Speeding Up Pairing Computations on Genus 2 Hyperelliptic Curves with Efficiently Computable Automorphisms

Abstract

Abstract. Pairings on the Jacobians of (hyper-)elliptic curves have received considerable attention not only as a tool to attack curve based cryptosystems but also as a building block for constructing cryptographic schemes with new and novel properties. Motivated by the work of Scott [34], we investigate how to use efficiently computable automorphisms to speed up pairing computations on two families of non-supersingular genus 2 hyperelliptic curves over prime fields. Our findings lead to new variants of Miller's algorithm in which the length of the main loop can be up to 4 times shorter than that of the original Miller's algorithm in the best case. We also implement the calculation of the Tate pairing on both a supersingular and a non-supersingular genus 2 curve with the same embedding degree of k = 4. Combining the new algorithm with known optimization techniques, we show that pairing computations on non-supersingular genus 2 curves over primes fields use up to 56.2% fewer field operations and run about 10% faster than supersingular genus 2 curves for the same security level.

Faster pairing computation on genus 2 hyperelliptic curves

“…Assume that the eigenvalue ofψ 2 on G p is p. To keep the consistency with notations in other lectures, we write G 2 for G p . Note that a λ (·,·) be the pairing in Theorem 2 proposed by Fan et al's [10]. Then the following theorem gives the main result in this paper.…”

“…Fan et al [10] speeded up pairing computation on genus 2 hyperelliptic curves with efficiently computable automorphisms and implemented their pairings on these hyperelliptic curves with the embedding degree 4. When the embedding degree k is divided by 4, we find more properties of automorphisms of these hyperelliptic curves.…”

“…Zhang [27] proposed Ate i pairing and twisted Ate pairing on hyperelliptic curves. Fan et al [10] implemented pairing computation on genus 2 hyperelliptic curves with 0020-0190/$ -see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2011.02.011 efficiently computable automorphisms.…”

“…T 10 The cost of computing f in Step 10 of Algorithm 1. T 11 The cost of computing f in Step 11 of Algorithm 1.…”

Criptografia de chave pública sem certificado

“…Desde então, o modelo recebeu cada vez mais atenção dos pesquisadores, que passaram a acres-2.2 centar melhorias em várias frentes: aumento da velocidade do cálculo de emparelhamento por Fan et al (2008), redução do tamanho da chave com Naccache (2007), e aumento do nível de segurança com demonstrações sem a hipótese de oráculos aleatórios, que é o caso do esquema proposto por Waters (2005). Paralelamente, surgiram inúmeros protocolos e aplicações interessantes; só para citar alguns:…”