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Computational Diffie–Hellman Assumption
The computational Diffie–Hellman (CDH) assumption is a computational hardness assumption about the Diffie–Hellman problem. The CDH assumption involves the problem of computing the discrete logarithm in cyclic groups. The CDH problem illustrates the attack of an eavesdropper in the Diffie–Hellman key exchange protocol to obtain the exchanged secret key. Definition Consider a cyclic group ''G'' of order ''q''. The CDH assumption states that, given :(g,g^a,g^b) \, for a randomly chosen generator ''g'' and random :a,b \in \,\, it is computationally intractable to compute the value :g^. \, Relation to Discrete Logarithms The CDH assumption is strongly related to the discrete logarithm assumption. If computing the discrete logarithm (base ''g'' ) in ''G'' were easy, then the CDH problem could be solved easily: Given :(g,g^a,g^b), \, one could efficiently compute g^ in the following way: * compute a by taking the discrete log of g^a to base g; * compute g^ by exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Hardness Assumption
In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where ''efficiently'' typically means "in polynomial time"). It is not known how to prove (unconditional) hardness for essentially any useful problem. Instead, computer scientists rely on reductions to formally relate the hardness of a new or complicated problem to a computational hardness assumption about a problem that is better-understood. Computational hardness assumptions are of particular importance in cryptography. A major goal in cryptography is to create cryptographic primitives with provable security. In some cases, cryptographic protocols are found to have information theoretic security; the one-time pad is a common example. However, information theoretic security cannot always be achieved; in such cases, cryptographers fall back to computational security. Roughly speaking, this means that these systems are secure ''assuming th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffie–Hellman Problem
The Diffie–Hellman problem (DHP) is a mathematical problem first proposed by Whitfield Diffie and Martin Hellman in the context of cryptography. The motivation for this problem is that many security systems use one-way functions: mathematical operations that are fast to compute, but hard to reverse. For example, they enable encrypting a message, but reversing the encryption is difficult. If solving the DHP were easy, these systems would be easily broken. Problem description The Diffie–Hellman problem is stated informally as follows: : Given an element ''g'' and the values of ''gx'' and ''gy'', what is the value of ''gxy''? Formally, ''g'' is a generator of some group (typically the multiplicative group of a finite field or an elliptic curve group) and ''x'' and ''y'' are randomly chosen integers. For example, in the Diffie–Hellman key exchange, an eavesdropper observes ''gx'' and ''gy'' exchanged as part of the protocol, and the two parties both compute the shared key ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm
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 mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Groups
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffie–Hellman Key Exchange
Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exchange * Diffie–Hellman protocol * Diffie–Hellman handshake is a mathematical method of securely exchanging cryptographic keys over a public channel and was one of the first public-key protocols as conceived by Ralph Merkle and named after Whitfield Diffie and Martin Hellman. DH is one of the earliest practical examples of public key exchange implemented within the field of cryptography. Published in 1976 by Diffie and Hellman, this is the earliest publicly known work that proposed the idea of a private key and a corresponding public key. Traditionally, secure encrypted communication between two parties required that they first exchange keys by some secure physical means, such as paper key lists transported by a trusted courier. The Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computationally Intractable
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm Assumption
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] |
Decisional Diffie–Hellman Assumption
The decisional Diffie–Hellman (DDH) assumption is a computational hardness assumption about a certain problem involving discrete logarithms in cyclic groups. It is used as the basis to prove the security of many cryptographic protocols, most notably the ElGamal and Cramer–Shoup cryptosystems. Definition Consider a (multiplicative) cyclic group G of order q, and with generator g. The DDH assumption states that, given g^a and g^b for uniformly and independently chosen a,b \in \mathbb_q, the value g^ "looks like" a random element in G. This intuitive notion can be formally stated by saying that the following two probability distributions are computationally indistinguishable (in the security parameter, n=\log(q)): * (g^a,g^b,g^), where a and b are randomly and independently chosen from \mathbb_q. * (g^a,g^b,g^c), where a,b,c are randomly and independently chosen from \mathbb_q. Triples of the first kind are often called DDH triplet or DDH tuples. Relation to other assumptions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantic Security
In cryptography, a semantically secure cryptosystem is one where only negligible information about the plaintext can be feasibly extracted from the ciphertext. Specifically, any probabilistic, polynomial-time algorithm (PPTA) that is given the ciphertext of a certain message m (taken from any distribution of messages), and the message's length, cannot determine any partial information on the message with probability non-negligibly higher than all other PPTA's that only have access to the message length (and not the ciphertext). S. Goldwasser and S. MicaliProbabilistic encryption & how to play mental poker keeping secret all partial information Annual ACM Symposium on Theory of Computing, 1982. This concept is the computational complexity analogue to Shannon's concept of perfect secrecy. Perfect secrecy means that the ciphertext reveals no information at all about the plaintext, whereas semantic security implies that any information revealed cannot be feasibly extracted. Goldreich, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ElGamal Encryption
In cryptography, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie–Hellman key exchange. It was described by Taher Elgamal in 1985. ElGamal encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. ElGamal encryption can be defined over any cyclic group G, like multiplicative group of integers modulo ''n''. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms. The algorithm ElGamal encryption consists of three components: the key generator, the encryption algorithm, and the decryption algorithm. Key generation The first party, Alice, generates a key pair as follows: * Generate an efficient description of a cyclic group G\, of order q\, with g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
