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Techniques for fast exponentiation (multiplication) in various groups have been extensively studied for use in cryptographic primitives. Specifically, the coding of the exponent (multiplier) plays an important role in the performances of the algorithms used. The crucial optimization relies in general on minimizing the Hamming weight of the exponent (multiplier). This can be performed optimally with nonadjacent representations. This paper introduces a compact encoding of non-adjacent representations that allows to skip the exponent recoding step. Furthermore, a straightforward technique for picking random numbers that alreadysatisfythe non-adjacence propertyis proposed. Several examples of application are given, in particular in the context of scalar multiplication on elliptic curves.

Compact Encoding of Non-adjacent Forms with Applications to Elliptic Curve Cryptography