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Primality Testing
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is '' trial division'': given an input number, n, check whether it is divisible by any prime number between 2 and \sqrt n (i.e., whether the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For any divisor p \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sieve Of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite number, composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with arithmetic progression, constant difference between them that is equal to that prime.Horsley, Rev. Samuel, F. R. S., "' or, The Sieve of Eratosthenes. Being an account of his method of finding all the Prime Numbers,''Philosophical Transactions'' (1683–1775), Vol. 62. (1772), pp. 327–347 This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes. The earliest known reference to the sieve (, ''kóskinon Eratosthén ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carmichael Number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation: : b^n\equiv b\pmod for all integers . The relation may also be expressed in the form: : b^\equiv 1\pmod for all integers b that are relatively prime to . They are infinite set, infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of Prime numbers, primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. The Carmichael numbers were named after the American mathematician Robert Daniel Carmichael, Robert Carmichael by N. G. W. H. Beeger, Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short. Overview Fermat's little theorem states that if p is a prime number, then for any integer , the number b^p-b is an i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pseudoprime
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes. Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically lower but still prohibitively high). But finding two large prime numbers as needed for this use is also expensive, so various probabilistic primality tests are used, some of which in rare cases inappropriately deliver composite numbers instead of primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Probable Prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer ''n'', choose some integer ''a'' that is not a multiple of ''n''; (typically, we choose ''a'' in the range ). Calculate . If the result is not 1, then ''n'' is composite. If the result is 1, then ''n'' is likely to be prime; ''n'' is then called a probable prime to base ''a''. A weak probable prime to base ''a'' is an integer that is a probable prime to base ''a'', but which is not a strong probable prime to base ''a'' (see below). For a fixed base ''a'', it is unusual fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sample Space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels ''S'', Ω, or ''U'' (for " universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E. If the outcome of an experiment is included in E, then event E has occurred. For example, if the experiment is tossing a single coin, the sample space is the set \, where the outcome H means that the coin is heads and the outcome T means that the coin is tails. The possible events are E=\, E=\, E = \, and E = \. For tossing two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result ( Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Samuel Wagstaff
Samuel Standfield Wagstaff Jr. (born 21 February 1945) is an American mathematician and computer scientist, whose research interests are in the areas of cryptography, parallel computation, and analysis of algorithms, especially number theoretic algorithms. He is currently a professor of computer science and mathematics at Purdue University who coordinates the Cunningham project, a project to factor numbers of the form ''b''''n'' ± 1, since 1983. He has authored/coauthored over 50 research papers and four books. He has an Erdős number of 1. Wagstaff received his Bachelor of Science in 1966 from Massachusetts Institute of Technology. His doctoral dissertation was titled, On Infinite Matroids, PhD in 1970 from Cornell University. Wagstaff was one of the founding faculty of Center for Education and Research in Information Assurance and Security ( CERIAS) at Purdue, and its precursor, the Computer Operations, Audit, and Security Technology (COAST A coast (coastl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carl Pomerance
Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least seven distinct prime factors. He joined the faculty at the University of Georgia, becoming full professor in 1982. He subsequently worked at Lucent Technologies for a number of years, and then became a distinguished professor at Dartmouth College. Contributions He has over 120 publications, including co-authorship with Richard Crandall of ''Prime numbers: a computational perspective'' (Springer-Verlag, first edition 2001, second edition 2005), and with Paul Erdős. He is the inventor of one of the integer factorization methods, the quadratic sieve algorithm, which was used in 1994 for the factorization of RSA-129. He is also one of the discoverers of the Adleman–Pomerance–Rumely primality test. Awards and honors He h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fibonacci Polynomial
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fibonacci Number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |