About: Counting points on elliptic curves

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An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA). While in number theory they have important consequences in the solving of Diophantine equations, with respect to cryptography, they enable us to make effective use of the difficulty of the discrete logarithm problem (DLP) for the group , of elliptic curves over a finite field , where q = pk and p is a prime. The DLP, as it has come to be known, is a widely used approach to public key cryptography, a

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rdfs:label	Counting points on elliptic curves (en) Подсчёт точек на эллиптических кривых (ru)
rdfs:comment	Подсчёт точек на эллиптических кривых — группа методов, которые позволяют эффективно вычислять точки на эллиптических кривых. Подсчёт точек на эллиптических кривых используется при изучении теории чисел, криптографии и создании цифровых подписей (см. Эллиптическая криптография и ECDSA). Уровень безопасности криптосистемы, построенной на эллиптической кривой над конечным полем , где q = pk, а p — простое число, определяется сложностью задачи дискретного логарифмирования (DLP) для данной эллиптической кривой . Ниже будут рассмотрены алгоритмы подсчёта точек на эллиптических кривых над полями больших характеристик, в частности, p > 3. Для кривых над полями небольших характеристик существуют более эффективные алгоритмы, основанные на p-адических методах. (ru) An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA). While in number theory they have important consequences in the solving of Diophantine equations, with respect to cryptography, they enable us to make effective use of the difficulty of the discrete logarithm problem (DLP) for the group , of elliptic curves over a finite field , where q = pk and p is a prime. The DLP, as it has come to be known, is a widely used approach to public key cryptography, a (en)
dcterms:subject	Elliptic curves
Wikipage page ID	21275881 (xsd:integer)
Wikipage revision ID	1123049653 (xsd:integer)
Link from a Wikipage to another Wikipage	Public key cryptography Quadratic residues Elliptic curve cryptography Elliptic curves Cryptography Classical modular curve Elliptic curve primality proving Computational complexity of mathematical operations Hasse's theorem on elliptic curves Las Vegas algorithm A. O. L. Atkin Finite field Noam Elkies Number theory Cardinality Pollard's rho algorithm for logarithms J-invariant Baby-step giant-step Chinese remainder theorem Lagrange's theorem (group theory) Diophantine equations Algorithms Elliptic curves Order (group theory) Randomized algorithm Schoof's algorithm Schoof–Elkies–Atkin algorithm Level of security Elliptic curve DSA Discrete logarithm problem Division polynomial Pollard kangaroo Pollard rho
Link from a Wikipage to an external page	http://www.mat.uniroma2.it/~schoof/ctg.pdf http://www.math.umn.edu/~musiker/schoof.pdf
sameAs	Counting points on elliptic curves Counting points on elliptic curves Counting points on elliptic curves Counting points on elliptic curves Counting points on elliptic curves
dbp:wikiPageUsesTemplate	dbt:Main dbt:Algebraic_curves_navbox
has abstract	An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA). While in number theory they have important consequences in the solving of Diophantine equations, with respect to cryptography, they enable us to make effective use of the difficulty of the discrete logarithm problem (DLP) for the group , of elliptic curves over a finite field , where q = pk and p is a prime. The DLP, as it has come to be known, is a widely used approach to public key cryptography, and the difficulty in solving this problem determines the level of security of the cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular p > 3. For curves over fields of small characteristic more efficient algorithms based on p-adic methods exist. (en) Подсчёт точек на эллиптических кривых — группа методов, которые позволяют эффективно вычислять точки на эллиптических кривых. Подсчёт точек на эллиптических кривых используется при изучении теории чисел, криптографии и создании цифровых подписей (см. Эллиптическая криптография и ECDSA). Уровень безопасности криптосистемы, построенной на эллиптической кривой над конечным полем , где q = pk, а p — простое число, определяется сложностью задачи дискретного логарифмирования (DLP) для данной эллиптической кривой . Ниже будут рассмотрены алгоритмы подсчёта точек на эллиптических кривых над полями больших характеристик, в частности, p > 3. Для кривых над полями небольших характеристик существуют более эффективные алгоритмы, основанные на p-адических методах. (ru)
prov:wasDerivedFrom	wikipedia-en:Counting_points_on_elliptic_curves?oldid=1123049653&ns=0
page length (characters) of wiki page	14155 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Counting_points_on_elliptic_curves
is Link from a Wikipage to another Wikipage of	Andrew Sutherland (mathematician) Elliptic-curve cryptography Elliptic curve Point counting Division polynomials Schoof's algorithm Point counting of elliptic curves
is Wikipage redirect of	Point counting of elliptic curves
is foaf:primaryTopic of	wikipedia-en:Counting_points_on_elliptic_curves

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