Two is the fastest prime: lambda coordinates for binary elliptic curves

In this work, we present new arithmetic formulas for a projective version of the affine point representation $$(x,x+y/x),$$(x,x+y/x), for $$x\ne 0,$$x≠0, which leads to an efficient computation of the scalar multiplication operation over binary elliptic curves. A software implementation of our formulas applied to a binary Galbraith–Lin–Scott elliptic curve defined over the field $$\mathbb {F}_{2^{254}}$$F2254 allows us to achieve speed records for protected/unprotected single/multi-core random-point elliptic curve scalar multiplication at the 127-bit security level. When executed on a Sandy Bridge 3.4 GHz Intel Xeon processor, our software is able to compute a single/multi-core unprotected scalar multiplication in 69,500 and 47,900 clock cycles, respectively, and a protected single-core scalar multiplication in 114,800 cycles. These numbers are improved by around 2 and 46 % on the newer Ivy Bridge and Haswell platforms, respectively, achieving in the latter a protected random-point scalar multiplication in 60,000 clock cycles.