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Schnorr Group
A Schnorr group, proposed by Claus P. Schnorr, is a large prime-order subgroup of \mathbb_p^\times, the multiplicative group of integers modulo p for some prime p. To generate such a group, generate p, q, r such that :p = qr + 1 with p, q prime. Then choose any h in the range 1 < h < p until you find one such that :. This value : is a generator of a subgroup of of order . Schnorr groups are useful in based cryptosystems including Schnorr signatures and [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claus P
Claus (sometimes Clas) is both a given name and a German, Danish, and Dutch surname. Notable people with the name include: Given name *Claus Schenk Graf von Stauffenberg (1907–1944), a German officer who, along with others, attempted to assassinate Hitler in 1944 *Claus von Amsberg, Prince Claus of the Netherlands, Jonkheer van Amsberg (1926–2002) * Claus von Bülow (born 1926), British socialite accused of attempting to murder his wife, Sunny von Bülow * Claus Clausen (other), three people of that name *Claus Bech Jørgensen (born 1976), Danish-born Faroese footballer * Claus Jacob (born 1969), German scientist *Claus Jørgensen (racewalker) (born 1974), Danish race walker * Claus Larsen (other), three people of that name *Claus Lundekvam (born 1973), Norwegian former footballer *Claus Moser, Baron Moser (born 1922), British statistician * Claus Nielsen (born 1964), Danish former football striker *Claus Norreen (born 1970), Danish musician with the band Aqua, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Group Of Integers Modulo N
In modular arithmetic, the integers coprime (relatively prime) to ''n'' from the set \ of ''n'' non-negative integers form a group under multiplication modulo ''n'', called the multiplicative group of integers modulo ''n''. Equivalently, the elements of this group can be thought of as the congruence classes, also known as ''residues'' modulo ''n'', that are coprime to ''n''. Hence another name is the group of primitive residue classes modulo ''n''. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo ''n''. Here ''units'' refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to ''n''. This quotient group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: , (\mathbb/n\mathbb)^\times, =\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Log
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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptosystems
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term '' cipher'' (sometimes ''cypher'') is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term ''cryptosystem'' is most often used when the key generation algorithm is important. For this reason, the term ''cryptosystem'' is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques. Formal definition Mathematically, a cryptosystem or encryption scheme can be defined as a tuple (\mathcal,\mathcal,\mathcal,\mathcal,\mathcal) with the following properties. # \mathcal is a set called the "plaintext space". Its elements are called plaintexts. # \mathcal is a set called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schnorr Signature
In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr. It is a digital signature scheme known for its simplicity, among the first whose security is based on the intractability of certain discrete logarithm problems. It is efficient and generates short signatures. It was covered by which expired in February 2008. Algorithm Choosing parameters *All users of the signature scheme agree on a group, G, of prime order, q, with generator, g, in which the discrete log problem is assumed to be hard. Typically a Schnorr group is used. *All users agree on a cryptographic hash function H: \^* \rightarrow \mathbb_q. Notation In the following, *Exponentiation stands for repeated application of the group operation *Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable) *Subtraction stands for subtraction on the set of congruence classe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Signature Algorithm
The Digital Signature Algorithm (DSA) is a public-key cryptosystem and Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. DSA is a variant of the Schnorr and ElGamal signature schemes. The National Institute of Standards and Technology (NIST) proposed DSA for use in their Digital Signature Standard (DSS) in 1991, and adopted it as FIPS 186 in 1994. Four revisions to the initial specification have been released. The newest specification isFIPS 186-4 from July 2013. DSA is patented but NIST has made this patent available worldwide royalty-free. A draft version of the specificatioFIPS 186-5indicates DSA will no longer be approved for digital signature generation, but may be used to verify signatures generated prior to the implementation date of that standard. Overview The DSA works in the framework of public-key cryptosystems and is based on the algebraic properties of mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Calculus
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete logarithm in (\mathbb/q\mathbb)^* where q is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm with respect to the discrete logarithms of small primes. Description Roughly speaking, the discrete log problem asks us to find an ''x'' such that g^x \equiv h \pmod, where ''g'', ''h'', and the modulus ''n'' are given. The algorithm (described in detail below) applies to the group (\mathbb/q\mathbb)^* where ''q'' is prime. It requires a ''factor base'' as input. This ''factor base'' is usually chosen to be the number −1 and the first ''r'' primes starting with 2. From the point of view o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birthday Attack
A birthday attack is a type of cryptographic attack that exploits the mathematics behind the birthday problem in probability theory. This attack can be used to abuse communication between two or more parties. The attack depends on the higher likelihood of collisions found between random attack attempts and a fixed degree of permutations (pigeonholes). With a birthday attack, it is possible to find a collision of a hash function in \sqrt = 2^, with 2^n being the classical preimage resistance security. There is a general (though disputed) result that quantum computers can perform birthday attacks, thus breaking collision resistance, in \sqrt = 2^. Although there are some digital signature vulnerabilities associated with the birthday attack, it cannot be used to break an encryption scheme any faster than a brute-force attack. Understanding the problem As an example, consider the scenario in which a teacher with a class of 30 students (n = 30) asks for everybody's birthday (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Subgroup Confinement Attack
In cryptography, a subgroup confinement attack, or small subgroup confinement attack, on a cryptographic method that operates in a large finite group is where an attacker attempts to compromise the method by forcing a key to be confined to an unexpectedly small subgroup of the desired group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide .... Several methods have been found to be vulnerable to subgroup confinement attack, including some forms or applications of Diffie–Hellman key exchange and DH-EKE. References * * * Cryptographic attacks Finite groups {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
