Safe-Error Attack on SPA-FA Resistant Exponentiations Using a HW Modular Multiplier

Abstract

The RSA is one of the most widely used algorithms nowadays in smart cards. The main part of RSA is the modular exponentiation composed of modular multiplications. Therefore most smart cards have a hardware modular multiplier to speed up the computation. However, secure implementation of a cryptographic algorithm in an embedded device such as a smart card has now become a big challenge since the advent of side channel analysis and fault attacks. In 2005 Giraud proposed an exponentiation algorithm, which is secure against Simple Power Analysis (SPA) and Fault Attacks (FA). Recently Boscher et al. proposed another SPA-FA resistant exponentiation algorithm. To the authors' best knowledge, only these two provide security against SPA and FA simultaneously in an exponentiation algorithm. Both algorithms are also secure against C safe-error attack and M safe-error attack when they are implemented in a software. However, when they are implemented with a hardware modular multiplier, and this is usual in a smart card, they could be vulnerable to another type of safe error attack. In this paper, we show how this attack is possible on both SPA-FA resistant exponentiation algorithms.