Pairing-Friendly Elliptic Curves of Prime Order

@inproceedings{Barreto2005PairingFriendlyEC,
  title={Pairing-Friendly Elliptic Curves of Prime Order},
  author={Paulo Barreto and Michael Naehrig},
  booktitle={Selected Areas in Cryptography},
  year={2005}
}
Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree $k \leqslant 6 $. More general methods produce curves over ${\mathbb F}_{p}$ where the bit length of p is often twice as large as that of the order r of the subgroup with embedding degree k; the best published results achieve ρ ≡ log(p)/log(r) ~ 5/4. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic… 

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References

SHOWING 1-10 OF 28 REFERENCES
Constructing Elliptic Curves with Prescribed Embedding Degrees
TLDR
Criteria for curves with larger k that generalize prior work by Miyaji et al. based on the properties of cyclotomic polynomials are examined, and efficient representations for the underlying algebraic structures are proposed.
Efficient Implementation of Pairing-Based Cryptosystems
TLDR
This paper describes how to construct ordinary (non-supersingular) elliptic curves containing groups with arbitrary embedding degree, and shows how to compute the Tate pairing on these groups efficiently.
Generating More MNT Elliptic Curves
TLDR
An alternative derivation of Miyaji et al.'s results for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6 is provided.
New Explicit Conditions of Elliptic Curve Traces for FR-Reduction
Elliptic curve cryptosystems([19],[25]) are based on the elliptic curve discrete logarithm problem(ECDLP). If elliptic curve cryptosystems avoid FRreduction([11],[17]) and anomalous elliptic curve
On Small Characteristic Algebraic Tori in Pairing-Based Cryptography
TLDR
This work transfers techniques recently developed for torus-based cryptography to pairing- based cryptography, resulting in more efficient computations, and lower bandwidth requirements, to illustrate the efficacy of this approach.
Constructing elliptic curves with given group order over large finite fields
TLDR
A procedure is developed for constructing elliptic curves with given group order over large finite fields that yields all representations of a given integer as a norm in an imaginary quadratic field.
On the Selection of Pairing-Friendly Groups
We propose a simple algorithm to select group generators suitable for pairing-based cryptosystems. The selected parameters are shown to favor implementations of the Tate pairing that are at once
Compressed Pairings
TLDR
This paper shows how to compress Pairing values, how to couple this technique with that of point compression, and how to benefit from the compressed representation to speed up exponentiations involving pairing values, as required in many pairing based protocols.
Supersingular Abelian Varieties in Cryptology
TLDR
This paper determines exactly which values can occur as the security parameters of supersingular abelian varieties (in terms of the dimension of theAbelian variety and the size of the finite field), and gives constructions of supersedication varieties that are optimal for use in cryptography.
Ordinary abelian varieties having small embedding degree
TLDR
This paper generalises the results of Miyaji, Nakabayashi and Takano by giving families corresponding to non-prime group orders with embedding degree suitable for pairing applications and considers the case of ordinary abelian varieties of dimension 2.
...
1
2
3
...