Avoiding Full Extension Field Arithmetic in Pairing Computations

The most costly operations encountered in pairing computations are those that take place in the full extension field Fpk">Fpk

${\mathbb{F}}_{p^k}$

. At high levels of security, the complexity of operations in Fpk">Fpk

${\mathbb{F}}_{p^k}$

dominates the complexity of the operations that occur in the lower degree subfields. Consequently, full extension field operations have the greatest effect on the runtime of Miller’s algorithm. Many recent optimizations in the literature have focussed on improving the overall operation count by presenting new explicit formulas that reduce the number of subfield operations encountered throughout an iteration of Miller’s algorithm. Unfortunately, almost all of these improvements tend to suffer for larger embedding degrees where the expensive extension field operations far outweigh the operations in the smaller subfields. In this paper, we propose a new way of carrying out Miller’s algorithm that involves new explicit formulas which reduce the number of full extension field operations that occur in an iteration of the Miller loop, resulting in significant speed ups in most practical situations of between 5 and 30 percent.