Fast Point Multiplication Algorithms for Binary Elliptic Curves with and without Precomputation

In this paper we introduce new methods for computing constant-time variable-base point multiplications over the Galbraith-Lin-Scott (GLS) and the Koblitz families of elliptic curves. Using a left-to-right double-and-add and a right-to-left halve-and-add Montgomery ladder over a GLS curve, we present some of the fastest timings yet reported in the literature for point multiplication. In addition, we combine these two procedures to compute a multicore protected scalar multiplication. Furthermore, we designed a novel regular τ -adic scalar expansion for Koblitz curves. As a result, using the regular recoding approach, we set the speed record for a single-core constant-time point multiplication on standardized binary elliptic curves at the 128-bit security level.