Research on efficient pairing implementation has focussed on reducing the loop length and on using higher-order twists. Existence of twists of degree larger than 2 is a very
restrictive criterion but luckily constructions for pairing-friendly elliptic curves with high-order twists exist. In fact, Freeman, Scott and Teske showed in their overview paper that often the best known methods of constructing pairing-friendly elliptic curves over fields of large prime characteristic produce curves that admit twists of degree 3, 4 or 6.
A few papers have presented explicit formulas for the doubling and the addition step in Miller’s algorithm, but the optimizations were all done for the Tate pairing with degree-2 twists, so the main usage of the high-order twists remained incompatible with more efficient formulas.
In this paper we present efficient formulas for curves with twists of degree 2, 3, 4 or 6. These formulas are significantly faster than their predecessors. We show how these faster formulas can be applied to Tate and ate pairing variants, thereby speeding up all practical suggestions for efficient pairing implementations over fields of large characteristic.
|Number of pages||16|
|Publication status||Published - 2009|
|Name||Cryptology ePrint Archive|