Exact Template Attacks with Spectral Computation

The so-called Gaussian template attacks (TA) is one of the optimal Side-Channel Analyses (SCA) when the measure-ments are captured with normal noise. In the SCA literature, several optimizations of its implementation are introduced, such as coalescence and spectral computation. The coalescence consists of averaging traces corresponding to the same plaintext value, thereby coalescing (synonymous: compacting) the dataset. Spec-tral computation consists of sharing the computational workload when estimating likelihood across key hypotheses. State-of-the-art coalescence leverages the Law of Large Num-bers (LLN) to compute the mean of equivalent traces. This approach comes with a drawback because the LLN is just an asymptotic approximation. So it does not lead to an exact Template Attack, especially for a few number of traces. In this paper, we introduce a way of calculating the TA exactly and with the same computational complexity (using the spectral approach), without using the LLN, regardless of the number of messages. For the experimental validation of this approach, we use the ANSSI SCA Database (ASCAD), with different numbers of messages and different amounts of samples per trace. Recall that this dataset concerns a software implementation of AES-128 bits, running on an ATMEGA-8515 microprocessor.