|
Pairing
In mathematics, a pairing is an ''R''- bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-modules. A pairing is any ''R''-bilinear map e:M \times N \to L. That is, it satisfies :e(r\cdot m,n)=e(m,r \cdot n)=r\cdot e(m,n), :e(m_1+m_2,n)=e(m_1,n)+e(m_2,n) and e(m,n_1+n_2)=e(m,n_1)+e(m,n_2) for any r \in R and any m,m_1,m_2 \in M and any n,n_1,n_2 \in N . Equivalently, a pairing is an ''R''-linear map :M \otimes_R N \to L where M \otimes_R N denotes the tensor product of ''M'' and ''N''. A pairing can also be considered as an ''R''-linear map \Phi : M \to \operatorname_ (N, L) , which matches the first definition by setting \Phi (m) (n) := e(m,n) . A pairing is called perfect if the above map \Phi is an isomorphism of ''R''-modules and the other evaluation map \Phi'\colon N\to \operatorname_(M,L) is an isomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing ''n'' of an elliptic curve ''E'', taking values in ''n''th roots of unity. More generally there is a similar Weil pairing between points of order ''n'' of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function. Formulation Choose an elliptic curve ''E'' defined over a field ''K'', and an integer ''n'' > 0 (we require ''n'' to be coprime to char(''K'') if char(''K'') > 0) such that ''K'' contains a primitive nth root of unity. Then the ''n''-torsion on E(\overline) is known to be a Cartesian product of two cyclic groups of order ''n''. The Weil pairing produces an ''n''-th root of unity :w(P,Q) \in \mu_n by mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity-based Encryption
Identity-based encryption (IBE), is an important primitive of identity-based cryptography. As such it is a type of public-key encryption in which the public key of a user is some unique information about the identity of the user (e.g. a user's email address). This means that a sender who has access to the public parameters of the system can encrypt a message using e.g. the text-value of the receiver's name or email address as a key. The receiver obtains its decryption key from a central authority, which needs to be trusted as it generates secret keys for every user. Identity-based encryption was proposed by Adi Shamir in 1984. He was however only able to give an instantiation of Identity-based cryptography, identity-based signatures. Identity-based encryption remained an open problem for many years. The pairing-based cryptography, pairing-based Boneh–Franklin scheme and Cocks IBE scheme, Cocks's encryption scheme based on quadratic residues both solved the IBE problem in 2001. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual System
In mathematics, a dual system, dual pair or a duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces, X and Y, over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb. In mathematics, duality is the study of dual systems and is important in functional analysis. Duality plays crucial roles in quantum mechanics because it has extensive applications to the theory of Hilbert spaces. Definition, notation, and conventions Pairings A or pair over a field \mathbb is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over \mathbb and a bilinear map b : X \times Y \to \mathbb called the bilinear map associated with the pairing, or more simply called the pairing's map or its bilinear form. The examples here only describe when \mathbb is either the real numbers or the complex numbers \Complex, but the mathematical theory is general. For every x \in X, define \begin b(x, \,\cdot\,) : \,& Y && \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Homomorphism
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ''R''-''linear map'' if for any ''x'', ''y'' in ''M'' and ''r'' in ''R'', :f(x + y) = f(x) + f(y), :f(rx) = rf(x). In other words, ''f'' is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If ''M'', ''N'' are right ''R''-modules, then the second condition is replaced with :f(xr) = f(x)r. The preimage of the zero element under ''f'' is called the kernel of ''f''. The set of all module homomorphisms from ''M'' to ''N'' is denoted by \operatorname_R(M, N). It is an abelian group (under pointwise addition) but is not necessarily a module unless ''R'' is commutative. The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorph ... [...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] |
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] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Product
In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: \operatorname^n(M, N) \otimes \operatorname^m(L, M) \to \operatorname^(L, N) induced by \operatorname(N, M) \otimes \operatorname(M, L) \to \operatorname(N, L),\, f \otimes g \mapsto g \circ f. Specifically, for an element \xi \in \operatorname^n(M, N) , thought of as an extension \xi : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_ \rightarrow M \rightarrow 0 , and similarly \rho : 0 \rightarrow M \rightarrow F_0\rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^m(L, M), we form the Yoneda (cup) product \xi \smile \rho : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_ \rightarrow F_0 \rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^(L, N). Note that the middle map E_ \rightarrow F_0 factors through the given maps to M. We extend this definition to include m, n = 0 using the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
