New fast algorithms for elliptic curve arithmetic in affine coordinates

Abstract

We present new algorithms computing 3P and 2P + Q by removing the same part of numerators and denominators of their formulas, given two points P and Q on elliptic curves defined over prime fields and binary fields in affine coordinates. Our algorithms save one or two field multiplications compared with ones presented by Ciet, Joye, Lauter, and Montgomery. Since 2P + Q takes (formula presented) proportion, 28.5% proportion, and 25.8% proportion of all point operations by non-adjacent form, binary/ternary approach and tree approach to compute scalar multiplications respectively, 3P occupies 42.9% proportion and 33.4% proportion of all point operations by binary/ternary approach and tree approach to compute scalar multiplications respectively, utilizing our new formulas of 2P + Q and 3P, scalar multiplications by using non-adjacent form, binary/ternary approach and tree approach are improved.