Order Masking with Linear and Non-linear Bijections

Hardware devices can be protected against side-channel attacks by introducing one random mask per sensitive variable. The computation throughout is unaltered if the shares (masked variable and mask) are processed concomitantly, in two distinct registers. Nonetheless, this setup can be attacked by a zero-offset second-order CPA attack. The countermeasure can be improved by manipulating the mask through a bijection F, aimed at reducing the dependency between the shares. Thus dth-order zero-offset attacks, that consist in applying CPA on the dth power of the centered side-channel traces, can be thwarted for d≥2 at no extra cost. We denote by n the size in bits of the shares and call F the transformation function, that is a bijection of $\mathbb{F}_2^n$. In this paper, we explore the functions F that thwart zero-offset HO-CPA of maximal order d. We mathematically demonstrate that optimal choices for F relate to optimal binary codes (in the sense of communication theory). First, we exhibit optimal linear F functions. Second, we note that for values of n for which non-linear codes exist with better parameters than linear ones. These results are exemplified in the case n=8, the optimal F can be identified:it is derived from the optimal rate 1/2 binary code of size 2n, namely the Nordstrom-Robinson (16, 256, 6) code. This example provides explicitly with the optimal protection that limits to one mask of byte-oriented algorithms such as AES or AES-based SHA-3 candidates. It protects against all zero-offset HO-CPA attacks of order d≤5. Eventually, the countermeasure is shown to be resilient to imperfect leakage models.