Side-Channel Attacks on Threshold Implementations using a Glitch Algebra

Threshold implementations allow to implement circuits us- ing secret sharing in a way to thwart side-channel attacks based on prob- ing or power analysis. It was proven they resist to attacks based on glitches as well. In this report, we show the limitations of these results. Concretely, this approach proves security against attacks which use the average power consumption of an isolated circuit. But there is no security provided against attacks using a non-linear function of the power traces (such as the mean of squares or the majority of a threshold function), and there is no security provided for cascades of circuits, even with the power mean. We take as an example the threshold implementation of the AND function by Nikova, Rechberger, and Rijmen with 3 and 4 shares. We further consider a proposal for higher-order by Bilgin et al.