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Topic: Hyperelliptic curve

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Hyperelliptic curve cryptography

	Cryptography using Hyperelliptic Curves
		The main goals of this project are the establishment of the algorithmic foundations and the implementation of a prototype.
		Hyperelliptic curves are generalisations of elliptic curves, ie, they are of a higher genus (elliptic curves have a genus equal to one).
		This cannot be simply read from the curve equation and there is still no known efficient algorithm to compute it for a general hyperelliptic curve.
	www.ercim.org /publication/Ercim_News/enw49/kux.html (447 words)

	Hyperelliptic curve - Wikipedia, the free encyclopedia
		A hyperelliptic function is a function from the function field of such a curve; or possibly on the Jacobian variety on the curve, these being two concepts that are the same for the elliptic function case, but different in this case.
		In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at the point at infinity.
		Hyperelliptic curves can be used in hyperelliptic curve cryptography in cryptosystems based on the discrete logarithm problem.
	en.wikipedia.org /wiki/Hyperelliptic_curve (419 words)

	Operations on Curves
		The degree of the hyperelliptic curve C or a pointset C of a hyperelliptic curve.
		The discriminant of the hyperelliptic curve C or a pointset C of a hyperelliptic curve.
		The genus of the hyperelliptic curve C or a pointset C of a hyperelliptic curve.
	www.math.niu.edu /help/math/magmahelp/text1023.html (1103 words)

	Hyperelliptic curve cryptography - Encyclopedia Glossary Meaning Explanation Hyperelliptic curve cryptography (Site not responding. Last check: 2007-10-19)
		Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insomuch as the algebraic geometry construct of a hyperelliptic curve with an appropriate group law provides an Abelian group on which to do arithmetic.
		Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA).
		Because the arithmetic on hyperelliptic curves is more complicated than that on elliptic curves, a properly implemented cryptosystem based on hyperelliptic curves can be more secure than elliptic curve based cryptosystems that have the same key size.
	www.encyclopedia-glossary.com /en/Hyperelliptic-curve-cryptography.html (254 words)

	[No title]
		Thomas Wollinger Friday, April 1, 2005 1:00pm in 320 S&T2 Abstract The hyperelliptic curve cryptosystem is one of the emerging cryptographic primitives of the last years.
		However, until recently the common belief in industry and in the research community was that hyperelliptic curves are out of scope for any practical application.
		We were able to reduce the complexity of the group operation for small genus hyperelliptic curves and we provide efficient algorithms for the computation of the hyperelliptic curve cryptosystem.
	bass.gmu.edu /seminars/wollinger.doc (431 words)

	Isomorphisms and Transformations
		A hyperelliptic curve isomorphism curve defined by the data of a linear fractional transformation t(x:z) = (ax + bz:cx + dz), a scale factor e, and a polynomial u(x) of degree at most g + 1, where g is the genus of the curve.
		Returns the hyperelliptic curve C' which is the codomain of the isomorphism specified by the data t, e and u, followed by the the isomorphism to the curve.
		The curve must be of genus at least one, and the automorphism group is defined to consist of those automorphisms which commute with the hyperelliptic involution, i.e.
	www.math.lsu.edu /magma/text1236.htm (1410 words)

	Hyperelliptic Cryptosystems in Practice
		The Hyperelliptic curve cryptosystem is one of the emerging cryptographic primitives of the last years.
		However, until recently the common belief in industry and in the research community was that hyperelliptic curves are out of scope for any practical application.
		We were able to show the practical use of hyperelliptic curve cryptosystems (HECC) by narrowing the performance gap between elliptic curve (EC) and hyperelliptic curve cryptosystems.
	www.gi-ev.de /fachbereiche/sicherheit/fg/krypto/tag/1/paar-pelzl-wollinger (316 words)

	Fraunhofer ITWM: Automorphism Groups of Hyperelliptic Function Fields
		In the project "Hyperelliptic Curve Cryptography" researchers at Fraunhofer ITWM developed an efficient method to compute the automorphism group* of an arbitrary hyperelliptic function field*.
		The Jacobians* of hyperelliptic function fields* have been suggested by Nean Koblitz in 1988 as groups* for cryptographic purposes, because the computation of the discrete logarithm* is believed to be hard in this kind of groups.
		Jacobians of hyperelliptic curves are a generalization of the groups of points on elliptic curves.
	www.itwm.fhg.de /mab/competences/Crypto/aut/index_en.php?printer=1 (306 words)

	Points on Hyperelliptic Curves (Site not responding. Last check: 2007-10-19)
		Given a point P on a hyperelliptic curve C_1, such that C is a base extension of C_1, returns the corresponding point on the hyperelliptic curve C. The curve C can be, e.g., the reduction of C_1 to finite characteristic (i.e.
		For a hyperelliptic curve C defined over a finite field returns all rational points on C. For a curve C over Q of the form y^2 = f(x) with integral coefficients, returns the set of points such that the naive height of the x-coordinate is less than Bound.
		Given a hyperelliptic curve C defined over a finite field, this function computes the zeta function of C. The zeta function is returned as an element of the function field in one variable over the integers.
	www.dtr.isy.liu.se /Magma/text781.html (589 words)

	Rational Points and Group Structure (Site not responding. Last check: 2007-10-19)
		Given the Jacobian J of a hyperelliptic curve defined over the rationals, this function returns a bound on the size of the rational torsion subgroup of the Jacobian.
		Given the Jacobian J of a hyperelliptic curve defined over a finite field and a prime p, this function returns the Sylow p-subgroup of the group of rational points of J, as an abstract abelian group A. The injection from A to J is also returned.
		Given the Jacobian J of a hyperelliptic curve defined over a finite field K, this function returns the group of rational points of J as an abstract abelian group A. The isomorphism from A to J(K) is returned as a second value.
	www.dtr.isy.liu.se /Magma/text784.html (346 words)

	Jacobians (Site not responding. Last check: 2007-10-19)
		The Jacobian of a hyperelliptic curve is implemented as the divisor class group of the curve.
		The hyperelliptic curve from which the Jacobian J was constructed.
		The dimension of the Jacobian J as an algebraic variety, equal to the genus of the curve C of which J is the Jacobian.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1137.htm (131 words)

	Points
		Given a point P on a hyperelliptic curve, returns a 3-element sequence consisting of the coordinates of the point P. Arithmetic of Points
		Given a hyperelliptic curve C defined over a finite field, returns the number of rational points on C. If the base field is small or there is no other good alternative, a naive point counting technique is used.
		For a hyperelliptic curve C defined over a finite field returns an indexed set of all rational points on C. For a curve C over Q of the form y^2 = f(x) with integral coefficients, returns the set of points such that the naive height of the x-coordinate is less than
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1213.htm (830 words)

	Operations on Curves
		The discriminant of the hyperelliptic curve C. Genus(C) : CrvHyp -> RngIntElt
		The genus of the hyperelliptic curve C. Igusa Invariants
		The Igusa invariants of the curve y^2 + hy = f are equal to the Igusa invariants of the polynomial h^2 + 4 * f except in characteristic 2, where the latter are not defined.
	cso.ulb.ac.be /~dleemans/magma/text1210.htm (1089 words)

	Rational Points and Group Structure
		Given a Jacobian J of a hyperelliptic curve defined over a finite field or the rationals, determine all rational points on the Jacobian J. In the case where the base field is Q, the corresponding curve must have genus two and be of the form y^2 = f(x), where f has integral coefficients.
		Given a Jacobian J of a hyperelliptic curve defined over the rationals and a finite field K at which J has good reduction, returns the Euler factor of the base extension of J to K. Abelian Group Structure
		Given the Jacobian J of a hyperelliptic curve defined over a finite field and a prime p, this function returns the Sylow p-subgroup of the group of rational points of J, as an abstract abelian group A. The injection from A to J is also returned as well as the generators of the p-Sylow subgroup.
	cso.ulb.ac.be /~dleemans/magma/text1217.htm (1531 words)

	Summer School on Elliptic and Hyperelliptic Curve Cryptography
		This Summer School on Elliptic and Hyperelliptic Curve Cryptography is part of the Thematic Program in Cryptography at the Fields Institute in Toronto.
		For elliptic curves we explain Schoof's algorithm as a method to count points on curves over prime fields and consider p-adic methods like AGM which are more efficient in the case of small characteristic fields.
		Elliptic curves I, Roger Oyono, was given as a flboard presentation.
	www.hyperelliptic.org /tanja/conf/summerschool06 (1102 words)

	Analytic Jacobians of Hyperelliptic Curves
		The analytic Jacobian of the curve is an abelian torus and is constructed as follows.
		It is known that the dimension of the vector space of holomorphic differentials is equal to the genus of the curve.
		This is equal to the genus of the curve C for which A is the Jacobian.
	www.math.lsu.edu /magma/text1245.htm (3528 words)

	Creation Functions (Site not responding. Last check: 2007-10-19)
		Given a hyperelliptic curve C defined over a field k, and a field K which is an extension of k, return a hyperelliptic curve C' over K using the natural inclusion of k in K to map the coefficients of C into elements of K. BaseChange(C, j) : Sch, Map -> Sch
		Given a hyperelliptic curve C defined over the rationals and a prime p, this function returns a minimal integral Weierstrass model C' of C with respect to the p-adic valuation.
		Given a hyperelliptic curve C defined over a field k of characteristic not equal to 2 and an element d that is coercible into k, return the quadratic twist of C by d.
	modular.fas.harvard.edu /docs/magma/htmlhelp/text1209.htm (1633 words)

	Uses of Class com.dragongate_technologies.saluki.HyperellipticCurve
		Constructs a random divisor div(a, b) on a hyperelliptic curve C
		Constructs a divisor div(a, b) on a hyperelliptic curve C
		Constructs a random divisor div(a, b) on a hyperelliptic curve C. Div_F2m_x(HyperellipticCurve C, Fq_x a, Fq_x b)
	www.dragongate-technologies.com /jSaluki/com/dragongate_technologies/saluki/class-use/HyperellipticCurve.html (80 words)

	[No title]
		The case g=1 is ECC since in that case the Jacobian is the same as the curve.
		We are considering hardware implementations of hyperelliptic curves of genus 2 and 3.
		She is now investigating genus 3 curves and will visit us for 2 months next year.These are examples of the kind of collaboration that we intend to pursue with Frey and his group in Germany.
	www.iccip.csl.uiuc.edu /research.html (1274 words)

	9.1 Hyperelliptic curves (Site not responding. Last check: 2007-10-19)
		It is clear that the vertex v does not lie on T so that projection gives a morphism on T which lands in the rational normal curve of degree d.
		The variety T is called a hyperelliptic curve, the involution is called the hyperelliptic involution and the morphism
		The number g = d - 1 is called the genus of the hyperelliptic curve.
	www.imsc.ernet.in /~kapil/crypto/notes/node48.html (510 words)

	Introduction (Site not responding. Last check: 2007-10-19)
		A hyperelliptic curve, which is taken to include the genus one case, is given by a nonsingular generalized Weierstrass equation
		Functionality for hyperelliptic curves includes optimized algorithms for working on genus two curves over Q, including heights on the Jacobian, and a datatype for the Kummer surface of the Jacobian.
		The initial development of machinery for hyperelliptic curves was undertaken by Michael Stoll support by members of the Magma group.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1230.htm (157 words)

	Creating a Hyperelliptic Curve from Invariants (Site not responding. Last check: 2007-10-19)
		The problem is to construct a curve of genus 2 from a given set of Igusa--Clebsch invariants.
		Construct a hyperelliptic curve of genus 2 from a set of Igusa-Clebsch invariants (see Subsection Igusa Invariants).
		Given a curve C of genus 2, attempt to find an isomorphic curve C' such that its coefficients are much smaller than those of C. The algorithm used is due to Van Wamelen [Wam99].
	www.math.niu.edu /help/math/magmahelp/text1024.html (166 words)

	Hyperelliptic Curve Cryptographic Software
		The curve itself is specified by the line; "equation = pow2Y(y,2) + pow2Y(y,1) + pow2X(x,5) + pow2X(x,3) + pow2X(x,1);", where equation is a Poly2XY instance.
		The curve itself is specified by the "equation" variable, where equation is a Poly2XY instance.
		The curve itself is specified by the "f" variable, which is the polynomial f(x) in "y^2 = f(x)".
	www.compapp.dcu.ie /~coheigeartaigh/crypto.html (2093 words)

	Bibliography
		The extended Euclidean algorithm on polynomials, and the computational efficiency of hyperelliptic cryptosystems, November 1999.
		Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves (Update).
		Improving Harley Algorithms for Jacobians of Genus 2 Hyperelliptic Curves.
	www.hecc.rub.de /tex/bib.html (539 words)

	Dragongate Technologies Ltd. - Products
		An Open Source C++ Elliptic Curve Cryptography Library.
		An Open Source Java Hyperelliptic Curve Cryptography Library.
		Hyperelliptic Curve Cryptography is still an experimental area so this library is only recommended for research and educational purposes.
	www.dragongate-technologies.com /products.html (641 words)

	Taylor & Francis Online
		The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive.
		The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography.
		For some special curves the discrete logarithm problem can be transferred to an easier one; the consequences are explained and suggestions for good choices are given.
	www.crcpress.com /shopping_cart/products/product_detail.asp?sku=C5181&pc=conventions/default.asp?m=7&y=2005 (467 words)

	Compact Representation of Domain Parameters of Hyperelliptic Curve Cryptosystems - Zhang, Liu, Kim (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		The domain parameters include the field over which the curve is defined, the curve itself, the order of the Jocobian and the base point.
		6 the discriminant of a hyperelliptic curve (context) - Lockhart - 1994
		1 Hyperelliptic cryptography (context) - Koblitz - 1989
	citeseer.ist.psu.edu /zhang02compact.html (514 words)

	[No title]
		# The integral to compute must be in the form: # / # (p(x)/q(x)y)dx # / # where y^2=f(x) defines the hyperelliptic curve.
		# # f : The defining polynomial of the curve: a squarefree univariate # polynomial in x of odd degree prime to q.
		# p0 : The defining polynomial of the curve: a squarefree univariate # polynomial of odd degree prime to q.
	www.cstp.umkc.edu /public/papers/place/maple/share/hint/hint.mpl (2236 words)

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