Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Mathematics of Computation
The Xedni Calculus and the Elliptic Curve Discrete LogarithmProblem
Designs, Codes and Cryptography
Analysis of the Xedni Calculus Attack
Designs, Codes and Cryptography
Analysis of the Weil Descent Attack of Gaudry, Hess and Smart
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Lifting Elliptic Curves and Solving the Elliptic Curve Discrete Logarithm Problem
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Factoring multivariate polynomials via partial differential equations
Mathematics of Computation
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three
Journal of Cryptology
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
An index calculus algorithm for plane curves of small degree
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
On the discrete logarithm problem on algebraic tori
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
Reducing elliptic curve logarithms to logarithms in a finite field
IEEE Transactions on Information Theory
Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Generating Genus Two Hyperelliptic Curves over Large Characteristic Finite Fields
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
On the Security of Pairing-Friendly Abelian Varieties over Non-prime Fields
Pairing '09 Proceedings of the 3rd International Conference Palo Alto on Pairing-Based Cryptography
Compact hardware for computing the tate pairing over 128-bit-security supersingular curves
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
CT-RSA'11 Proceedings of the 11th international conference on Topics in cryptology: CT-RSA 2011
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
Improving the complexity of index calculus algorithms in elliptic curves over binary fields
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
On polynomial systems arising from a weil descent
ASIACRYPT'12 Proceedings of the 18th international conference on The Theory and Application of Cryptology and Information Security
On efficient pairings on elliptic curves over extension fields
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
High-Performance scalar multiplication using 8-dimensional GLV/GLS decomposition
CHES'13 Proceedings of the 15th international conference on Cryptographic Hardware and Embedded Systems
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We propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over F"q"^"3 in heuristic asymptotic running time O@?(q^4^/^3); and an elliptic problem over F"q"^"4 or a genus 2 problem over F"q"^"2 in heuristic asymptotic running time O@?(q^3^/^2).