Preventing differential analysis in GLV elliptic curve scalar multiplication

Abstract

In [2], Gallant, Lambert and Vanstone proposed a very efficient algorithm to compute Q = kP on elliptic curves having non-trivial efficiently computable endomorphisms. Cryptographic protocols are sensitive to implementations, indeed as shown in [6,7] information about the secret can be revealed analysing external leakage of the support, typically a smart card. Several software countermeasures have been proposed to protect the secret. However, speed computation is needed for practical use. In this paper, we propose & method to protect scalar multiplication on elliptic curves against Differential Analysis, that benefits from the speed of the Gallant, Lambert and Vanstone method. It can be viewed as a two-dimensional analogue of Coron's method [1] of randomising the exponent k. We propose two variants of this method (one linear and one affine), the second one slightly more effective, whereas the first one offers "two in one", combining point-blinding and exponent randomisation, which have hitherto been dealt separately. For instance, for at most a mere 37.5% (resp. 25%) computation speed loss on elliptic curves over fields with 160 (resp. 240) bits the computation of kP can take on 2 40 different consumption patterns. © Springer-Verlag 2003.