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We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, which uses affine arithmetic and is resistant against simple power attacks. Also, using a modification of known techniques the algorithm can be made immune against differential power attacks. The algorithm uses Montgomery’s trick and a precomputed table consisting of multiples of the base point. Consequently, the algorithm is useful in a scenario where the base point is fixed, like Elgamal encryption or signature generation. Under such circumstances, for hyperelliptic curves, the algorithm compares favourably with other known algorithms over all fields. For elliptic curves, under similar circumstances, the algorithm performs better than other algorithms over prime fields. The increase in speed is due to a proper application of Montgomery’s trick to efficiently perform the simultaneous inversion of several field elements.

Application of Montgomery’s Trick to Scalar Multiplication for Elliptic and Hyperelliptic Curves Using a Fixed Base Point