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Doubling-oriented Doche–Icart–Kohel Curve
In mathematics, the doubling-oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written. It is a special case of the Weierstrass form and it is also important in elliptic-curve cryptography because the doubling speeds up considerably (computing as composition of 2- isogeny and its dual). It was introduced by Christophe Doche, Thomas Icart, and David R. Kohel in ''Efficient Scalar Multiplication by Isogeny Decompositions.''Christophe Doche, Thomas Icart, and David R. Kohel, ''Efficient Scalar Multiplication by Isogeny Decompositions'' Definition Let K be a field and let a\in K. Then, the doubling-oriented Doche–Icart–Kohel curve with parameter ''a'' in affine coordinates is represented by: y^2=x^3+ax^2+16ax. Equivalently, in projective coordinates: ZY^2=X^3+aZX^2+16aXZ^2, with x=\frac and y=\frac . Since this curve is a special case of the Weierstrass form, transformations to the most common form of elliptic curve (Weierstrass for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Oriented
Doubling may refer to: Mathematics * Arithmetical doubling of a count or a measure, expressed as: ** Multiplication by 2 ** Increase by 100%, i.e. one-hundred percent ** Doubling the cube (i. e., hypothetical geometric construction of a cube with twice the volume of a given cube) * Doubling time, the length of time required for a quantity to double in size or value * Doubling map, a particular infinite two-dimensional geometrical construction Music * The composition or performance of a melody with itself or itself transposed at a constant interval such as the octave, third, or sixth, Voicing (music)#Doubling * The assignment of a melody to two instruments in an arrangement * The playing of two (or more) instruments alternately by a single player, e.g. ''Flute, doubling piccolo'' ** Musicians who play more than one woodwind instrument are called woodwind doublers or reed players * Double tracking, a recording technique in which a musical part (or vocal) is recorded twice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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] |
Elliptic Curve Cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys to provide equivalent security, compared to cryptosystems based on modular exponentiation in Galois fields, such as the RSA cryptosystem and ElGamal cryptosystem. Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme. They are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. History The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005. In 1999, NIST recommended fifteen elliptic curves. Specifically, FIPS 186 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogeny
In mathematics, particularly in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that . Such an isogeny then provides a group homomorphism between the groups of -valued points of and , for any field over which is defined. The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny. Degree of isogeny Let be isogeny between two algebraic groups. This mapping induces a pullback mapping between their rational function fields. Since the mapping is nontrivial, it is a field embedding and \operatorname f^* is a subfield of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Abelian Variety
In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field (mathematics), field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. Definition Let ''A'' be an abelian variety over a field ''k''. We define \operatorname^0 (A) \subset \operatorname (A) to be the subgroup of the Picard group consisting of line bundles ''L'' such that m^*L \cong p^*L \otimes q^*L, where m, p, q are the multiplication and projection maps A \times_k A \to A respectively. An element of \operatorname^0(A) is called a degree 0 line bundle on ''A''. To ''A'' one then associates a dual abelian variety ''A''v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a ''k''-variety ''T'' is defined to be a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. ''Parameter'' has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. Modelization When a system theory, system is modeled by equations, the values that describe the system are called ''parameters''. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and t ... [...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. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called '' points'', which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through points in general position, a -dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines wi ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation By Squaring
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add. Basic method Recursive version The method is based on the observation that, for any integer n > 0, one has: x^n= \begin x \, ( x^)^, & \mbox n \mbox \\ (x^)^ , & \mbox n \mbox \end If the exponent is zero then the answer is 1. If the exponent is negative then we can reuse the previous formula by rewriting the value using a positive exponent. That is, x^n = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
