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Tate Pairing
In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by and extended by . applied the Tate pairing over finite fields to cryptography. See also * Weil pairing Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg * Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with ... References * * * * Pairing-based cryptography Elliptic curve cryptography Elliptic curves {{Crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairing-based Cryptography
Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group with a mapping e :G_1 \times G_2 \to G_T to construct or analyze cryptographic systems. Definition The following definition is commonly used in most academic papers. Let F_q be a Finite field over prime q, G_1, G_2 two additive cyclic groups of prime order q and G_T another cyclic group of order q written multiplicatively. A pairing is a map: e: G_1 \times G_2 \rightarrow G_T , which satisfies the following properties: ; Bilinearity: \forall a,b \in F_q^*, P\in G_1, Q\in G_2:\ e\left(aP, bQ\right) = e\left(P, Q\right)^ ; Non-degeneracy: e \neq 1 ; Computability: There exists an efficient algorithm to compute e. Classification If the same group is used for the first two groups (i.e. G_1 = G_2), the pairing is called ''symmetric'' and is a mapping from two elements of one group to an element from a second group. Some researchers classify pairing instantiations int ... [...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] |
Abelian Varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local field k, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology: :\displaystyle H^r(k,M)\times H^(k,M')\rightarrow H^2(k,\mathbb_m)=\Q/ \Z where M is a finite group scheme, M' its dual \operatorname(M,G_m), and \mathbb_m is the multiplicative group. For a local field of characteristic p>0, the statement is similar, except that the pairing takes values in H^2(k, \mu) = \bigcup_ \tfrac \Z/\Z. The statement also holds when k is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case. Global Tate duality Given a finite group scheme M over a global field k, global Tate duality relates the cohomology of M with that of M' = \operatorname(M,G_m) using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Pairing
Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg *Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with the surname Weill, Weyl * Doctor Weil (Mega Man Zero), a fictional character from the ''Mega Man'' Zero video game series * Weil-Marbach, now the Marbach Stud in Baden-Württemberg See also * Weill (other) * Weil, Gotshal & Manges Weil, Gotshal & Manges LLP is an American international law firm with approximately 1,100 attorneys, headquartered in New York City. With a gross annual revenue in excess of $1.8 billion, it is among the world's largest law firms according to ..., law firm founded in the United States * Weil's disease {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...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 compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.Commercial National Security Algorithm Suite and Quantum Computing FAQ U.S. National Security Agency, January 2016. Elliptic curves are applicable for , s, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
