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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 multi-core protected scalar multiplication. Furthermore, we designed a novel regular \(\tau \)-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.

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