Statistical Moment or Cumulative Distribution? Evaluations and Improvements on Side-channel Analysis

Side-channel techniques, such as Distance-of-Means and Kolmogorov-Smirnov analysis, provide valuable insights about leakages from the indirect perspectives of statistical moment and cumulative distribution function (CDF), circumventing the direct and costly estimation of leakage probability densities and thus enabling broad application. Though both the perspectives are informative, their relationship remains unclear – the question of ”which one is better and under what circumstances?” is left open. In this paper, we introduce the probability-probability (PP) plot as a common framework for explaining the mathematical foundations of the CDF-based techniques, facilitating an intuitive understanding of different variant strategies. Then, we propose a novel distinguisher based on the Mann-Kendall test, where the key identification task is reformulated as a goodness-of-fit test checking whether a key-dependent sequence originates from a random binomial distribution. Compared to the existing methods, it delivers substantial gains within the CDF-based family. Finally, we explore the symmetry and dual counterpart of CDF in mathematics, developing an interesting technique based on the inverse cumulative distribution function (ICDF). We present a general discussion of its bridging role and, on this basis, establish the relationships among moment-based, ICDF-based, and CDF-based techniques, thereby enabling the evaluation of CDF-based techniques using metrics originally proposed for the moment-based family. Our theoretical analysis accounts for the empirical observations in practice, and further shows that the improvement achieved by the Mann–Kendall distinguisher is near-optimal.