Comparing Approaches to Rank Estimation for Side-Channel Security Evaluations

Abstract

Rank estimation is an important tool for side-channel evaluations laboratories. It allows determining the remaining security after an attack has been performed, quantified as the time complexity required to brute force the key given the leakages. Several solutions to rank estimation have been introduced in the recent years. In this paper, we first clarify the connections between these solutions, by organizing them according to their (maximum likelihood or weak maximum likelihood) strategy and whether they take as argument a side-channel distinguishers’ output or some evaluation metrics. This leads us to introduce new combinations of these approaches, and to discuss the use of weak maximum likelihood strategies for suboptimal but highly parallel enumeration. Next, we show that the different approaches to rank estimation can also be implemented with different mixes of very similar tools (e.g. histograms, convolutions, combinations and subsampling). Eventually, we provide various experiments allowing to discuss the pros and cons of these different approaches, hence consolidating the literature on this topic.

A Bound from the Metric-Based Maximum Likelihood

We start from the observation that if the lists of probabilities obtained for each S-box are independent, $rank(k_0)=d_0$, $\dots$, $rank(k_{N_s'})=d_{N_s'}$ and $rank(k)=d$, then $d\ge \prod _i d_i$. When estimating a master key success rate, it implies that:

$$\widehat{SR}[d] \le \sum _{d_i, \prod _i d_i\le d} \varDelta SR_0[d_0]\times \dots \times \varDelta SR_{N_s'}[d_{N_s'}].$$

Algorithm 4 just computes such a sum of products of derivative success rates.

Note that the independence condition for the S-box probabilities in this bound is not the same as the usual independent leakages considered, e.g. in leakage-resilience cryptography [7], which is a purely physical condition. Here, we only need that the adversary considers the subkeys independently.