Constructing elliptic curves over finite fields using double eta-quotients
type: Item
instance of: scholarly article
title: Constructing elliptic curves over finite fields using double eta-quotients
zbMATH Open document ID: 1072.11039
author: Andreas Enge
author: Reinhard Schertz
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-07-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10670
review text: For several applications (cryptography, factorizing, prime proving) one needs elliptic curves \(E\) over a finite field \(\mathbb F\), where e.g. \(| \mathbb F| \) and/or \(| E(\mathbb F)| \) have large prime factors. Usually this is accomplished by reducing a global elliptic curve with complex multiplication by \(\mathfrak o\), which in turn is obtained from the ring class field \(K\) of the imaginary quadratic order \(\mathfrak o\). The known methods to construct \(K\) by singular values of modular functions may fail for special types of discriminants, or yield minimal polynomials with too large coefficients. For their new approach, the authors investigate a class of modular functions for \(\Gamma_0 (N)\) (\(N=p_1 p_2\) with primes \(p_1,p_2\)), so-called double eta quotients \[ \mathfrak w_{p_1,p_2} (z) = \frac {\eta(z/p_1) \eta(z/p_2)} {\eta(z) \eta(z/N)}, \] where \(\eta\) denotes Dedekind's eta-function. These yield an explicit construction of the ring class field (without restrictions on the discriminant) and class polynomials with rather small coefficients. In general, these double eta quotients do no longer generate an extension field of \(\mathbb C (j)\) of genus \(0\), but the authors suggest a way how to use the modular polynomial giving the algebraic relation between \(\mathfrak w_{p_1,p_2}\) and \(j\) (with coefficients from \(\mathbb Z\)) to obtain the \(j\)-invariant of the elliptic curve over \(\mathbb F\). The paper concludes with a numerical example of cryptographic size.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11F20
zbMATH DE Number: 2188531
zbMATH Keywords: Weber function
zbMATH Keywords: ring class field
zbMATH Keywords: modular function
describes a project that uses: NTL
describes a project that uses: MPFR
describes a project that uses: ECPP
describes a project that uses: MPC
MaRDI profile type: Publication
OpenAlex ID: W2019135503
cites work: Elliptic Curves and Primality Proving
cites work: Q5521595
cites work: Q3996190
cites work: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
cites work: Q5735284
cites work: Elliptic curves and their applications to cryptography. An introduction
cites work: Q4737490
cites work: Generalised Weber functions
cites work: Modular curves of composite level
cites work: Q4737492
cites work: MPFR
cites work: Elliptic Curve Cryptosystems
cites work: Q4847937
cites work: Factoring integers with elliptic curves
cites work: Q3718617
cites work: Construction and Application of a Class of Modular Functions (II)†
cites work: Q913858
cites work: Weber's class invariants revisited
cites work: Q3710637
DOI: 10.5802/JTNB.460
description: scientific article