Provable Order Amplification for Code-based Masking: How to Avoid Non-linear Leakages due to Masked Operations

Code-based masking schemes have been shown to provide higher theoretical security guarantees than the Boolean masking. In particular, one interesting feature put forward at CARDIS 2016 and then analyzed at CARDIS 2017 is the so-called security order amplification: under the assumption that the leakage function is linear, it guarantees that an implementation performing only linear operations will have a security order in the bounded moment leakage model larger than d - 1, where d is the number of shares. The main question regarding this feature is its practical relevance. First of all, concrete block ciphers do not only perform linear operations. Second, it may be that actual leakage functions are not perfectly linear (raising questions regarding what happens when one deviates from such assumptions). In this paper, we show that the issue of only linear operations can be provably avoided, and that it is possible to obtain security order amplification for any functionality to implement. We then show that (not so) slightly non-linear leakage functions do not annihilate the nice properties (i.e., that the code-based schemes we consider remain interesting compared to the Boolean masking). We conclude with a performance evaluation of the proposals, showing that the performance overheads are moderate for a reasonable number of shares (we studied when the number of the shares d = 2; 3; 4). Additionally, our results could be specified to the case of provable security for low entropy masking, which can be considered as a side bonus of our contributions. We give some preliminary results on how to construct low entropy masking schemes with provable high security order against linear leakage