Efficient computation of pairings on Jacobi quartic elliptic curves
type: Item
instance of: scholarly article
title: Efficient computation of pairings on Jacobi quartic elliptic curves
zbMATH Open document ID: 1311.14032
author: Sylvain Duquesne
author: Nadia El Mrabet
author: Emmanuel Fouotsa
published in: Journal of Mathematical Cryptology
publication date: 2015-01-22T00:00:00Z
review text: The article is devoted to describe and analyze an algorithm for the computation of pairings on the Jacobi quartic elliptic curve \(E_d\) defined by \(Y^2=dX^4+Z^4\) over a finite field \(\mathbb{F}_q\) of characteristic \(p\geq 5\) not congruent to 3 modulo 4. Let \(E\) be an elliptic curve defined over \(\mathbb{F}_q\) and denote by \(E(\mathbb{F}_q)\) the group of \(\mathbb{F}_q\)-rational points of \(E\). Let \(r\) be a prime divisor of the group order \(\#E(\mathbb{F}_q)\) and \(k\) the embedding degree of \(E\) with respect to \(r\), that is, the smallest integer for which \(r\) divides \(q^k-1\). The set of \(r\)-torsion points \(E(\overline{\mathbb{F}}_q)[r]\) with coordinates in an algebraic closure \(\overline{\mathbb{F}}_q\) of \(\mathbb{F}_q\) is contained in \(E(\mathbb{F}_{q^k})\). The reduced pairing \(e_r:E(\mathbb{F}_q)[r]\times E(\mathbb{F}_{q^k})[r]\to\mu_r\) is a bilinear and non-degenerate map on the group \(\mu_r\) of \(r\)th roots of unity in \(\mathbb{F}^*_{q^k}\). The paper under review is concerned with the computation of the Tate pairing \(e_r\), as well as Ate, twisted Ate and optimal pairings, on the Jacobi quartic elliptic curve \(E_d\). For this purpose, the authors consider the Miller algorithm, an efficient method which requires the computation of addition and doubling in \(E(\mathbb{F}_{q^k})[r]\) ([\textit{V. Miller}, \url{http://crypto.stanford.edu/miller/miller.pdf}]). In [Chin. J. Electronics 20, No. 4, 655--661 (2011), \url{http://eprint.iacr.org/2010/475.pdf}], H. Wang et al. presented a geometric interpretation of the group law on Jacobi quartic curves which leads to an improvement of Miller's algorithm for this kind of curves. In the article under review a further interpretation of the group law is obtained, which is based on the birational equivalence between Jacobi quartic curves and Weierstrass curves. This allows the authors to improve the results of H. Wang et al. [loc. cit.], for the case of the curves \(E_d\). The auth…
reviewed by: Guillermo Matera
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 68W40
Mathematics Subject Classification ID: 94A60
zbMATH DE Number: 6389265
zbMATH Keywords: Jacobi quartic curves
zbMATH Keywords: Tate pairing
zbMATH Keywords: Ate pairing
zbMATH Keywords: twists
zbMATH Keywords: Miller algorithm
zbMATH Keywords: computation
MaRDI profile type: Publication
full work available at URL: jmc-2013-0033
OpenAlex ID: W2050216354
DOI: 10.1515/JMC-2013-0033
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description: scientific article; zbMATH DE number 6389265