The arithmetic of Jacobian groups of superelliptic cubics

DOI10.1090/S0025-5718-04-01699-0zbMath1094.14049MaRDI QIDQ4821055

Andreas Enge, Abdolali Basiri, Nicolas Gürel, Jean-Charles Faugère

Publication date: 7 October 2004

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

zbMATH Keywords

JacobianGröbner basisreduced divisorsuperelliptic curve

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) Algebraic coding theory; cryptography (number-theoretic aspects) (11T71) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Special algebraic curves and curves of low genus (14H45) Jacobians, Prym varieties (14H40) Curves over finite and local fields (11G20) Computational aspects of algebraic curves (14Q05)

Related Items (6)

Group arithmetic in \(C_{3,5}\) curves ⋮ A generic approach to searching for Jacobians ⋮ Index calculus in class groups of non-hyperelliptic curves of genus three ⋮ Fast Jacobian Group Operations for C_3,4 Curves over a Large Finite Field ⋮ On Jacobian group arithmetic for typical divisors on curves ⋮ Isomorphism classes of Picard curves over finite fields

Cites Work

• Hyperelliptic cryptosystems
• Efficient computation of zero-dimensional Gröbner bases by change of ordering
• An addition algorithm in Jacobian of \(C_{ab}\) curves.
• Advances in cryptology - EUROCRYPT 2000. 19th international conference on the theory and application of cryptographic techniques, Bruges, Belgium, May 14--18, 2000. Proceedings
• Computing Riemann-Roch spaces in algebraic function fields and related topics.
• Ideal arithmetic and infrastructure in purely cubic function fields
• Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve
• Arithmetic on superelliptic curves
• Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time
• A general framework for subexponential discrete logarithm algorithms
• Computing in the Jacobian of a Hyperelliptic Curve
• Computing discrete logarithms in real quadratic congruence function fields of large genus
• The arithmetic of certain cubic function fields
• Algorithmic Number Theory
• The extended Euclidean algorithm on polynomials, and the computational efficiency of hyperelliptic cryptosystems
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