Chapter

Public Key Cryptography – PKC 2010

Volume 6056 of the series Lecture Notes in Computer Science pp 209-223

Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

  • Robert GrangerAffiliated withClaude Shannon Institute School of Computing, Dublin City University
  • , Michael ScottAffiliated withClaude Shannon Institute School of Computing, Dublin City University

Abstract

This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of \(\mathbb{F}_{q^6}^{\times}\), for \(q \equiv 1 \bmod{6}\). Our result arises from considering the Weil restriction of scalars of this group from \(\mathbb{F}_{q^6}\) to \(\mathbb{F}_{q^2}\), and provides efficiency improvements for both pairing-based and torus-based cryptographic protocols. In particular we argue that such fields are ideally suited for the latter when the field characteristic satisfies \(p \equiv 1 \pmod{6}\), and since torus-based techniques can be applied to the former, we present a compelling argument for the adoption of a single approach to efficient field arithmetic for pairing-based cryptography.

Keywords

Pairing-based cryptography torus-based cryptography finite field arithmetic