In a theoretical context of side-channel attacks, optimal bounds between success rate and guessing entropy are derived with a simple majorization (Schur-concavity) argument. They are further theoretically refined for different versions of the classical Hamming weight leakage model, in particular assuming a priori equiprobable secret keys and additive white Gaussian measurement noise. Closed-form expressions and numerical computation are given. A study of the impact of the choice of the substitution box with respect to side-channel resistance reveals that its nonlinearity tends to homogenize the expressivity of success rate and guessing entropy. The intriguing approximate relation GE = 1/SR is observed in the case of 8-bit bytes and low noise.

At Eurocrypt 2015, Duc et al. conjectured that the success rate of a side-channel attack targeting an intermediate computation encoded in a linear secret-sharing, a.k.a. masking with d+1 shares, could be inferred by measuring the mutual information between the leakage and each share separately. This way, security bounds can be derived without having to mount the complete attack. So far, the best proven bounds for masked encodings were nearly tight with the conjecture, up to a constant factor overhead equal to the field size, which may still give loose security guarantees compared to actual attacks. In this paper, we improve upon the state-of-the-art bounds by removing the field size loss, in the cases of Boolean masking and arithmetic masking modulo a power of two. As an example, when masking in the AES field, our new bound outperforms the former ones by a factor 256. Moreover, we provide theoretical hints that similar results could hold for masking in other fields as well.