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{{Short description|Type of elliptic curve}} ...some operations (see the last sections), it is close in speed to [[Edwards curve]]s. Twisted Hessian curves were introduced by [[Daniel J. Bernstein|Bernste ...

...up addition and certain related operations that are used in elliptic curve cryptography algorithms. ...omputational operations. The rows of the table are for different models of elliptic curves. These are the operations considered: ...

...sion points. They play a central role in the study of [[counting points on elliptic curves]] in [[Schoof's algorithm]]. * The division polynomials form a generic [[elliptic divisibility sequence]] over the ring <math>\mathbb{Q}[x,y,A,B]/(y^2-x^3-Ax ...

{{Short description|Type of elliptic curve}} ...bling oriented.svg|300px|right|thumb|A Doubling-oriented Doche-Icart-Kohel curve of equation <math>y^2=x^3-x^2-16x</math>]] ...

...have higher order twists such as '''cubic''' and '''quartic twists'''. The curve and its twists have the same [[j-invariant]]. ...id=2326034 |issn=0012-7094}}</ref> and when generalized to [[hyperelliptic curve]]s, the study of the [[Sato–Tate conjecture]].<ref name="LLG2019">{{Cite jo ...

| genre = [[elliptic-curve cryptography|Elliptic-curve]] cryptographic library ...Longa |first2=Patrick |title=FourQ: four-dimensional decompositions on a Q-curve over the Mersenne prime |year=2015 |url=https://eprint.iacr.org/2015/565 |a ...

{{Short description|Type of elliptic curve}} ...date=March 2022}}; it is a particular type of [[elliptic curve|Weierstrass curve]]. At certain conditions some [[operation (mathematics)|operations]], as ad ...

...ic-curve cryptography]]. Their vertices represent [[supersingular elliptic curve]]s over [[finite field]]s and their edges represent [[Isogeny|isogenies]] b ...ath>\ell</math>, and considering the class of all [[supersingular elliptic curve]]s defined over the [[finite field]] <math>\mathbb{F}_{p^2}</math>. There a ...

[[Image:Twisted Edwards curve.svg|300px|right|thumb|A twisted Edwards curve of equation <math>10x^2+y^2=1+6x^2y^2</math>]] ...tician)|Harold M. Edwards]]. Elliptic curves are important in [[public key cryptography]] and twisted Edwards curves are at the heart of an electronic signature sc ...

| name = Elliptic curve only hash (ECOH) '''The elliptic curve only hash (ECOH)''' algorithm was submitted as a candidate for SHA-3 in the ...

...]] of the cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular ''p''&nbsp;>&nbs ==Approaches to counting points on elliptic curves== ...

...perties studied, by [[Morgan Ward]]<ref name="Ward">Morgan Ward, Memoir on elliptic divisibility sequences, ''Amer. J. Math.'' '''70''' (1948), 31&ndash;74.</r ...have applications to other areas of mathematics including [[logic]] and [[cryptography]]. ...

{{Short description|Family of elliptic curves used in cryptography}} ...leted image removed: [[Image:Edwards curve.jpg|300px|right|thumb|An Edward curve of equation <math>x^2+y^2=1-300x^2y^2</math>]] --> ...

{{Short description|Type of elliptic curve}} In [[mathematics]], the '''Montgomery curve''' is a form of [[elliptic curve]] introduced by [[Peter Montgomery (mathematician)|Peter L. Montgomery]] in ...

...The Jacobi curve also offers faster arithmetic compared to the Weierstrass curve. The Jacobi curve can be of two types: the '''Jacobi intersection''', that is given by an int ...

...|Lenstra]] heuristics, and [[elliptic curve]]s and their applications to [[cryptography]].{{cn|date=January 2023}} ...Elliptic_Curves.html?id=vPL19KNdm2wC ''Elliptic Curves: Number theory and cryptography''], CRC Press, 2003, 2nd edn. 2008 ...

...arious [[cryptographic primitive]]s in private key as well as [[public-key cryptography]]. Their result is the construction of an efficient [[pseudorandom function The '''Naor–Reingold''' function can be used as the basis of many [[cryptography|cryptographic]] schemes including [[Symmetric-key algorithm|symmetric encr ...

...n as '''Boneh&ndash;Lynn&ndash;Shacham'''<ref name=BLS2004/> (BLS), is a [[cryptography|cryptographic]] [[signature scheme]] which allows a user to verify that a s ...secret key is an integer in <math>[0,q-1]</math>. Working in an elliptic curve group provides some defense against [[index calculus]] attacks (with the ca ...

...ecurity proof of Sakai-Kasahara's identity-based encryption scheme|journal=Cryptography ePrint Archive|volume=2005/226|url=http://eprint.iacr.org/2005/226.pdf}}</r * There is a bilinear map, a Tate-Lichtenbaum [[Pairing#Pairings in cryptography|pairing]], <math>\textstyle e(,)</math> from E to G. This means that for <m ...

...ess of the [[discrete logarithm]] problem and the computational [[Elliptic curve Diffie–Hellman]]. ===Elliptic curves cryptography over a finite field=== ...