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Tripling-oriented Doche–Icart–Kohel Curve
The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve. At certain conditions some operations, as adding, doubling or tripling points, are faster to compute using this form. The Tripling oriented Doche–Icart–Kohel curve, often called with the abbreviation 3DIK has been introduced by Christophe Doche, Thomas Icart, and David R. Kohel in Definition Let K be a field of characteristic different form 2 and 3. An elliptic curve in tripling oriented Doche–Icart–Kohel form is defined by the equation: : T_a\ :\ y^2 = x^3 + 3a(x+1)^2 with a\in K. A general point ''P'' on T_ has affine coordinates (x,y). The "point at infinity" represents the neutral element for the group law and it is written in projective coordinates as O = (0:1:0). The negation of a point ''P'' = (''x'', ''y'') with respect to this neutral element is −''P''&n ... [...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] |
Additive Inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself. The additive inverse of is denoted by unary minus: (see also below). For example, the additive inverse of 7 is −7, because , and the additive inverse of −0.3 is 0.3, because . Similarly, the additive inverse of is which can be simplified to . The additive inverse of is , because . The additive inverse is defined as its inverse element under the binary operation of addition (see also below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: . Common examples For a number (and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel J
Daniel is a masculine given name and a surname of Hebrew origin. It means "God is my judge"Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 68. (cf. Gabriel—"God is my strength"), and derives from two early biblical figures, primary among them Daniel from the Book of Daniel. It is a common given name for males, and is also used as a surname. It is also the basis for various derived given names and surnames. Background The name evolved into over 100 different spellings in countries around the world. Nicknames (Dan, Danny) are common in both English and Hebrew; "Dan" may also be a complete given name rather than a nickname. The name "Daniil" (Даниил) is common in Russia. Feminine versions (Danielle, Danièle, Daniela, Daniella, Dani, Danitza) are prevalent as well. It has been particularly well-used in Ireland. The Dutch names "Daan" and "Daniël" are also variations of Daniel. A related surname developed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PKC (conference) , a local authority in Scotland
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PKC may refer to: * Paroxysmal kinesogenic choreoathetosis, a neurological disorder *Protein kinase C, a family of enzymes *Public-key cryptography, a cryptographic system using pairs of keys ** PKC (conference) *Petropavlovsk-Kamchatsky Airport or Yelizovo Airport, Kamchatka Krai, Russia (IATA code PKC) * PKC Group, a Finnish company * Perth and Kinross Council Perth and Kinross Council ( gd, Comhairle Pheairt is Cheann Rois) is the local government council for the Perth and Kinross council area of Scotland. It employs around 6,000 people. The council was created in 1996, under the '' Local Governme ... [...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] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is Holomorphic function, holomorphic away from a simple pole at the Cusp (singularity), cusp such that :j\left(e^\right) = 0, \quad j(i) = 1728 = 12^3. Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a Cube (algebra), cube, also since 1728 (number), 1728 = 12^3. The given functions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation Solving
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as ''unknowns''. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. Solving an equation ''numerically'' means that only numbers are admitted as solutions. Solving an equation ''symbolically'' means that expressions can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a " continuous function" in topology, a "linear transformation" in linear algebra, etc. Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Curves
In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant. Applications of twists include cryptography, the solution of Diophantine equations, and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture. Quadratic twist First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form: : y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \, Given d\neq 0 not a square in K, the quadratic twist of E is the curve E^d, defined by the equation: : dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \, or equivalently : y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6. \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate System
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the ''number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
