Group arithmetic in C3,5C3,5 curves

In this paper we present a fast addition algorithm in the Jacobian of a C3,5C3,5 curve over a finite field FqFq. We give formulae for D1⊕D2=−(D1+D2)D1⊕D2=−(D1+D2) which require 2I+264M+10S2I+264M+10S when D1≠D2D1≠D2 and 2I+297M+13S2I+297M+13S when D1=D2D1=D2; and for the computation of −D   which require 2I+41M+3S2I+41M+3S. The ⊕ operation is sufficient to compute scalar multiplications after performing a single (initial) −D  . Computing the scalar multiplication [k]D[k]D, based on the previous fact combined with our algorithm for computing D1⊕D2D1⊕D2, is to date the fastest one performing this operation for C3,5C3,5 curves. These formulae can be easily combined to compute the full group addition and doubling in 3I+308M+13S3I+308M+13S and 3I+341M+16S3I+341M+16S respectively, which compares favorably with previously presented formulae.