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Damgård–Jurik Cryptosystem
The Damgård–Jurik cryptosystemIvan Damgård, Mads JurikA Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System Public Key Cryptography 2001: 119-136 is a generalization of the Paillier cryptosystem. It uses computations modulo n^ where n is an RSA modulus and s a (positive) natural number. Paillier's scheme is the special case with s=1. The order \varphi(n^) ( Euler's totient function) of Z^*_ can be divided by n^s. Moreover, Z^*_ can be written as the direct product of G \times H. G is cyclic and of order n^s, while H is isomorphic to Z^*_n. For encryption, the message is transformed into the corresponding coset of the factor group G\times H/H and the security of the scheme relies on the difficulty of distinguishing random elements in different cosets of H. It is semantically secure if it is hard to decide if two given elements are in the same coset. Like Paillier, the security of Damgård–Jurik can be proven under the decisional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Damgård
Ivan Bjerre Damgård (born 1956) is a Danish cryptographer and currently a professor at the Department of Computer Science, Aarhus University, Denmark. Academic background In 1983, he obtained a master's degree in mathematics (with minors in music and computer science) at Aarhus University. He began his PhD studies in 1985 at the same university, and was for a period a guest researcher at CWI in Amsterdam in 1987. He earned his PhD degree in May, 1988, with the thesis ''Ubetinget beskyttelse i kryptografiske protokoller'' (Unconditional protection in cryptographic protocols) and has been employed at Aarhus University ever since. Damgård became full professor in 2005. Research Damgård co-invented the Merkle–Damgård construction, which is used in influential cryptographic hash functions such as SHA-2, SHA-1 and MD5. He discovered the structure independently of Ralph Merkle and published it in 1989. Ivan Damgård is one of the founders of the Cryptomathic Cryptomath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paillier Cryptosystem
The Paillier cryptosystem, invented by and named after Pascal Paillier in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The problem of computing ''n''-th residue classes is believed to be computationally difficult. The decisional composite residuosity assumption is the intractability hypothesis upon which this cryptosystem is based. The scheme is an additive homomorphic cryptosystem; this means that, given only the public key and the encryption of m_1 and m_2, one can compute the encryption of m_1+m_2. Algorithm The scheme works as follows: Key generation #Choose two large prime numbers p and q randomly and independently of each other such that \gcd(pq, (p-1)(q-1))=1. This property is assured if both primes are of equal length.Jonathan Katz, Yehuda Lindell, "Introduction to Modern Cryptography: Principles and Protocols," Chapman & Hall/CRC, 2007 #Compute n=pq and \lambda=\operatorname(p-1,q-1). lcm means Least Common Multiple. #Select random inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA (algorithm)
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ) (the British signals intelligence agency) by the English mathematician Clifford Cocks. That system was declassified in 1997. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers. The security of RSA relies on the practical difficulty of factoring the product of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Group
Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used in the production of goods and services Science and technology Biology * Coagulation factors, substances essential for blood coagulation * Environmental factor, any abiotic or biotic factor that affects life * Enzyme, proteins that catalyze chemical reactions * Factor B, and factor D, peptides involved in the alternate pathway of immune system complement activation * Transcription factor, a protein that binds to specific DNA sequences Computer science and information technology * Factor (programming language), a concatenative stack-oriented programming language * Factor (Unix), a utility for factoring an integer into its prime factors * Factor, a substring, a subsequence of consecutive symbols in a st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantically Secure
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] |
Decisional Composite Residuosity Assumption
The decisional composite residuosity assumption (DCRA) is a mathematical assumption used in cryptography. In particular, the assumption is used in the proof of the Paillier cryptosystem. Informally, the DCRA states that given a composite n and an integer z, it is hard to decide whether z is an n-residue modulo n^2. ''I.e.'' whether there exists a y such that : z \equiv y^n \pmod. \, See also * Quadratic residuosity problem * Higher residuosity problem In cryptography, most public key cryptosystems are founded on problems that are believed to be intractable. The higher residuosity problem (also called the n th-residuosity problem) is one such problem. This problem is ''easier'' to solve than ... References * P. Paillier, ''Public-Key Cryptosystems Based on Composite Degree Residuosity Classes'', Eurocrypt 1999. {{DEFAULTSORT:Decisional Composite Residuosity Assumption Computational hardness assumptions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
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] |
Relative Prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
