Squaring in cyclotomic subgroups

Abstract

We propose new squaring formulae for cyclotomic subgroups of the multiplicative group of certain finite fields. Our formulae use a compressed representation of elements having the property that decompression can be performed at a very low cost. The squaring formulae lead to new exponentiation algorithms in cyclotomic subgroups which outperform the fastest previouslyknown exponentiation algorithms when the exponent has low Hamming weight. Our algorithms can be adapted to accelerate the final exponentiation step of pairing computations. ? 2012 American Mathematical Society.