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Topic: Prime factorization algorithm

Related Topics

Integer factorization

Prime number

Fundamental theorem of arithmetic

1 E4

Euclid number

Factorization

List of number theory topics

Mersenne prime

Table of prime factors

Composite number

Prime number theorem

Sieve of Eratosthenes

Illegal prime

Riemann zeta function

Euclidean algorithm

	Prime number - Wikipedia, the free encyclopedia
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
		A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime.
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.
	en.wikipedia.org /wiki/Prime_number (3273 words)

	Prime factorization algorithm - Wikipedia, the free encyclopedia
		A prime factorization algorithm is any algorithm by which an integer (whole number) is "decomposed" into a product of factors that are prime numbers.
		This article gives a simple example of an algorithm, which works well for numbers whose prime factors are small; faster algorithms for numbers with larger prime factors are discussed in the article on integer factorization.
		Factorizer Windows software to decompose numbers up to 2,147,483,646 into their prime constituents.
	en.wikipedia.org /wiki/Prime_factorization_algorithm (519 words)

	Learn more about Fundamental theorem of arithmetic in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		Once the prime factorizations of two numbers are known, their greatest common divisor and least common multiple can be found quickly.
		However if the prime factorizations are not known, the use of Euclid's algorithm generally requires much less calculation than factoring the two numbers.
		Another proof of the uniqueness of the prime factorization of a given integer uses infinite descent: Assume that a certain integer can be written as (at least) two different products of prime numbers, then there must exist a smallest integer s with such a property.
	www.onlineencyclopedia.org /f/fu/fundamental_theorem_of_arithmetic.html (830 words)

	A simple factorization algorithm for univariate polynomials
		The algorithm was named the binary factoring algorithm since it determines factors to a polynomial modulo 2^n for successive values of n, effectively adding one binary digit to the solution in each iteration.
		The algorithm presented here is similar in approach to applying the Berlekamp algorithm to factor modulo a small prime, and then factoring modulo powers of this prime (using the solutions found modulo the small prime by the Berlekamp algorithm) by applying Hensel lifting.
		It is not a requirement of the algorithm that the algorithm being worked with is square-free, but it speeds up computations to work with the square-free part of the polynomial if the only thing sought after is the set of factors.
	yacas.sourceforge.net /Algochapter3.html (2777 words)

	[No title]
		The most famous example of such an algorithm is Shor's prime factorization algorithm, which will be discussed along with his less familiar algorithm used for computing discrete logarithms.
		Prime Factorization Algorithm Many elements of society depend upon the current computational speed of computers, and if they were suddenly able to factor a large number into two primes in polynomial time, certain issues might arise.
		The best algorithm for factoring arbitrary integers, both in theory and practice, is of exponential time complexity, resulting in prohibitively long calculations.
	www.cs.caltech.edu /~justinw/final.doc (2814 words)

	COURSE DESCRIPTION
		Algorithms for: arithmetic operations, binomial coefficients, prime factorization, gcd.
		Algorithm case studies are employed to identify design principles and illustrate analyses of algorithm efficiency relevant to software and hardware design.
		Using the number of integer addition, multiplication and division operations as an efficiencey measure, algorithms are given and analyzed for: exponentiation, logarithms, binomial coefficients, prime factorization, gcd, and multiplicative inverses module a prime.
	engr.smu.edu /cse/3353/outline.htm (870 words)

	Factorization
		After that, if a composite factor of at least 25 digits remains, MPQS is used; it is the best method available for factoring integers of more than about 40 decimal digits especially for products of two primes of roughly equal size.
		A combination of algorithms (Cunningham, trial division, Pollard rho, ECM and MPQS) is used to attempt to find the complete factorization of n, where n is a non-zero integer.
		The function will always return the complete prime factorization (in the form of a factorization sequence) of the number n (but it may take very long before it completes); it should be pointed out, however, that the primes appearing in the factorization are only probable primes and a rigorous primality prover has not been applied.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text527.htm (2385 words)

	Institut für Numerische Simulation
		Prime factors therefore represent a fundamental (and unique) decomposition of a given positive integer.
		While RSA-576 is a much smaller number than the 6,320,430-digit monster Mersenne prime announced earlier this week, its factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization").
		The factorization was accomplished using a prime factorization algorithm known as the general number field sieve (GNFS).
	www.ins.uni-bonn.de /news/RSA576a.html (764 words)

	Prime_number
		− 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,...
		+ 1 is prime for n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154...
		Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes.
	www.plasmatvwholesaler.com /search.php?title=Prime_number (2847 words)

	Factorization using the Elliptic Curve Method
		The execution time depends on the magnitude of the second largest prime factor and on your computer speed.
		In order to do it, run the factorization in the first computer from curve 1, run it in the second computer from curve 10000, in the third computer from curve 20000, and so on.
		When the number to be factorized is in the range 31-90 digits, after computing some curves in order to find small factors, the program switches to SIQS (if the checkbox located below the applet enables it), which is an algorithm that is much faster than ECM when the number has two large prime factors.
	www.alpertron.com.ar /ECM.HTM (618 words)

	NTU Info Centre: Prime factorization algorithm (Site not responding. Last check: 2007-10-21)
		A '''prime factorization algorithm''' is any [[algorithm]] (a step-by-step process) by which an [[integer]] (whole number) is "decomposed" into a product of [[divisorfactor]]s that are [[prime number]]s.
		== A simple factorization algorithm == === Description === We can describe a [[recursionrecursive]] algorithm to perform such factorizations: given a number ''n'' * if ''n'' is prime, this is the factorization, so stop here.
		For example, for an 18-[[decimal digitdigit]] (or 60 [[Binary numeral systembit]]) number, all primes below about 1,000,000,000 may need to be tested, which is taxing even for a computer.
	www.nowtryus.com /article:Prime_factorization_algorithm?source=true (464 words)

	Shor's Algorithm for Quantum Factorization
		In contrast to finding and multiplying of large prime numbers, no efficient classical algorithm for the factorization of large number is known.
		This section describes Shor's algorithm from a functional point of view which means that it doesn't deal with the implementation for a specific hardware architecture.
		Since there are efficient classical algorithms to factorize pure prime powers (and of course to recognize a factor of 2), an efficient probabilistic algorithm for factorization can be found if the period
	tph.tuwien.ac.at /~oemer/doc/quprog/node18.html (1444 words)

	Citations: Theorems on factorization and primality testing - Pollard (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		, which efficiently finds any prime factors p of a composite number n provided the prime factors of (p 1) are all less than or equal to a small bound B. Such algo rithms are sometimes referred to as special purpose factoring algorithms [32] For a general number n that is not known to satisfy....
		For the case of unknown factorization of the group order, note that no known discrete logarithm algorithm for general (not smooth) groups requires knowledge of the factored group order.
		We partially factor N Gamma 1 by one of the methods whose running time depend on the size of the prime factor to be extracted, such as 1 2 JAMES MCKEE AND RICHARD PINCH Pollard s p Gamma 1 or ae methods
	citeseer.ist.psu.edu /context/11766/0 (1292 words)

	Integers as Prime or Composite
		Examples of twin primes are: 3 and 5, 5 and 7, 11 and 13, 17 and 19,....
		There is a relationship between the prime factors and the number of factors; it involves the exponents.
		For example: The factors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} and the factors of 12 are {1, 2, 3, 4, 6, 12} and so the factors 30 and 12 have in common are {1, 2, 3, 6}.
	www.andrews.edu /~calkins/math/webtexts/numb03.htm (1425 words)

	ipedia.com: Prime factorization algorithm Article (Site not responding. Last check: 2007-10-21)
		A prime factorization algorithm is an algorithm by which an integer is "decomposed" into a product of factorss that are prime numbers.
		The first prime 1573 is divisible by is 11.
		Similarly the first prime 143 is divisible is 11.
	www.ipedia.com /prime_factorization_algorithm.html (314 words)

	Project Proposal for CS 20c for Jonathan Chang, Kevin Ko, Peter Yi (Site not responding. Last check: 2007-10-21)
		Prime factorization has been one of the most studied problems in computing and mathematics, especially now since it has taken such a large role in cryptography.
		Already, it has been shown that Prime factorization is "hard" under the conventional notion of the Turing Machine.
		By demonstrating, in practice, how a factorization would work, we can not only elucidate in detail the specifics of the algorithm but also the problems and current questions associated with prime factorization and quantum computing in general.
	www.cs.caltech.edu /~coder/project/proposal.html (485 words)

	University of Calgary: Calendar: Courses: Pure Mathematics
		Factor groups and rings, polynomial rings, field extensions, finite fields, Sylow theorems, solvable groups.
		Block designs and extremal set theory, efficiency of algorithms, complexity theory, trees and sorting, graphs and transversals of families of sets, networks and the max-flow min-cut theorem, dynamic programming, recursion.
		Such problems include: modular exponentiation, primality testing, integer factoring, solution of polynomial congruences, quadratic partitions or primes, invariant computation in certain algebraic number fields, etc. Emphasis will be placed on practical techniques and their computational complexity.
	www.ucalgary.ca /pubs/calendar/current/What/Courses/PMAT.htm (1149 words)

	Prime Numbers (Site not responding. Last check: 2007-10-21)
		A factorization of a number a is a way of writing a as a product of smaller numbers.
		A number p is prime if it is bigger than 1 and its only divisors are 1 and itself.
		A prime factorization of a number is a way of writing it as a product of prime numbers.
	www.math.okstate.edu /~wrightd/crypt/lecnotes/node7.html (332 words)

	[No title]
		Moreover, it doesn't always work; if you try to factor 2047, the cycles are the same length modulo each of the prime factors, so you get only the factor 2047 coming out.
		Moreover, it doesn't always work; if you try to factor 2047, the Although it was me who proposed the Pollard Rho method in this thread, I am no longer really convinced that it works fast enough for such small numbers, for the following reason: A trial division works with the precision of the hardware, i.e.
		The number of steps is on the order of sqrt(2nd largest prime factor), per the text.
	www.math.niu.edu /~rusin/known-math/98/pollard_rho (945 words)

	Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
		This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems.
		Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer.
		These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
	epubs.siam.org /sam-bin/dbq/article/29317 (214 words)

	Letter P
		Later, inPrimary School, by changing labels, a SPO becomes a (CPL) factor lattice (PL),on a "square-free" number, say, 30; a MPO becomes a (DL) factor lattice on a "nonsquare-free",say, 12.
		A prime factorization algorithm which is implementable in one- ortwo-step form.
		Implicitly, this means that, by finding a prime and eliminating its mulltiples, sieving also finds the next prime.
	members.fortunecity.com /jonhays/letterP.htm (1165 words)

	sci.crypt: Re: [Newbie] Prime factorization question
		conclusively that a number is prime or not.
		Primes which are far away from the next lower prime
		Thus a prime which is say 20 away from the next lower prime
	www.derkeiler.com /Newsgroups/sci.crypt/2003-10/0853.html (491 words)

	Quantum computers - Science Forums and Debate
		I don´t think many of the algorithms written for quantum computers have yet been implemented (if the factorization of large numbers had been done one would have certainly heard about it) because it´s quite easy to design an algorithm theoretically when one doesn´t have to care about the huge problems in implementation.
		It is an ingenious method of encryption, I made a an RSA cryptosystem once, the mathematics involved is not extremely difficult, but the fact is to crack it you need know how to factorize large numbers into prime factors, thats what Shore's algorithm does.
		Using 100 digit or so prime numbers, RSA develops a function that is "impossible" to invert.
	www.scienceforums.net /forums/showthread.php?t=5002 (1255 words)

	Letter C
		We use a ten as a base (PL) and five is a prime factor (PL) of ten.
		Hence, we have an algorithm for detecting a power of two as factor of a number in terms of certain last digits of the number.
		We use a ten as a base (PL) and two is a prime factor (PL) of ten.
	members.fortunecity.com /jonhays/letterC.htm (7206 words)

	Physics Help and Math Help - Physics Forums - Help with prime factorization proof
		I have to prove that if ab is divisible by the prime p, and a is not divisible by p, then b is divisible by p.
		I have not seen to many proofs regarding prime factorization.
		If ab has a factor p and a don't, then b has the factor.
	www.physicsforums.com /printthread.php?t=15404 (576 words)

	Quantum Computing (Site not responding. Last check: 2007-10-21)
		The mathematics behind the function are very complicated, but basically the algorithm attempts to amplify the correct answer.
		This algorithm was recently developed and poses a great threat to digital encryption and security systems.
		Now factoring a 400-digit number will not take in excess of several thousand years, but less that one year.
	www2.hmc.edu /~karukstis/chem21a/presentations/alexanderlaurich.htm (2272 words)

	Quantum Computers at Times to Come (Site not responding. Last check: 2007-10-21)
		If N is a composite of primes then the period is shorter, knowledge of the period leads to the prime factors of N
		The final measurement of x is the measuring of one of the 'Braggs Peaks' of this scattering process (this gives an unknown multiple of the fundamental period of x), there are simple number theories existing that allow this to be used to deduce the periodicity of the fundamental period.
		Another algorithm ready to be run on a quantum computer is quantum teleportation using Bell measurements (an unknown quantum state is transported to a remote system) This has use as a repeater fro a noisy quantum link, an dmy be used to help prove quantum theory.
	www.timestocome.com /personal/quantum.html (1872 words)

	CCIS > Undergraduate > Course Descriptions
		Discusses the counting techniques and arguments needed to estimate the size of sets, the growth of functions, and the space-time complexity of algorithms.
		Upon completion of this course, a student should be able to : use algebraic expressions with one or more variables describe sets using set operations and set-builder notation do arithmetic modulo n be able to compute gcd and lcm of pairs of integers or of integers expressed abstractly (e.g.
		This course provides the mathematical foundation for many courses in computer and information science including, data structures, algorithms, theory of computation, compilers, computer security, and operating systems.
	www.ccs.neu.edu /course/CSU200Charter.html (292 words)

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