In number theory, integer factorization is the process of breaking down a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer. In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

When the numbers are very large, no efficient integer factorization algorithm is publicly PIE PEI PEI EIPE IPEIEP IEPIEIPEIP]] such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... This article is about an algorithm for public-key encryption. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. ... This article or section does not cite its references or sources. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ...

Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, i.e. the product of two distinct primes. When they are both large, randomly chosen, and about the same size (but not too close), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical. In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ...

Prime decomposition

By the fundamental theorem of arithmetic, every positive integer greater than one has a unique prime factorization. However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence. In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...

Given an algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm. In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...

Practical applications

The hardness of this problem is at the heart of several important cryptographic systems. A fast integer factorization algorithm would mean that the RSA public-key algorithm is insecure. Some cryptographic systems, such as the Rabin public-key algorithm and the Blum Blum Shub pseudo-random number generator can make a stronger guarantee - any means of breaking them can be used to build a fast integer factorization algorithm, so if integer factorization is hard then they are strong. In contrast, it may turn out that there are attacks on the RSA problem more efficient than integer factorization, though none are currently published. This article is about an algorithm for public-key encryption. ... PKC, see PKC (disambiguation) Public-key cryptography is a form of modern cryptography which allows users to communicate securely without previously agreeing on a shared secret key. ... The Rabin cryptosystem is an asymmetric cryptographic technique, which like RSA is based on the difficulty of factorization. ... Blum Blum Shub (BBS) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub (Blum et al, 1986). ... A pseudorandom number generator (PRNG) is an algorithm that generates a sequence of numbers, the elements of which are approximately independent of each other. ... In cryptography, the RSA problem is the task of finding eth roots modulo a composite number N whose factors are not known. ...

A similar hard problem with cryptographic applications is the discrete logarithm problem. In abstract algebra and its applications, the discrete logarithms are defined in group theory in analogy to ordinary logarithms. ...

Even in the absence of cryptographic systems based on its hardness, integer factorization also has many positive applications in algorithms. For example, once an integer n is placed in its prime factorization representation, it enables the rapid computation of multiplicative functions on n. It can also be used to save storage, since any multiset of prime numbers can be stored without loss of information as its product; this was exploited, for example, by the Arecibo message. In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... In mathematics, a multiset (or bag) is a generalization of a set. ... Arecibo Observatory This is the message with color added to highlight its separate parts. ...

Current state of the art

A team at the German Federal Agency for Information Technology Security (BSI) holds the record for factorization of semiprimes in the series proposed by the RSA Factoring Challenge sponsored by RSA Security. On May 9, 2005, this team announced factorization of RSA-200, a 663-bit number (200 decimal digits), using the general number field sieve. Building in Bonn The Bundesamt für Sicherheit in der Informationstechnik (abbreviated BSI - in English: Federal Office for Information Security) is the German government agency in change of managing computer and communication security for the German government. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... The RSA Factoring Challenge is a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. ... RSA, The Security Division of EMC Corporation (NYSE: EMC), is headquartered in Bedford, Massachusetts, and maintains offices in Ireland, the United Kingdom, Singapore, and Japan. ... Wikinews has news related to this article: Two hundred digit number factored In mathematics, RSA-200 is one of the RSA numbers, large semiprimes that are part of the RSA Factoring Challenge. ... In mathematics, the general number field sieve (GNFS) is the most efficient algorithm known for factoring integers larger than 100 digits. ...

The same team later announced factorization of RSA-640, a smaller number containing 193 decimal digits (640 bits), on November 4, 2005. In mathematics, RSA-640 is one of the RSA numbers - large semiprimes that are part of the RSA Factoring Challenge. ...

Both factorizations required several months of computer time using the combined power of 80 AMD Opteron CPUs. Advanced Micro Devices, Inc. ... The Opteron is AMDs x86 server processor line, and was the first processor to implement the AMD64 instruction set architecture (known generically as x86-64). ...

Difficulty and complexity

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i.e., that can factor it in time O(b^k) for some constant k. There are published algorithms that are faster than O((1+ε)^b) for all positive ε, i.e., sub-exponential. This article is about the unit of information. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

The best published asymptotic running time is for the general number field sieve (GNFS) algorithm, which, for a b-bit number n, is: In mathematics, the general number field sieve (GNFS) is the most efficient algorithm known for factoring integers larger than 100 digits. ...

For an ordinary computer, GNFS is the best published algorithm for large n (more than about 100 digits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. This will have significant implications for cryptography if a large quantum computer is ever built. Shor's algorithm takes only O(b³) time and O(b) space on b-bit number inputs. In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15. The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Peter Shor Peter W. Shor (born August 14, 1959) is an American theoretical computer scientist most famous for his work on quantum computation, in particular for devising a quantum algorithm for factoring exponentially faster than the best currently-known algorithm running on a classical computer (see Shors algorithm). ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ...

When discussing what complexity classes the integer factorization problem falls into, it's necessary to distinguish two slightly different versions of the problem: In computational complexity theory, a complexity class is a set of problems of related complexity. ...

• The function problem version: given an integer N, find an integer d with 1 < d < N that divides N (or conclude that N is prime). This problem is trivially in FNP and it's not known whether it lies in FP or not. This is the version solved by most practical implementations.
• The decision problem version: given an integer N and an integer M with 1 ≤ M ≤ N, does N have a factor d with 1 < d < M? This version is useful because most well-studied complexity classes are defined as classes of decision problems, not function problems. This is a natural decision version of the problem, analogous to those frequently used for optimization problems, because it can be combined with binary search to solve the function problem version in a logarithmic number of queries.

It is not known exactly which complexity classes contain the decision version of the integer factorization problem. It is known to be in both NP and co-NP. This is because both YES and NO answers can be trivially verified given the prime factors (we can verify their primality using the AKS primality test, and that their product is N by multiplication). It is known to be in BQP because of Shor's algorithm. It is suspected to be outside of all three of the complexity classes P, NP-Complete, and co-NP-Complete. If it could be proved that it is in either NP-Complete or co-NP-Complete, that would imply NP = co-NP. That would be a very surprising result, and therefore integer factorization is widely suspected to be outside both of those classes. Many people have tried to find classical polynomial-time algorithms for it and failed, and therefore it is widely suspected to be outside P. In computational complexity theory, a function problem is a problem other than a decision problem, that is, a problem requiring a more complex answer than just YES or NO. Notable examples include the traveling salesman problem which asks for the route taken by the salesman, and the integer factorization problem... In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is a bit of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: A binary relation P(x,y... In computational complexity theory, the complexity class FP is the set of function problems which can be solved by a deterministic Turing machine in polynomial time; it is the function problem version of the decision problem class P. Roughly speaking, it is the class of functions that can be efficiently... In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In computer science, an optimization problem is the problem to find among all feasible solutions for some problem the best one. ... In computer science, binary search or binary chop is a search algorithm for finding a particular value in a linear array, by ruling out half of the data at each step. ... As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... In computational complexity theory, NP (Non-deterministic Polynomial time) is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. ... In computational complexity theory, co-NP is a complexity class. ... The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The... BQP, in computational complexity theory, stands for Bounded error, Quantum, Polynomial time. It denotes the class of problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/4 for all instances. ... Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... In complexity theory, the complexity class Co-NP-complete is the set of problems that are the hardest problems in Co-NP, in the sense that they are the ones most likely not to be in P. If you can find a way to solve a Co-NP-complete problem...

In contrast, the decision problem "is N a composite number?" (or equivalently: "is N a prime number?") appears to be much easier than the problem of actually finding the factors of N. Specifically, the former can be solved in polynomial time (in the number n of digits of N) with the AKS primality test. In addition, there are a number of probabilistic algorithms that can test primality very quickly if one is willing to accept the small possibility of error. The ease of primality testing is a crucial part of the RSA algorithm, as it is necessary to find large prime numbers to start with. A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The... A randomized algorithm or probabilistic algorithm is an algorithm which employs a degree of randomness as part of its logic. ... A primality test is an algorithm for determining whether an input number is prime. ... This article is about an algorithm for public-key encryption. ...

Factoring algorithms

Special-purpose

A special-purpose factoring algorithm's running time depends on the properties of its unknown factors: size, special form, etc. Exactly what the running time depends on varies between algorithms.

• Trial division
• Pollard's rho algorithm
• Pollard's p − 1 algorithm
• Williams' p+1 algorithm
• Lenstra elliptic curve factorization
• Fermat's factorization method
• Special number field sieve

Trial division is the simplest and easiest to understand of the integer factorization algorithms. ... Pollards rho algorithm is a special-purpose integer factorization algorithm. ... Pollards p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. ... In computational number theory, Williams p + 1 algorithm is an integer factorization algorithm invented by H. C. Williams. ... The Lenstra elliptic curve factorization or the elliptic curve factorization method (ECM) is a fast, sub-exponential running time algorithm for integer factorization which employs elliptic curves. ... With Fermats factoring method, one tries to represent an odd integer as the difference of two squares: . That difference is algebraically factorable as ; if neither factor equals one, it is a proper factorization of N. Furthermore, each odd number has such a representation. ... The special number field sieve (SNFS) is a special-purpose integer factorization algorithm. ...

General-purpose

A general-purpose factoring algorithm's running time depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method. The RSA numbers, listed by security company RSA Security, are certain large semiprime numbers (i. ... In number theory, a congruence of squares modulo an integer n is an equality . Such a relationship carries information useful in trying to factor the integer n: finding a congruence of squares modulo n is something sought after in integer factorization. ...

• Dixon's algorithm
• Continued fraction factorization (CFRAC)
• Quadratic sieve
• General number field sieve
• Shanks' square forms factorization (SQUFOF)

In number theory, Dixons factorization method (also Dixons algorithm) is a general-purpose integer factorization algorithm. ... In number theory, the continued fraction factorization method is an integer factorization algorithm. ... The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). ... In mathematics, the general number field sieve (GNFS) is the most efficient algorithm known for factoring integers larger than 100 digits. ... Input: N, the integer to be factored, which must be neither a prime number nor a perfect square. ...

Other notable algorithms

• Shor's algorithm, for quantum computers

Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ...

External links

• Richard P. Brent, "Recent Progress and Prospects for Integer Factorisation Algorithms", Computing and Combinatorics", 2000, pp.3-22. download
• Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P." Annals of Mathematics 160(2): 781-793 (2004). August 2005 version PDF
• [1] is a public-domain integer factorization program for Windows. It claims to handle 80-digit numbers. See also the web site for this program MIRACL
• http://www.alpertron.com.ar/ECM.HTM is an integer factorization Java applet that uses the Elliptic Curve Method and the Self Initializing Quadratic Sieve.
• The RSA Challenge Numbers - a factoring challenge.
• Eric W. Weisstein, “RSA-640 Factored,” MathWorld Headline News, November 8, 2005, http://mathworld.wolfram.com/news/2005-11-08/rsa-640/
• Qsieve, a suite of programs for integer factorization. It contains several factorization methods like Elliptic Curve Method and MPQS.
• Classical and Pollard p-1, summary of the algorithms and C source code

Manindra Agrawal (मणीन्द्र अग्रवाल) (born 20 May 1966 in Allahabad) is a professor of computer science at the Indian Institute of Technology, Kanpur. ...

References

• Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.5.4: Factoring into Primes, pp. 379–417.
• Richard Crandall and Carl Pomerance (2001). Prime Numbers: A Computational Perspective, 1st edition, Springer. ISBN 0-387-94777-9.  Chapter 5: Exponential Factoring Algorithms, pp. 191–226. Chapter 6: Subexponential Factoring Algorithms, pp. 227–284. Section 7.4: Elliptic curve method, pp. 301–313.