On the (Im)possibility of Preventing Differential Computation Analysis with Internal Encodings

White-box cryptography aims at protecting implementations of cryptographic algorithms against a very powerful attacker who controls the execution environment. The first defensive brick traditionally embedded in such implementations consists of encodings, which are bijections supposed to conceal sensitive data manipulated by the white-box. Several previous works have sought to evaluate the relevance of encodings to protect white-box implementations against grey-box attacks such as Differential Computation Analysis (DCA). However, these works have been either probabilistic or partial in nature. In particular, while they showed that DCA succeeds with high probability against AES white-box implementations protected by random encodings, they did not refute the existence of a particular class of encodings that could prevent the attack. One could thus wonder if carefully crafting specific encodings instead of drawing random bijections could be a solution.This article bridges the gap between preceding research efforts and investigates this question. We first focus on the protection of the S-box output and we show that no 4-bit encoding can actually protect this sensitive value against side-channel attacks. We then argue that the use of random 8-bit encodings is both necessary and sufficient, but that this assertion holds exclusively for the S-box output. Indeed, while we define a class of 8-bit encodings that actually prevents a classical DCA targeting the MixColumns output, we also explain how to adapt this attack and exploit the correlation traces in order to defeat even these specific encodings. Our work thus rules out the existence of a set of practical encodings that could be used to protect an AES white-box implementation against DCA-like attacks.