Ordinary Pairing-Friendly Genus 2 Hyperelliptic Curves with Absolutely Simple Jacobians

Abstract

We present a method for producing pairing-friendly, simple, ordinary Jacobian varieties of genus 2 hyperelliptic curves defined over a prime field F p . The proposed method heavily relies on the construction of a suitable p-Weil number and a corresponding quartic CM-field. Our Jacobians are absolutely simple and for this special class of Jacobians we give the first examples in the literature with ρ-values below 4, while previous results had in general ρ-values between 6 and 8. These examples derive from "families" of pairing-friendly Jacobians, which are basically polynomial representations of the Jacobian parameters.An asymmetric pairing is a bilinear, non-degenerate, efficiently computable map e : G 1 × G 2 −→ G T , where G 1 , G 2 , G T are cyclic groups of prime order r with G 1 = G 2 . A crucial cryptographic requirement is that the discrete logarithm problem (DLP) is computationally infeasible in all pairing groups G 1 , G 2 , G T . We call G 1 , G 2 the source groups and G T the target group. Initially the source groups were set as r-order subgroups of ordinary elliptic curves over a finite field, while the target group was an r-order subgroup of a finite field.Since elliptic curves are genus 1 algebraic curves, an obvious question is whether pairings on higher genus curves can be also used in implementations. In this case G 1 and G 2 consist of elements in the Jacobian variety of a genus g hyperelliptic curve defined over a finite field. By [Ber06, Lan06], this can be an advantageous choice especially when g = 2, since genus 2 curves and their Jacobians: 1. are competitive to elliptic curves in performance and security [Ber06, Lan06]. 2. result in efficient Tate pairing calculations [FL06]. 3. have efficient CM-constructions [LS13, Wen02] and point operations [Can87]. 4. have points with smaller size.This is our motivation for constructing "pairing-friendly", ordinary Jacobians of genus 2 hyperelliptic curves over a prime field F p , called the base field.

Constructing suitable parameters for pairing-based cryptography

“…In Section 5.4 we further investigated the construction of non-absolutely simple abelian varieties of dimension 2, taken from our work in [Fotiadis and Konstantinou, 2017b]. Examples in this case are very rare and the existing ones have in general large ρ-values.…”

“…We summarize the most famous methods and emphasize on our results on nonabsolutely simple, ordinary g-dimensional abelian varieties [Fotiadis and Konstantinou, 2017a]. In the case of absolutely simple abelian varieties, we are restricted in 2-dimensional Jacobians of genus 2 hyperlliptic curves were we describe our results obtained by our most recent work [Fotiadis and Konstantinou, 2017b].…”

“…All these are examples of non-absolutely simple abelian varieties. Additionally we study the case of absolutely simple abelian varieties and present the best examples so far in the literature in terms of the ρ-value [Fotiadis and Konstantinou, 2017b].…”

“…This study is presented in Section 5.3. Finally, in Section 5.4 we present a more recent work of ours based on constructing 2-dimensional Jacobian varieties [Fotiadis and Konstantinou, 2017b].…”

“…In this section we specialize on the construction of pairing-friendly two-dimensional Jacobians defined over prime fields F p [Fotiadis and Konstantinou, 2017b]. All families presented in Section 5.3 correspond to Jacobian varieties of dimensions 2, 3 and 4.…”