A Stochastic Model for Differential Side Channel Cryptanalysis

Abstract

Abstract. This contribution presents a new approach to optimize the efficiency of differential side channel cryptanalysis against block ciphers by advanced stochastic methods. We approximate the real leakage function within a suitable vector subspace. Under appropriate conditions profiling requires only one test key. For the key extraction we present a 'minimum principle' that solely uses deterministic data dependencies and the 'maximum likelihood principle' that additionally incorporates the characterization of the noise revealed during profiling. The theoretical predictions are accompanied and confirmed by experiments. We demonstrate that the adaptation of probability densities is clearly advantageous regarding the correlation method, especially, if multiple leakage signals at different times can be jointly evaluated. Though our efficiency at key extraction is limited by template attacks profiling is much more efficient which is highly relevant if the designer of a cryptosystem is bounded by the number of measurements in the profiling step.

Templates vs. Stochastic Methods

“…The Stochastic Model [7] assumes that the physical observable I t (x, k) at time t is composed of two parts, a data-dependent part h t (x, k) as a function of known data x and subkey k and a noise term R t with zero mean:…”

“…For this paper, we use the maximum likelihood based approach of [7] and skip the minimum principle as it is already proven to be less efficient in [7]. Profiling processes N = N 1 +N 2 samples representing a known subkey k and known data x 1 , x 2 , .…”

“…We chose the vector subspace F 9 , i.e., bitwise coefficients at the S-Box outcome as selection function as suggested by [7]. The base vectors…”

“…Table 1 summarizes the fundamental differences in the approaches of both attacks. Following the notation in [7], Templates estimate the data-dependent part h t itself, whereas the Stochastic model approximates the linear part of h t in the chosen vector subspace (e.g., F 9 ) and is not capable of including non-linear parts. Templates build a covariance matrix for each key dependency whereas the Stochastic Model generates only one covariance matrix, hereby neglecting possible multivariate key dependent noise terms.…”

“…The methods under consideration are the Template Attack [3] and the Stochastic Model [7]. Both methods include a profiling step for the estimation of a key dependent multivariate probability density of the physical observable.…”