Cover and Decomposition Index Calculus on Elliptic Curves Made Practical - Application to a Previously Unreachable Curve over $\mathbb{F}_{p^6}$

@inproceedings{Joux2012CoverAD,
  title={Cover and Decomposition Index Calculus on Elliptic Curves Made Practical - Application to a Previously Unreachable Curve over \$\mathbb\{F\}\_\{p^6\}\$},
  author={Antoine Joux and Vanessa Vitse},
  booktitle={EUROCRYPT},
  year={2012}
}
We present a new "cover and decomposition" attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm algorithm. This attack applies, at least theoretically, to all composite degree extension fields, and is particularly well-suited for curves defined over Fp6. We give a real-size example of discrete logarithm computations on a curve over a 151-bit degree 6 extension field, which would not have been… 
Cover and Decomposition Index Calculus on Elliptic Curves made practical. Application to a seemingly secure curve over Fp6
We present a new “cover and decomposition” attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm
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References

SHOWING 1-10 OF 41 REFERENCES
Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields
TLDR
A variation of Faugère’s Gröbner basis algorithm F4, which significantly speeds up the relation computation, and it is shown how this index calculus also applies to oracle-assisted resolutions of the static Diffie–Hellman problem on these elliptic curves.
Constructive and destructive facets of Weil descent on elliptic curves
TLDR
It is shown that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves.
Generalising the GHS attack on the elliptic curve discrete logarithm problem
  • F. Hess
  • Mathematics, Computer Science
  • 2004
TLDR
The Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) is generalised to arbitrary Artin-Schreier extensions and a formula for the characteristic polynomial of Frobenius of the obtained curves is given.
On the discrete logarithm problem in elliptic curves
  • C. Diem
  • Mathematics
    Compositio Mathematica
  • 2010
Abstract We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕ and natural numbers (ni)i∈ℕ with ni⟶∞ and ni/log
Weak Fields for ECC
We demonstrate that some finite fields, including \(\mathbb{F}_{{2}^{210}}\), are weak for elliptic curve cryptography in the sense that any instance of the elliptic curve discrete logarithm problem
An Index Calculus Algorithm for Plane Curves of Small Degree
  • C. Diem
  • Mathematics, Computer Science
    ANTS
  • 2006
TLDR
It is concluded that on heuristic grounds, “almost all” instances of the DLP in degree 0 class groups of (non-hyperelliptic) curves of a fixed genus g ≥3 (represented initially by plane models of bounded degree) can be solved in an expected time of $\tilde{O}(q^{2 -2/(g-1)})$.
Elliptic curve cryptosystems
TLDR
The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.
Scholten Forms and Elliptic/Hyperelliptic Curves with Weak Weil Restrictions
TLDR
This paper shows explicitly the classes of elliptic and hyperelliptic curves of low genera dened over extension elds, which have weak coverings, and how to construct such curves from these curves and analyze density of the curves for them such construction is possible.
Extending the GHS Weil Descent Attack
TLDR
The Weil descent attack due to Gaudry, Hess and Smart (GHS) is extended to a much larger class of elliptic curves and it is shown that a larger proportion than previously thought of elliptIC curves over F2155 should be considered weak.
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