Equations System coming from Weil descent and subexponential attack for algebraic curve cryptosystem
Koh-ichi Nagao
Abstract
Faugére et al. shows that the decomposition problem of a point of
elliptic curve over binary field reduces to solving low degree equations system
over coming from Weil descent. Using this method, the discrete logarithm problem of elliptic curve over
reduces to linear constrains, i.e., solving equations system using linear algebra
of monomial modulo field equations, and its complexity is expected to be subexponential of input size . However, it is pity that at least using linear constrains, it is exponential.
Petit et al. shows that assuming first fall degree assumption,
from which the complexity of solving low degree equations system using Gröbner basis computation is subexponential, its total complexity is heuristically subexponential.
On the other hands, the author shows that the decomposition problem of Jacobian of plane curve over also essentially reduces to solving low degree equations system over coming from Weil descent. In this paper, we revise (precise estimation of first fall degree) the results of Petit et al. and show that the discrete logarithm problem of elliptic curve over small characteristic field is subexponential of input size , and the discrete logarithm problem of Jacobian of small genus curve over small characteristic field is also subexponential of input size , under first fall degree assumption.
Note: Revise 6 NOV.
In \S 3, we estimate the degree of the for some
monomial .
For our aim, the solution of the equations system
must equals to
.
However, it is not true. Let ,
and take new
by
(not monomial but polynomial), we have this property and all lemmas still hold. }
@misc{cryptoeprint:2013/549,
author = {Koh-ichi Nagao},
title = {Equations System coming from Weil descent and subexponential attack for algebraic curve cryptosystem},
howpublished = {Cryptology {ePrint} Archive, Paper 2013/549},
year = {2013},
url = {https://eprint.iacr.org/2013/549}
}
Note: In order to protect the privacy of readers, eprint.iacr.org
does not use cookies or embedded third party content.