Group arithmetic in C3, 5C3, 5 curves

Abstract In this paper we present a fast addition algorithm in the Jacobian of a C 3 , 5 curve over a finite field F q . We give formulae for D 1 ⊕ D 2 = − ( D 1 + D 2 ) which require 2 I + 264 M + 10 S when D 1 ≠ D 2 and 2 I + 297 M + 13 S when D 1 = D 2 ; and for the computation of −D which require 2 I + 41 M + 3 S . The ⊕ operation is sufficient to compute scalar multiplications after performing a single (initial) −D. Computing the scalar multiplication [ k ] D , based on the previous fact combined with our algorithm for computing D 1 ⊕ D 2 , is to date the fastest one performing this operation for C 3 , 5 curves. These formulae can be easily combined to compute the full group addition and doubling in 3 I + 308 M + 13 S and 3 I + 341 M + 16 S respectively, which compares favorably with previously presented formulae.

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