Four-Dimensional Gallant–Lambert–Vanstone Scalar Multiplication

The GLV method of Gallant, Lambert, and Vanstone (CRYPTO 2001) computes any multiple kP of a point P of prime order n lying on an elliptic curve with a low-degree endomorphism Φ (called GLV curve) over 𝔽_p as kP = k_1P + k_2(P) with max{ |k_1|,|k_2| }≤ C_1√(n) for some explicit constant C 1 >0. Recently, Galbraith, Lin, and Scott (EUROCRYPT 2009) extended this method to all curves over 𝔽_p^2 which are twists of curves defined over 𝔽_p . We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over 𝔽_p^2 , a four-dimensional decomposition together with fast endomorphisms Φ , Ψ over 𝔽_p^2 acting on the group generated by a point P of prime order n , resulting in a proven decomposition for any scalar k ∈[1, n ] given by kP=k_1P+ k_2(P)+ k_3(P) + k_4(P) with max_i (|k_i| )< C_2 n^1/4 for some explicit C 2 >0. Remarkably, taking the best C 1 , C 2 , we obtain C 2 / C 1 <412, independently of the curve, ensuring in theory an almost constant relative speedup. In practice, our experiments reveal that the use of the merged GLV–GLS approach supports a scalar multiplication that runs up to 1.5 times faster than the original GLV method. We then improve this performance even further by exploiting the Twisted Edwards model and show that curves originally slower may become extremely efficient on this model. In addition, we analyze the performance of the method on a multicore setting and describe how to efficiently protect GLV-based scalar multiplication against several side-channel attacks. Our implementations improve the state-of-the-art performance of scalar multiplication on elliptic curves over large prime characteristic fields for a variety of scenarios including side-channel protected and unprotected cases with sequential and multicore execution.