A duality in space usage between left-to-right and right-to-left exponentiation

Abstract

Most exponentiation algorithms are categorised as being left-to-right or right-to-left because of the order in which they use the digits of the exponent. There is clear value in having a canonical way of transforming an algorithm in one direction into an algorithm in the opposite direction: it may lead to new algorithms, different implementations of existing algorithms, improved side-channel resistance, greater insights. There is already an historic duality between left-to-right and right-to-left exponentiation algorithms which shows they take essentially the same time, but it does not treat the space issues that are always so critical in resource constrained embedded crypto-systems. To address this, here is presented a canonical duality which preserves both time and space. As an example, this is applied to derive a new, fast yet compact, left-to-right algorithm which makes optimal use of recently developed composite elliptic curve operations. © 2012 Springer-Verlag.