Estimating the Signal-to-Noise Ratio Under Repeated Sampling of the Same Centered Signal: Applications to Side-Channel Attacks on a Cryptoprocessor

This paper introduces an estimator of the signal-to-noise ratio in the framework where a noisy source emits the same signal a number n of times. The estimator has the structure of a U-statistic from which derives many desirable properties: it is unbiased, consistent and, being a Rao-Blackwellisation of existing proposals, is closer to optimal variance-wise. However, its variance is numerically difficult to evaluate and two approximations are obtained to facilitate its use in practice. These allow quantifying the improvement in variance, which is found to be substantial as the estimator needs roughly one-third of the data previously required to perform similarly. Moreover, a simulation shows that the estimator is approximately normally distributed for n as small as 10, which allows for accurate inference. The estimator is then applied to data arising in a cryptanalysis, where the numerical security of a cryptoprocessor is tested against a side-channel attack. This problem is a representative of situations where the signal-to-noise ratio must be precisely estimated for small n. We derive a rigorous data-driven approach that is shown to much enhance the efficiency of standard side-channel attacks.