In side-channel analysis (SCA), Points-of-Interest (PoIs), i.e., the informative sample points remain sparsely scattered across the whole side-channel trace. Several works in the SCA literature have demonstrated that the attack efficacy could be significantly improved by combining information from the sparsely occurring PoIs. In Deep Learning (DL), a common approach for combining the information from the sparsely occurring PoIs is softmax attention. This work studies the training instability of the softmax attention-based CNN models on long traces. We show that the softmax attention-based CNN model incurs an unstable training problem when applied to longer traces (e.g., traces having a length greater than$10K$sample points). We also explore the use of batch normalization and multi-head softmax attention to make the CNN models stable. Our results show that the use of a large number of batch normalization layers and/or multi-head softmax attention (replacing the vanilla softmax attention) can make the models significantly more stable, resulting in better attack efficacy. Moreover, we found our models to achieve similar or better results (up to 85% reduction in the minimum number of the required traces to reach the guessing entropy 1) than the state-of-the-art results on several synchronized and desynchronized datasets. Finally, by plotting the loss surface of the DL models, we demonstrate that using multi-head softmax attention instead of vanilla softmax attention in the CNN models can make the loss surface significantly smoother.

Many profiling differential power analysis (DPA) attacks estimate the multivariate probability distribution using a profiling step, and thus, can optimally combine the leakages of multiple sample points. Though there exist several approaches like filtering or principal component analysis for combining the leakages of multiple sample points in nonprofiling DPA, their optimality has been rarely studied. We study the issue of optimally combining the leakages of multiple sample points in nonprofiling DPA attacks using a linear function. In this paper, we introduce a multivariate leakage model based on some observations obtained by profiling the power traces of Advanced Encryption Standard (AES) encryption on Virtex-5 field programmable gate array (FPGA) device. Then, we use the introduced multivariate leakage model to propose optimal combining functions for nonprofiling DPA. The theoretical claims are supported by experimental evidence. We have also discussed different sides of the proposed combining functions in various practical scenarios.