Public Key Cryptography – PKC 2010

Volume 6056 of the series Lecture Notes in Computer Science pp 224-242

Faster Pairing Computations on Curves with High-Degree Twists

  • Craig CostelloAffiliated withInformation Security Institute, Queensland University of Technology
  • , Tanja LangeAffiliated withDepartment of Mathematics and Computer Science, Technische Universiteit Eindhoven
  • , Michael NaehrigAffiliated withDepartment of Mathematics and Computer Science, Technische Universiteit Eindhoven


Research on efficient pairing implementation has focussed on reducing the loop length and on using high-degree twists. Existence of twists of degree larger than 2 is a very restrictive criterion but luckily constructions for pairing-friendly elliptic curves with such 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-degree 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.


Pairings Miller functions explicit formulas Tate pairing ate pairing twists Weierstrass curves