Fast computation of elliptic curve isogenies in characteristic two

In [1] and [4] we defined the elliptic curve over the ring F3d [ε], ε2 = 0. In this work, we will study the elliptic curve over the ring A = F2d [ε], where d is a positive integer and ε2= 0. More precisely we will establish a group homomorphism between the abulia group (Ea,b,c(F2d ), +) and (F2d, +).

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On the solutions of an elliptic curve over a field of characteristic two

Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252) ◽

10.1109/isit.1998.708679 ◽

2002 ◽

Cited By ~ 2

Author(s):

I.F. Blake ◽

G. Seroussi ◽

R.M. Roth

Keyword(s):

Elliptic Curve ◽

Characteristic Two

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Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000000723 ◽

2002 ◽

Vol 5 ◽

pp. 127-174 ◽

Cited By ~ 14

Author(s):

Markus Maurer ◽

Alfred Menezes ◽

Edlyn Teske

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Finite Fields ◽

Discrete Logarithm ◽

Discrete Logarithm Problem ◽

Signature Scheme ◽

Extension Degree ◽

Characteristic Two ◽

Weil Descent

AbstractIn this paper, the authors analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP) for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field F2N, where N is in [100,600], elliptic curve parameters are identified such that: (i) there should exist a cryptographically interesting elliptic curve E over F2N with these parameters; and (ii) the GHS attack is more efficient for solving the ECDLP in E(F2N) than for solving the ECDLP on any other cryptographically interesting elliptic curve over F2N. The feasibility of the GHS attack on the specific elliptic curves is examined over F2176, F2208, F2272, F2304 and F2368, which are provided as examples in the ANSI X9.62 standard for the elliptic curve signature scheme ECDSA. Finally, several concrete instances are provided of the ECDLP over F2N, N composite, of increasing difficulty; these resist all previously known attacks, but are within reach of the GHS attack.

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Computation of Mordell–Weil bases for ordinary elliptic curves in characteristic two

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000175 ◽

2014 ◽

Vol 17 (A) ◽

pp. 1-13

Author(s):

G. Moehlmann

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Upper Bound ◽

Duality Theorem ◽

Homogeneous Spaces ◽

Integral Point ◽

Selmer Groups ◽

Global Function ◽

Global Function Fields ◽

Characteristic Two

AbstractIn this paper we consider ordinary elliptic curves over global function fields of characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an $S$-integral point on $A$. To illustrate our methods, a detailed example is presented.

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