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Edwards Curves
In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form. Definition The equation of an Edwards curve over a field ''K'' which does not have characteristic 2 is: : x^2 + y^2 = 1 + d x^2 y^2 \, for some scalar d\in K\setminus\. Also the following form with parameters ''c'' and ''d'' is called an Edwards curve: : x^2 + y^2 = c^2(1 + dx^2 y^2) \, where ''c'', ''d'' ∈ ''K'' with ''cd''(1 − ''c''4·''d'') ≠ 0. Every Edwards curve is birationally equivalent to an elliptic curve in Montgomery form, and thus admits an algebraic group law once one chooses a point to serve as a neutral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwards Curve
In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form. Definition The equation of an Edwards curve over a field ''K'' which does not have characteristic 2 is: : x^2 + y^2 = 1 + d x^2 y^2 \, for some scalar d\in K\setminus\. Also the following form with parameters ''c'' and ''d'' is called an Edwards curve: : x^2 + y^2 = c^2(1 + dx^2 y^2) \, where ''c'', ''d'' ∈ ''K'' with ''cd''(1 − ''c''4·''d'') ≠ 0. Every Edwards curve is birationally equivalent to an elliptic curve in Montgomery form, and thus admits an algebraic group law once one chooses a point to serve as a neutral el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Costs Of Operations In Elliptic Curves
Elliptic curve cryptography is a popular form of public key encryption that is based on the mathematical theory of elliptic curves. Points on an elliptic curve can be added and form a group under this addition operation. This article describes the computational costs for this group addition and certain related operations that are used in elliptic curve cryptography algorithms. Abbreviations for the operations The next section presents a table of all the time-costs of some of the possible operations in elliptic curves. The columns of the table are labelled by various computational operations. The rows of the table are for different models of elliptic curves. These are the operations considered : DBL - Doubling ADD - Addition mADD - Mixed addition: addition of an input that has been scaled to have ''Z''-coordinate 1. mDBL - Mixed doubling: doubling of an input that has been scaled to have ''Z'' coordinate 1. TPL - Tripling. DBL+ADD - Combined double and add step To see how addin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Edwards Curve
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Daniel J. Bernstein, Bernstein, Birkner, Joye, Tanja Lange, Lange and Peters in 2008. The curve set is named after mathematician Harold Edwards (mathematician), Harold M. Edwards. Elliptic curves are important in public key cryptography and twisted Edwards curves are at the heart of an electronic signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes. Definition Each twists of curves, twisted Edwards curve is a Twists of curves, twist of an Edwards curve. A twisted Edwards curve E_ over a field (mathematics), field \mathbb with \operatorname(\mathbb) \neq 2 is an wikt:affine, affine plane curve defined by the equation: : E_: a x^2+y^2= 1+dx^2y^2 where a, d are distinct non-zero elements of \mathbb. The special case a = 1 is ''untwisted'', because the curve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Point On Edwards Curve
A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ''The Double'' (1934 film), a German crime comedy film * ''The Double'' (1971 film), an Italian film * ''The Double'' (2011 film), a spy thriller film * ''The Double'' (2013 film), a film based on the Dostoevsky novella * '' Kamen Rider Double'', a 2009–10 Japanese television series ** Kamen Rider Double (character), the protagonist in a Japanese television series of the same name Food and drink * Doppio, a double shot of espresso * Dubbel, a strong Belgian Trappist beer or, more generally, a strong brown ale * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doublin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Two Points On An Edwards Curve
Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the addition of a sequence of numbers * 3SUM, a term from computational complexity theory * Band sum, a way of connecting mathematical knots * Connected sum, a way of gluing manifolds * Digit sum, in number theory * Direct sum, a combination of algebraic objects ** Direct sum of groups ** Direct sum of modules ** Direct sum of permutations ** Direct sum of topological groups * Einstein summation, a way of contracting tensor indices * Empty sum, a sum with no terms * Indefinite sum, the inverse of a finite difference * Kronecker sum, an operation considered a kind of addition for matrices * Matrix addition, in linear algebra * Minkowski addition, a sum of two subsets of a vector space * Power sum symmetric polynomial, in commutative algebra * Prefix sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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