# Integer factorization

In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number; for example, given a number like 91, the challenge is to find a number such as 7 which divides it. When the numbers are very large, no efficient algorithm is known; a recent effort which factored a 200 digit number (RSA-200) took eighteen months and used over half a century of computer time. The supposed difficulty of this problem is at the heart of certain algorithms in cryptography 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.

## Prime decomposition

By the fundamental theorem of arithmetic, every integer has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm.

## 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 was 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 known.

A similar hard problem with cryptographic applications is the discrete logarithm problem.

## 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.

The same team later announced factorization of RSA-640, a smaller number containing 193 decimal digits (640 bits), on November 4, 2005.

Both factorizations required several months of computer time using the combined power of 80 AMD Opteron CPUs.

### Difficulty and complexity

If a large, n-bit number is the product of two primes that are roughly the same size, then no algorithm is known that can factor in polynomial time. That means there is no known algorithm that can factor it in time O(nk) for any constant k. There are algorithms, however, that are faster than Θ(en). In other words, the best known algorithms are sub-exponential, but super-polynomial. In particular, the best known asymptotic running time is for the general number field sieve (GNFS) algorithm, which is:

${\displaystyle \Theta \left(\exp \left(\left({\frac {64}{9}}n\right)^{\frac {1}{3}}(\log n)^{\frac {2}{3}}\right)\right).}$

For an ordinary computer, GNFS is the best known algorithm for large n. 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((log n)3) time and O(log n) space. In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15.

It is not known exactly which complexity classes contain the integer factorization problem. The decision-problem form of it ("does N have a factor less than M?") is known to be in both NP and co-NP. This is because both YES and NO answers can be checked if given the prime factors along with their primality proofs. 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, 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, therefore it is widely suspected to be outside P. Another problem in NP but not believed to be in P or NP-complete is the graph isomorphism problem.

Interestingly, 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), according to a recent preprint given in the references, below. 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 easiness of primality testing is a crucial part of the RSA algorithm, as it is necessary to find large prime numbers to start with.

## 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.

### 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.