Constructing hyperelliptic curves of genus 2 suitable for cryptography

DOI10.1090/S0025-5718-02-01422-9zbMath1013.11023MaRDI QIDQ4780381

Annegret Weng

Publication date: 19 November 2002

Published in: Mathematics of Computation (Search for Journal in Brave)

zbMATH Keywords

hyperelliptic curveJacobianCM-methodMestre's algorithm

Mathematics Subject Classification ID

Number-theoretic algorithms; complexity (11Y16) Curves over finite and local fields (11G20) Complex multiplication and moduli of abelian varieties (11G15) Computational aspects of algebraic curves (14Q05) Applications to coding theory and cryptography of arithmetic geometry (14G50)

Related Items

Fast cryptography in genus 2 ⋮ Class invariants for quartic CM fields ⋮ Evaluating Igusa functions ⋮ Construction of CM Picard curves ⋮ A quasi-linear time algorithm for computing modular polynomials in dimension 2 ⋮ A new proof of a Thomae-like formulafor non hyperelliptic genus 3 curves ⋮ Hilbert theta series and invariants of genus 2 curves ⋮ A CM construction for curves of genus 2 with \(p\)-rank 1 ⋮ Computing genus 2 curves from invariants on the Hilbert moduli space ⋮ Generating pairing-friendly parameters for the CM construction of genus 2 curves over prime fields ⋮ Algebraic Description of Jacobians Isogeneous to Certain Prym Varieties with Polarization (1,2) ⋮ Isogeny graphs of ordinary abelian varieties ⋮ Algebraic curves and cryptography ⋮ A generic approach to searching for Jacobians ⋮ Fast genus 2 arithmetic based on Theta functions ⋮ The Birch and Swinnerton-Dyer conjecture for an elliptic curve over \(\mathbb{Q}(\sqrt[4{5})\)] ⋮ Pairing-based algorithms for Jacobians of genus 2 curves with maximal endomorphism ring ⋮ On counting and generating curves over small finite fields ⋮ Curves, Jacobians, and cryptography ⋮ Computing Igusa class polynomials ⋮ Constructing abelian surfaces for cryptography via Rosenhain invariants

Uses Software

• ECPP

Cites Work

• Primality of the number of points on an elliptic curve over a finite field
• Hyperelliptic cryptosystems
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• 2-dimensional simple factors of \(J_ 0(N)\)
• Computing Frobenius maps and factoring polynomials
• On evaluation of \(L\)-functions over real quadratic fields
• Arithmetic variety of moduli for genus two
• Elliptic Curves and Primality Proving
• An improved algorithm for computing logarithms over<tex>GF(p)</tex>and its cryptographic significance (Corresp.)
• Examples of genus two CM curves defined over the rationals
• Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one
• Hyperelliptic Simple Factors of J₀(N) with Dimension at Least 3
• Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura
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