Английская Википедия:Integer factorization

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In number theory, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors, in which case it is called a composite number, or it is not, in which case it is called a prime number. For example, Шаблон:Math is a composite number because Шаблон:Math, but Шаблон:Math is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example Шаблон:Math. Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem.

To factorize a small integer Шаблон:Mvar using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers Шаблон:Math, Шаблон:Math, Шаблон:Math, and so on, up to the square root of Шаблон:Mvar. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involves testing whether each factor is prime each time a factor is found.

When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature.[1] 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 two thousand bits long, randomly chosen, and about the same size (but not too close, for example, 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 integer 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

Файл:PrimeDecompositionExample.svg
Prime decomposition of Шаблон:Math as Шаблон:Math

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (By convention, 1 is the empty product.) Testing whether the integer is prime can be done in polynomial time, for example, by the AKS primality test. If composite, however, the polynomial time tests give no insight into how to obtain the factors.

Given a general algorithm for integer factorization, any integer can be factored into its constituent prime factors by repeated application of this algorithm. The situation is more complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if Шаблон:Math where Шаблон:Math are very large primes, trial division will quickly produce the factors 3 and 19 but will take Шаблон:Math divisions to find the next factor. As a contrasting example, if Шаблон:Math is the product of the primes Шаблон:Math, Шаблон:Math, and Шаблон:Math, where Шаблон:Math, Fermat's factorization method will begin with Шаблон:Math which immediately yields Шаблон:Math and hence the factors Шаблон:Math and Шаблон:Math. While these are easily recognized as composite and prime respectively, Fermat's method will take much longer to factor the composite number because the starting value of Шаблон:Math for Шаблон:Math is a factor of 10 from Шаблон:Math.

Current state of the art

Шаблон:See also

Among the Шаблон:Math-bit numbers, the most difficult to factor in practice using existing algorithms are those semiprimes whose factors are of similar size. For this reason, these are the integers used in cryptographic applications.

In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power.[2] The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.[3]

The largest such semiprime yet factored was RSA-250, an 829-bit number with 250 decimal digits, in February 2020. The total computation time was roughly 2700 core-years of computing using Intel Xeon Gold 6130 at 2.1 GHz. 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

No algorithm has been published that can factor all integers in polynomial time, that is, that can factor a Шаблон:Math-bit number Шаблон:Math in time Шаблон:Math for some constant Шаблон:Math. Neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist and hence that the problem is not in class P.[4][5] The problem is clearly in class NP, but it is generally suspected that it is not NP-complete, though this has not been proven.[6]

There are published algorithms that are faster than Шаблон:Math for all positive Шаблон:Math, that is, sub-exponential. Шаблон:As of, the algorithm with best theoretical asymptotic running time is the general number field sieve (GNFS), first published in 1993,[7] running on a Шаблон:Math-bit number Шаблон:Math in time:

<math>\exp\left( \left(\left(\tfrac83\right)^\frac23 + o(1)\right)\left(\log n\right)^\frac13\left(\log \log n\right)^\frac23\right).</math>

For current computers, GNFS is the best published algorithm for large Шаблон:Math (more than about 400 bits). 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 quantum computation becomes scalable. Shor's algorithm takes only Шаблон:Math time and Шаблон:Math space on Шаблон:Math-bit number inputs. In 2001, Shor's algorithm was implemented for the first time, by using NMR techniques on molecules that provide seven qubits.[8]

It is not known exactly which complexity classes contain the decision version of the integer factorization problem (that is: does Шаблон:Math have a factor smaller than Шаблон:Math besides 1?). It is known to be in both NP and co-NP, meaning that both "yes" and "no" answers can be verified in polynomial time. An answer of "yes" can be certified by exhibiting a factorization Шаблон:Math with Шаблон:Math. An answer of "no" can be certified by exhibiting the factorization of Шаблон:Math into distinct primes, all larger than Шаблон:Math; one can verify their primality using the AKS primality test, and then multiply them to obtain Шаблон:Math. The fundamental theorem of arithmetic guarantees that there is only one possible string of increasing primes that will be accepted, which shows that the problem is in both UP and co-UP.[9] It is known to be in BQP because of Shor's algorithm.

The problem is suspected to be outside 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 to be either NP-complete or co-NP-complete, this would imply NP = co-NP, a very surprising result, and therefore integer factorization is widely suspected to be outside both these classes.

In contrast, the decision problem "Is Шаблон:Math a composite number?" (or equivalently: "Is Шаблон:Math a prime number?") appears to be much easier than the problem of specifying factors of Шаблон:Math. The composite/prime problem can be solved in polynomial time (in the number Шаблон:Math of digits of Шаблон:Math) with the AKS primality test. In addition, there are several probabilistic algorithms that can test primality very quickly in practice if one is willing to accept a 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. The parameters which determine the running time vary among 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.[10] For example, naive 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,[10] has a running time which 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.

Other notable algorithms

Heuristic running time

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

<math>L_n\left[\tfrac12,1+o(1)\right]=e^{(1+o(1))\sqrt{(\log n)(\log \log n)}}</math>

in little-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,[11] Seysen,[12] and Lenstra,[13] which they proved only assuming the unproved generalized Riemann hypothesis.

Rigorous running time

The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance[14] to have expected running time Шаблон:Math by replacing the GRH assumption with the use of multipliers. The algorithm uses the class group of positive binary quadratic forms of discriminant Шаблон:Math denoted by Шаблон:Math. Шаблон:Math is the set of triples of integers Шаблон:Math in which those integers are relative prime.

Schnorr–Seysen–Lenstra algorithm

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

Denote by Шаблон:Math the set of all primes Шаблон:Math with Kronecker symbol Шаблон:Math. By constructing a set of generators of Шаблон:Math and prime forms Шаблон:Math of Шаблон:Math with Шаблон:Math in Шаблон:Math a sequence of relations between the set of generators and Шаблон:Math are produced. The size of Шаблон:Math can be bounded by Шаблон:Math for some constant Шаблон:Math.

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

Let Шаблон:Math be the number to be factored. Шаблон:Ordered list x^{r(x)}\right).\left(\prod_{q \in P_\Delta} f^{t(q)}_{q}\right) = 1.</math> | Construct an ambiguous form Шаблон:Math that is an element Шаблон:Math of order dividing 2 to obtain a coprime factorization of the largest odd divisor of Шаблон:Math in which Шаблон:Math or Шаблон:Math or Шаблон:Math. | If the ambiguous form provides a factorization of Шаблон:Math then stop, otherwise find another ambiguous form until the factorization of Шаблон:Math is found. In order to prevent useless ambiguous forms from generating, build up the 2-Sylow group Шаблон:Math of Шаблон:Math. }} 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 Шаблон:Math.[14]

See also

Notes

Шаблон:Reflist

References

External links

Шаблон:Computational hardness assumptions Шаблон:Number theoretic algorithms Шаблон:Divisor classes Шаблон:Authority control

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  1. ↑ Шаблон:Citation
  2. ↑ Шаблон:Cite web
  3. ↑ Шаблон:Cite journal
  4. ↑ Шаблон:Citation
  5. ↑ Шаблон:Citation
  6. ↑ Шаблон:Citation. See in particular p. 583.
  7. ↑ Шаблон:Cite book
  8. ↑ Шаблон:Cite journal
  9. ↑ Шаблон:Cite web
  10. ↑ 10,0 10,1 Шаблон:Cite book
  11. ↑ Шаблон:Cite journal
  12. ↑ Шаблон:Cite journal
  13. ↑ Шаблон:Cite journal
  14. ↑ 14,0 14,1 Шаблон:Cite journal
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