Scalar multiplication [ n ]P is the kernel and the most time-consuming operation in elliptic curve cryptosystems. In order to improve scalar multiplication, in this paper, we propose a tripling algorithm using Lopez and Dahab projective coordinates, in which there are 3 field multiplications and 3 field squarings less than that in the Jacobian projective tripling algorithm. Furthermore, we map P to φε-1 ( P), and compute [ n ]φε-1 ( P) on elliptic curve Eε , which is faster than computing [ n ]P on E, where φε is an isomorphism. Finally we calculate φε ([ n ]φε-1 ( P))= [ n ]P. Combined with our efficient point tripling formula, this method leads scalar multiplication using double bases to achieve about 23% improvement, compared with Jacobian projective coordinates.
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