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Random Self-reducibility
Random self-reducibility (RSR) is the rule that a good algorithm for the average case implies a good algorithm for the worst case. RSR is the ability to solve all instances of a problem by solving a large fraction of the instances. Definition If for a function ''f'' evaluating any instance ''x'' can be reduced in polynomial time to the evaluation of ''f'' on one or more random instances ''yi'', then it is self-reducible (this is also known as a ''non-adaptive uniform self-reduction''). In a random self-reduction, an arbitrary worst-case instance ''x'' in the domain of ''f'' is mapped to a random set of instances ''y''1, ..., ''yk''. This is done so that ''f''(''x'') can be computed in polynomial time, given the coin-toss sequence from the mapping, ''x'', and ''f''(''y''1), ..., ''f''(''yk''). Therefore, taking the average with respect to the induced distribution on ''yi'', the average-case complexity of ''f'' is the same (within polynomial factors) as the worst-case randomized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (computer Science)
In computer programming, a function or subroutine is a sequence of program instructions that performs a specific task, packaged as a unit. This unit can then be used in programs wherever that particular task should be performed. Functions may be defined within programs, or separately in libraries that can be used by many programs. In different programming languages, a function may be called a routine, subprogram, subroutine, method, or procedure. Technically, these terms all have different definitions, and the nomenclature varies from language to language. The generic umbrella term ''callable unit'' is sometimes used. A function is often coded so that it can be started several times and from several places during one execution of the program, including from other functions, and then branch back (''return'') to the next instruction after the ''call'', once the function's task is done. The idea of a subroutine was initially conceived by John Mauchly during his work on ENIAC, ... [...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 produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) which ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-Complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no two x_j the same, a polynomial function p(x) is said to interpolate the data if p(x_j)=y_j for each j\in\. Two common explicit formulas for this polynomial are the Lagrange polynomials and Newton polynomials. Applications Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permanent Is Sharp-P-complete , Buddhist concept
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{{disambiguation ...
Permanent may refer to: Art and entertainment * ''Permanent'' (film), a 2017 American film * ''Permanent'' (Joy Division album) * "Permanent" (song), by David Cook Other uses *Permanent (mathematics), a concept in linear algebra *Permanent (cycling event) *Permanent wave, a hairstyling process See also *Permanence (other) *''Permanently'', a 2000 album by Mark Wills *Endless (other) *Eternal (other) *Forever (other) *Impermanence Impermanence, also known as the philosophical problem of change, is a philosophical concept addressed in a variety of religions and philosophies. In Eastern philosophy it is notable for its role in the Buddhist three marks of existence. It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant. Definition The permanent of an matrix is defined as \operatorname(A)=\sum_\prod_^n a_. The sum here extends over all elements σ of the symmetric group ''S''''n''; i.e. over all permutations of the numbers 1, 2, ..., ''n''. For example, \operatorname\begina&b \\ c&d\end=ad+bc, and \operatorname\begina&b&c \\ d&e&f \\ g&h&i \end=aei + bfg + cdh + ceg + bdi + afh. The definition of the permanent of ''A'' differs from that of the determinant of ''A'' in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per ''A'', perm ''A'', or Per ''A'', sometimes with parentheses around the argument. Minc uses Per(''A'') for the permanent of rectangular m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Residuosity Problem
The quadratic residuosity problem (QRP) in computational number theory is to decide, given integers a and N, whether a is a quadratic residue modulo N or not. Here N = p_1 p_2 for two unknown primes p_1 and p_2, and a is among the numbers which are not obviously quadratic non-residues (see below). The problem was first described by Gauss in his '' Disquisitiones Arithmeticae'' in 1801. This problem is believed to be computationally difficult. Several cryptographic methods rely on its hardness, see . An efficient algorithm for the quadratic residuosity problem immediately implies efficient algorithms for other number theoretic problems, such as deciding whether a composite N of unknown factorization is the product of 2 or 3 primes. Precise formulation Given integers a and T, a is said to be a ''quadratic residue modulo T'' if there exists an integer b such that :a \equiv b^2 \pmod T. Otherwise we say it is a quadratic non-residue. When T = p is a prime, it is customary to us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy. The fields of ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
