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Pairingfriendly Hyperelliptic Curves of type y 2 = x 5 + ax
 In 2008 Symposium on Cryptography and Information Security (SCIS 2008
, 2008
"... Abstract. An explicit construction of pairingfriendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman. In this paper, we give other explicit constructions of pairingfriendly hyperelliptic curves. Our methods are based on the closed formulae for the order of the Jacobia ..."
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Abstract. An explicit construction of pairingfriendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman. In this paper, we give other explicit constructions of pairingfriendly hyperelliptic curves. Our methods are based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type y 2 = x 5 + ax over a finite prime field Fp which are given by E. Furukawa, M. Haneda, M. Kawazoe and T. Takahashi. We present two methods in this paper. One is an analogue of the CocksPinch method and the other is a cyclotomic method. Our methods construct a pairingfriendly hyperelliptic curve y 2 = x 5 + ax over Fp whose Jacobian has a prescribed embedding degree with respect to some prime number ℓ. Curves constructed by the analogue of the CocksPinch method satisfy p ≈ ℓ 2, whereas p ≈ ℓ 4 in Freeman’s construction. Moreover, for the case of embedding degree 24, we can construct a cyclotomic family with p ≈ ℓ 3/2.
CONSTRUCTING PAIRINGFRIENDLY HYPERELLIPTIC CURVES USING WEIL RESTRICTION
"... Abstract. A pairingfriendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large primeorder subgroup. In this paper we construct pairingfriendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simpl ..."
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Abstract. A pairingfriendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large primeorder subgroup. In this paper we construct pairingfriendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become pairingfriendly over a finite extension of Fq. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the CocksPinch and BrezingWeng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded ρvalue for simple, nonsupersingular abelian surfaces. 1.
A Generalized BrezingWeng Algorithm for Constructing PairingFriendly Ordinary Abelian Varieties
"... Abstract. We give an algorithm that produces families of Weil numbers for ordinary abelian varieties over finite fields with prescribed embedding degree. The algorithm uses the ideas of Freeman, Stevenhagen, and Streng to generalize the BrezingWeng construction of pairingfriendly elliptic curves. ..."
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Abstract. We give an algorithm that produces families of Weil numbers for ordinary abelian varieties over finite fields with prescribed embedding degree. The algorithm uses the ideas of Freeman, Stevenhagen, and Streng to generalize the BrezingWeng construction of pairingfriendly elliptic curves. We discuss how CM methods can be used to construct these varieties, and we use our algorithm to give examples of pairingfriendly ordinary abelian varieties of dimension 2 and 3 that are absolutely simple and have smaller ρvalues than any previous such example.
D.: Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and PairingFriendly Constructions
 PairingBased CryptographyPairing 2012. LNCS
, 2013
"... Abstract. The use of elliptic and hyperelliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve. Recently, Satoh proposed a probabilistic polynomial time algorithm to test whether the Jacobian – over a finite field Fq – of a hyperelliptic curve of the form ..."
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Abstract. The use of elliptic and hyperelliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve. Recently, Satoh proposed a probabilistic polynomial time algorithm to test whether the Jacobian – over a finite field Fq – of a hyperelliptic curve of the form Y 2 = X 5 + aX 3 + bX (with a, b ∈ F ∗ q) has a large prime factor. His approach is to obtain candidates for the zeta function of the Jacobian over F ∗ q from its zeta function over an extension field where the Jacobian splits. We extend and generalize Satoh’s idea to provide explicit formulas for the zeta function of the Jacobian of genus 2 hyperelliptic curves of the form Y 2 = X 5 +aX 3 +bX and Y 2 = X 6 +aX 3 +b (with a, b ∈ F ∗ q). Our results are proved by elementary (but intricate) polynomial rootfinding techniques. Hyperelliptic curves with small embedding degree and large primeorder subgroup are key ingredients for implementing pairingbased cryptographic systems. Using our closed formulas for the Jacobian order, we propose two algorithms which complement those of Freeman and Satoh to produce genus 2 pairingfriendly hyperelliptic curves. Our method relies on techniques initially proposed to produce pairingfriendly elliptic curves (namely, the CocksPinch method and the BrezingWeng method). We show that the previous security considerations about embedding degree are valid for an elliptic curve and can be lightened for a Jacobian. We demonstrate this method by constructing several interesting curves with ρvalues around 4 with a CocksPinchlike method and around 3 with a BrezingWenglike method.
PAIRINGS ON HYPERELLIPTIC CURVES
, 2009
"... We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techni ..."
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We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research.
Constructing PairingFriendly Genus 2 Curves with Split Jacobian
, 2012
"... Genus 2 curves with simple but not absolutely simple jacobians can be used to construct pairingbased cryptosystems more efficient than for a generic genus 2 curve. We show that there is a full analogy between methods for constructing ordinary pairingfriendly elliptic curves and simple abelian va ..."
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Genus 2 curves with simple but not absolutely simple jacobians can be used to construct pairingbased cryptosystems more efficient than for a generic genus 2 curve. We show that there is a full analogy between methods for constructing ordinary pairingfriendly elliptic curves and simple abelian varieties, which are iogenous over some extension to a product of elliptic curves. We extend the notion of complete, complete with variable discriminant, and sparse families introduced in by Freeman, Scott and Teske [11] for elliptic curves, and we generalize the CocksPinch method and the BrezingWeng method to construct families of each type. To realize abelian surfaces as jacobians we use of genus 2 curves of the form y 2 = x 5 + ax 3 + bx or y 2 = x 6 + ax 3 + b, and apply the method of Freeman and Satoh [10]. As applications we find some families of abelian surfaces with recorded ρvalue ρ = 2 for embedding degrees k = 3, 4, 6, 12, or ρ = 2.1 for k = 27, 54. We also give variablediscriminant families with best ρvalues.
A CM CONSTRUCTION FOR CURVES OF GENUS 2 WITH pRANK 1
"... We construct Weil numbers corresponding to genus2 curves with prank 1 over the finite field Fp2 of p2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2valued points of the Jacobian has prime order, while another allo ..."
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We construct Weil numbers corresponding to genus2 curves with prank 1 over the finite field Fp2 of p2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over Fp2 out of necessity: we show that curves of prank 1 over Fp for large p cannot be efficiently constructed using explicit CM constructions.
Generating Pairingfriendly Parameters for the CM Construction of Genus 2 Curves over Prime Fields
"... Abstract. We present two contributions in this paper. First, we give a quantitative analysis of the scarcity of pairingfriendly genus 2 curves. This result is an improvement relative to prior work which estimated the density of pairingfriendly genus 2 curves heuristically. Second, we present a met ..."
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Abstract. We present two contributions in this paper. First, we give a quantitative analysis of the scarcity of pairingfriendly genus 2 curves. This result is an improvement relative to prior work which estimated the density of pairingfriendly genus 2 curves heuristically. Second, we present a method for generating pairingfriendly parameters for which ρ ≈ 8, where ρ is a measure of efficiency in pairingbased cryptography. This method works by solving a system of equations given in terms of coefficients of the Frobenius element. The algorithm is easy to understand and implement. 1