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Pollard's Rho Algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized. Core ideas The algorithm is used to factorize a number n = pq, where p is a non-trivial factor. A polynomial modulo n, called g(x) (e.g., g(x) = (x^2 + 1) \bmod n), is used to generate a pseudorandom sequence. It is important to note that g(x) must be a polynomial. A starting value, say 2, is chosen, and the sequence continues as x_1 = g(2), x_2 = g(g(2)), x_3 = g(g(g(2))), etc. The sequence is related to another sequence \. Since p is not known beforehand, this sequence cannot be explicitly computed in the algorithm. Yet, in it lies the core idea of the algorithm. Because the number of possible values for these sequences is finite, both the \ sequence, which is mod n, and \ sequence will eventually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Algorithms
''Introduction to Algorithms'' is a book on computer programming by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The book has been widely used as the textbook for algorithms courses at many universities and is commonly cited as a reference for algorithms in published papers, with over 10,000 citations documented on CiteSeerX. The book sold half a million copies during its first 20 years. Its fame has led to the common use of the abbreviation "CLRS" (Cormen, Leiserson, Rivest, Stein), or, in the first edition, "CLR" (Cormen, Leiserson, Rivest). In the preface, the authors write about how the book was written to be comprehensive and useful in both teaching and professional environments. Each chapter focuses on an algorithm, and discusses its design techniques and areas of application. Instead of using a specific programming language, the algorithms are written in pseudocode. The descriptions focus on the aspects of the algorithm itself, its ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard's Kangaroo Algorithm
In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime ''p'', it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group. Algorithm Suppose G is a finite cyclic group of order n which is generated by the element \alpha, and we seek to find the discrete logarithm x of the element \beta to the base \alpha. In other words, one seeks x \in Z_n such that \alpha^x = \beta. The lambda algorithm allows one to search for x in some interval ,\ldots,bsubset Z_n. One may search the entire range of poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard's Rho Algorithm For Logarithms
Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute \gamma such that \alpha ^ \gamma = \beta, where \beta belongs to a cyclic group G generated by \alpha. The algorithm computes integers a, b, A, and B such that \alpha^a \beta^b = \alpha^A \beta^B. If the underlying group is cyclic of order n, by substituting \beta as a^ and noting that two powers are equal if and only if the exponents are equivalent modulo the order of the base, in this case modulo n, we get that \gamma is one of the solutions of the equation (B-b) \gamma = (a-A) \pmod n. Solutions to this equation are easily obtained using the extended Euclidean algorithm. To find the needed a, b, A, and B the algorithm uses Floyd's cycle-finding algorithm to find a cycle in the sequence x_i = \alpha^ \beta^, where the function f: x_i \maps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birthday Paradox
In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 50% in a group of only 23 people. The birthday paradox is a veridical paradox: it appears wrong, but is in fact true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the comparisons of birthdays will be made between every possible pair of individuals. With 23 individuals, there are (23 × 22) / 2 = 253 pairs to consider, much more than half the number of days in a year. Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UNIVAC 1110
The UNIVAC 1100/2200 series is a series of compatible 36-bit computer systems, beginning with the UNIVAC 1107 in 1962, initially made by Sperry Rand. The series continues to be supported today by Unisys Corporation as the ClearPath Dorado Series. The solid-state 1107 model number was in the same sequence as the earlier vacuum-tube computers, but the early computers were not compatible with the solid-state successors. Architecture Data formats * Fixed-point, either integer or fraction **Whole word – 36-bit (ones' complement) **Half word – two 18-bit fields per word (unsigned or ones' complement) **Third word – three 12-bit fields per word (ones' complement) **Quarter word – four 9-bit fields per word (unsigned) **Sixth word – six 6-bit fields per word (unsigned) *Floating point **Single precision – 36 bits: sign bit, 8-bit characteristic, 27-bit mantissa **Double precision – 72 bits: sign bit, 11-bit characteristic, 60-bit mantissa *Alphanumeric ** FIELDATA – UNIV ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UNIVAC
UNIVAC (Universal Automatic Computer) was a line of electronic digital stored-program computers starting with the products of the Eckert–Mauchly Computer Corporation. Later the name was applied to a division of the Remington Rand company and successor organizations. The BINAC, built by the Eckert–Mauchly Computer Corporation, was the first general-purpose computer for commercial use, but it was not a success. The last UNIVAC-badged computer was produced in 1986. History and structure J. Presper Eckert and John Mauchly built the ENIAC (Electronic Numerical Integrator and Computer) at the University of Pennsylvania's Moore School of Electrical Engineering between 1943 and 1946. A 1946 patent rights dispute with the university led Eckert and Mauchly to depart the Moore School to form the Electronic Control Company, later renamed Eckert–Mauchly Computer Corporation (EMCC), based in Philadelphia, Pennsylvania. That company first built a computer called BINAC (BINary Automat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2''k'' + 1 is prime and ''k'' > 0, then ''k'' must be a power of 2, so 2''k'' + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are ''F''0 = 3, ''F''1 = 5, ''F''2 = 17, ''F''3 = 257, and ''F''4 = 65537 ; heuristics suggest that there are no more. Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 ( caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is ''quadratic''. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of iterated function values : x_0,\ x_1=f(x_0),\ x_2=f(x_1),\ \dots,\ x_i=f(x_),\ \dots must eventually use the same value twice: there must be some pair of distinct indices and such that . Once this happens, the sequence must continue periodically, by repeating the same sequence of values from to . Cycle detection is the problem of finding and , given and . Several algorithms for finding cycles quickly and with little memory are known. Robert W. Floyd's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of memory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory (in particular factorisation), random number generators, computer architecture, and analysis of algorithms. In 1973, he published a root-finding algorithm (an algorithm for solving equations numerically) which is now known as Brent's method. In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of \pi. At the same time, he showed that all the elementary functions (such as log(''x''), sin(''x'') etc.) can be evaluated to high precision in the same time as \pi (apart from a small constant factor) using the arithmetic-geometric mean of Carl Friedrich Gauss. In 1979 he showed that the first 75 million complex zeros of the Rieman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
