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Cryptographic Multilinear Map
A cryptographic n-multilinear map is a kind of multilinear map, that is, a function e:G_1\times \cdots \times G_n \rightarrow G_T such that for any integers a_1, \ldots, a_n and elements g_i \in G_i , e(g_1^,\ldots,g_n^)=e(g_1,\ldots,g_n)^, and which in addition is efficiently computable and satisfies some security properties. It has several applications on cryptography, as key exchange protocols, identity-based encryption ID-based encryption, or identity-based encryption (IBE), is an important primitive of ID-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 ( ..., and broadcast encryption. There exist constructions of cryptographic 2-multilinear maps, known as bilinear maps, however, the problem of constructing such multilinear maps for n > 2 seems much more difficult and the security of the proposed candidates is still unclear. Definition For ''n'' = 2 In this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are vector spaces (or modules over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then f(v_1, \ldots, v_i, \ldots, v_n) is a linear function of v_i. A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra. If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating ''k''-linear maps. The latter coincide if the underlying ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Key Exchange
Key exchange (also key establishment) is a method in cryptography by which cryptographic keys are exchanged between two parties, allowing use of a cryptographic algorithm. If the sender and receiver wish to exchange encrypted messages, each must be equipped to encrypt messages to be sent and decrypt messages received. The nature of the equipping they require depends on the encryption technique they might use. If they use a code, both will require a copy of the same codebook. If they use a cipher, they will need appropriate keys. If the cipher is a symmetric key cipher, both will need a copy of the same key. If it is an asymmetric key cipher with the public/private key property, both will need the other's public key. Channel of exchange Key exchange is done either in-band or out-of-band. The key exchange problem The key exchange problem describes ways to exchange whatever keys or other information are needed for establishing a secure communication channel so that no one else ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ID-based Encryption
ID-based encryption, or identity-based encryption (IBE), is an important primitive of ID-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. ID-based encryption was proposed by Adi Shamir in 1984. He was however only able to give an instantiation of identity-based signatures. Identity-based encryption remained an open problem for many years. The pairing-based Boneh–Franklin scheme and Cocks's encryption scheme based on quadratic residues both solved the IBE problem in 2001. Usage Identity-based systems allow any party to generate a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broadcast Encryption
Broadcast encryption is the cryptographic problem of delivering encrypted content (e.g. TV programs or data on DVDs) over a broadcast channel in such a way that only qualified users (e.g. subscribers who have paid their fees or DVD players conforming to a specification) can decrypt the content. The challenge arises from the requirement that the set of qualified users can change in each broadcast emission, and therefore revocation of individual users or user groups should be possible using broadcast transmissions, only, and without affecting any remaining users. As efficient revocation is the primary objective of broadcast encryption, solutions are also referred to as revocation schemes. Rather than directly encrypting the content for qualified users, broadcast encryption schemes distribute keying information that allows qualified users to reconstruct the content encryption key whereas revoked users find insufficient information to recover the key. The typical setting considered is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinearity
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V and w_1, w_2 \in W, then B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) and B(v, w_1 + w_2) = B(v ... [...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, and the term 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm Problem
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b''''k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ''a'' is an integer ''k'' such that . In number theory, the more commonly used term is index: we can write ''x'' = ind''r'' ''a'' (mod ''m'') (read "the index of ''a'' to the base ''r'' modulo ''m''") for ''r''''x'' ≡ ''a'' (mod ''m'') if ''r'' is a primitive root of ''m'' and gcd(''a'',''m'') = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Definition Let ''G'' be any group. Denote its group operation by mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security ( data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synonymo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
