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 Can integer factorization be done in polynomial time?

In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.

When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort by several researchers concluded in 2009, factoring a 232-digit number (RSA-768), utilizing hundreds of machines over a span of 2 years.[1] The presumed difficulty of this problem is at the heart of widely used 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.

Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than 2000 bits long, randomly chosen, and about the same size (but not too close, e.g. to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.

Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

## Prime decomposition

This image demonstrates the prime decomposition of 864. A shorthand way of writing the resulting prime factors is 25 × 33

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (A special case for 1 is not needed using an appropriate notion of the empty product.) However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm. However, this is not the case with a special-purpose factorization algorithm, since it may not apply to the smaller factors that occur during decomposition, or may execute very slowly on these values. For example, if N is the number (2521 − 1) × (2607 − 1), then trial division will quickly factor 10 × N as 2 × 5 × N, but will not quickly factor N into its factors.

## Current state of the art

The most difficult integers to factor in practice using existing algorithms are those that are products of two large primes of similar size, and for this reason these are the integers used in cryptographic applications. The largest such semiprime yet factored was RSA-768, a 768-bit number with 232 decimal digits, on December 12, 2009.[1] This factorization was a collaboration of several research institutions, spanning two years and taking the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron. Like all recent factorization records, this factorization was completed with a highly optimized implementation of the general number field sieve run on hundreds of machines.

### 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(bk) for some constant k. There are published algorithms that are faster than O((1+ε)b) for all positive ε, i.e., sub-exponential.

The best published asymptotic running time is for the general number field sieve (GNFS) algorithm, which, for a b-bit number n, is:

$O\left(\exp\left(\left(\begin{matrix}\frac{64}{9}\end{matrix} b\right)^{1\over3} (\log b)^{2\over3}\right)\right).$

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 seven-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15.[2]

When discussing what complexity classes the integer factorization problem falls into, it's necessary to distinguish two slightly different versions of the problem:

• 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 verified in polynomial time given the prime factors (we can verify their primality using the AKS primality test, and that their product is N by multiplication). The fundamental theorem of arithmetic guarantees that there is only one possible string that will be accepted (providing the factors are required to be listed in order), which shows that the problem is in both UP and co-UP.[3] 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. It is therefore a candidate for the NP-intermediate complexity class. 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 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 in practice if one is willing to accept the vanishingly 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.

## Factoring algorithms

### Special-purpose

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

An important subclass of special-purpose factoring algorithms is the Category 1 or First Category algorithms, whose running time depends on the size of smallest prime factor. Given an integer of unknown form, these methods are usually applied before general-purpose methods to remove small factors.[4] For example, trial division is a Category 1 algorithm.

### General-purpose

A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm (after Maurice Kraitchik),[4] has a 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.

## Heuristic running time

In number theory, there are many integer factoring algorithms that heuristically have expected running time

$L_n\left[1/2,1+o(1)\right]=e^{(1+o(1))(\log n)^{\frac{1}{2}}(\log \log n)^{\frac{1}{2}}}$

in o and L-notation. Some examples of those algorithms are the elliptic curve method and the quadratic sieve. Another such algorithm is the class group relations method proposed by Schnorr,[5] Seysen,[6] and Lenstra[7] that is proved under of the Generalized Riemann Hypothesis (GRH).

## Rigorous running time

The Schnorr-Seysen-Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance[8] to have expected running time $L_n\left[1/2,1+o(1)\right]$ by replacing the GRH assumption with the use of multipliers. The algorithm uses the class group of positive binary quadratic forms of discriminant Δ denoted by GΔ. GΔ is the set of triples of integers (a, b, c) in which those integers are relative prime.

### Schnorr-Seysen-Lenstra Algorithm

Given is an integer n that will be factored, where n is an odd positive integer greater than a certain constant. In this factoring algorithm the discriminant Δ is chosen as a multiple of n, Δ= -dn, where d is some positive multiplier. The algorithm expects that for one d there exist enough smooth forms in GΔ. Lenstra and Pomerance show that the choice of d can be restricted to a small set to guarantee the smoothness result.

Denote by PΔ the set of all primes q with Kronecker symbol $\left(\tfrac{\Delta}{q}\right)=1$. By constructing a set of generators of GΔ and prime forms fq of GΔ with q in PΔ a sequence of relations between the set of generators and fq are produced. The size of q can be bounded by $c_0(\log|\Delta|)^2$ for some constant $c_0$.

The relation that will be used is a relation between the product of powers that is equal to the neutral element of GΔ. These relations will be used to construct a so-called ambiguous form of GΔ, which is an element of GΔ of order dividing 2. By calculating the corresponding factorization of Δ and by taking a gcd, this ambiguous form provides the complete prime factorization of n. This algorithm has these main steps:

Let n be the number to be factored.

1. Let Δ be a negative integer with Δ = -dn, where d is a multiplier and Δ is the negative discriminant of some quadratic form.
2. Take the t first primes $p_1=2,p_2=3,p_3=5, \dots ,p_t$, for some $t\in{\mathbb N}$.
3. Let $f_q$ be a random prime form of GΔ with $\left(\tfrac{\Delta}{q}\right)=1$.
4. Find a generating set X of GΔ
5. Collect a sequence of relations between set X and {fq : qPΔ} satisfying: $\left(\prod_{x \in X_{}} x^{r(x)}\right).\left(\prod_{q \in P_\Delta} f^{t(q)}_{q}\right) = 1$
6. Construct an ambiguous form (a, b, c) that is an element fGΔ of order dividing 2 to obtain a coprime factorization of the largest odd divisor of Δ in which Δ = -4a.c or a(a - 4c) or (b - 2a).(b + 2a)
7. If the ambiguous form provides a factorization of n then stop, otherwise find another ambiguous form until the factorization of n is found. In order to prevent useless ambiguous forms from generating, build up the 2-Sylow group S2(Δ) of G(Δ).

To obtain an algorithm for factoring any positive integer, it is necessary to add a few steps to this algorithm such as trial division, and the Jacobi sum test.

### Expected running time

The algorithm as stated is a probabilistic algorithm as it makes random choices. Its expected running time is at most $L_n\left[1/2,1+o(1)\right]$.[8]

## Notes

1. ^ a b Kleinjung, et al (2010-02-18). Factorization of a 768-bit RSA modulus. International Association for Cryptologic Research. Retrieved 2010-08-09.
2. ^ LIEVEN M. K. VANDERSYPEN, et al (2007-12-27). NMR quantum computing: Realizing Shor's algorithm. Nature. Retrieved 2010-08-09.
3. ^ Lance Fortnow (2002-09-13). "Computational Complexity Blog: Complexity Class of the Week: Factoring".
4. ^ a b David Bressoud and Stan Wagon (2000). A Course in Computational Number Theory. Key College Publishing/Springer. pp. 168–69. ISBN 978-1-930190-10-8.
5. ^ Schnorr, Claus P. (1982). "Refined analysis and improvements on some factoring algorithms". Journal of Algorithms 3 (2): 101–127. doi:10.1016/0196-6774(82)90012-8.
6. ^ Seysen, Martin (1987). "A probabilistic factorization algorithm with quadratic forms of negative discriminant". Mathematics of Computation 48 (178): 757–780. doi:10.1090/S0025-5718-1987-0878705-X.
7. ^ Lenstra, Arjen K (1988). "Fast and rigorous factorization under the generalized Riemann hypothesis". Indagationes Mathematicae 50: 443–454.
8. ^ a b H.W. Lenstra, and C. Pomerance; Pomerance, Carl (July 1992). "A Rigorous Time Bound for Factoring Integers" (PDF). Journal of the American Mathematical Society 5 (3): 483–516. doi:10.1090/S0894-0347-1992-1137100-0.

## References

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