Computing the endomorphism ring of an ordinary elliptic curve over a finite field

DOI10.1016/j.jnt.2009.11.003zbMath1225.11085arXiv0902.4670OpenAlexW2081640523MaRDI QIDQ2430982

Andrew V. Sutherland, Gaetan Bisson

Publication date: 8 April 2011

Published in: Journal of Number Theory (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0902.4670

zbMATH Keywords

finite fieldsendomorphism ringordinary elliptic curves

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Curves over finite and local fields (11G20) Algebraic number theory computations (11Y40) Applications to coding theory and cryptography of arithmetic geometry (14G50)

Related Items (28)

Fast computation of elliptic curve isogenies in characteristic two ⋮ A quasi-linear time algorithm for computing modular polynomials in dimension 2 ⋮ Improved algorithm for the isogeny problem for ordinary elliptic curves ⋮ Distorting the volcano ⋮ -adic images of Galois for elliptic curves over (and an appendix with John Voight) ⋮ Subrings of \(p\)-power index in endomorphism rings of simple abelian varieties over finite fields ⋮ Computing the endomorphism ring of an ordinary abelian surface over a finite field ⋮ Computing isogeny volcanoes of composite degree ⋮ Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation ⋮ Modular polynomials via isogeny volcanoes ⋮ A low-memory algorithm for finding short product representations in finite groups. ⋮ On the computation of the endomorphism rings of abelian surfaces ⋮ On the elliptic curve endomorphism generator ⋮ Pairing the volcano ⋮ Computational problems in supersingular elliptic curve isogenies ⋮ Accelerating the CM method ⋮ Computing separable isogenies in quasi-optimal time ⋮ A Subexponential Algorithm for Evaluating Large Degree Isogenies ⋮ Analogues of Vélu’s formulas for isogenies on alternate models of elliptic curves ⋮ Smoothness testing of polynomials over finite fields ⋮ Computing Hilbert class polynomials with the Chinese remainder theorem ⋮ Identification protocols and signature schemes based on supersingular isogeny problems ⋮ A remark on the group structure of 2-isogenous elliptic curves in towers of finite fields ⋮ Computing endomorphism rings of elliptic curves under the GRH ⋮ Computing modular polynomials and isogenies of rank two Drinfeld modules over finite fields ⋮ Computing $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves ⋮ Computing endomorphism rings of abelian varieties of dimension two ⋮ Spanning the isogeny class of a power of an elliptic curve

Uses Software

• gmp
• NTL
• ECPP

Cites Work

• Binary quadratic forms. An algorithmic approach
• Expander graphs based on GRH with an application to elliptic curve cryptography
• On the number of positive integers \(\leq x\) and free of prime factors \(>y\)
• Factoring integers with elliptic curves
• PRIMES is in P
• Counting points on elliptic curves over finite fields
• A $p$-adic algorithm to compute the Hilbert class polynomial
• Computing modular polynomials in quasi-linear time
• Computing Hilbert class polynomials with the Chinese remainder theorem
• Elliptic Curves and Primality Proving
• On the coefficients of the transformation polynomials for the elliptic modular function
• A Rigorous Subexponential Algorithm For Computation of Class Groups
• A Rigorous Time Bound for Factoring Integers
• Modular polynomials via isogeny volcanoes
• Do All Elliptic Curves of the Same Order Have the Same Difficulty of Discrete Log?
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