General number field sieve

In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10¹⁰⁰. Heuristically, its complexity for factoring an integer n (consisting of ⌊log₂ n⌋ + 1 bits) is of the form

${\displaystyle {\begin{aligned}&\exp \left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)\\[5pt]={}&L_{n}\left[1/3,(64/9)^{1/3}\right]\end{aligned}}}$

in O and L-notations.^[1] It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).

The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler

. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.

The size of the input to the algorithm is log₂ n or the number of bits in the binary representation of n. Any element of the order n^c for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.

Number fields

Suppose f is a k-degree polynomial over ${\textstyle \mathbb {Q} }$ (the rational numbers), and r is a complex root of f. Then, f(r) = 0, which can be rearranged to express r^k as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of r with exponent e ≥ k. For example, if f(x) = x² + 1 and r is the imaginary unit i, then i² + 1 = 0, or i² = −1. This allows us to define the complex product:

${\displaystyle {\begin{aligned}(a+bi)(c+di)&=ac+(ad+bc)i+(bd)i^{2}\\[4pt]&=(ac-bd)+(ad+bc)i.\end{aligned}}}$

In general, this leads directly to the algebraic number field ${\textstyle \mathbb {Q} [r]}$, which can be defined as the set of complex numbers given by:

${\displaystyle a_{k-1}r^{k-1}+\cdots +a_{1}r^{1}+a_{0}r^{0},{\text{ where }}a_{0},\ldots ,a_{k-1}\in \mathbb {Q} .}$

The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of r with exponent e ≥ k as described above, yielding a value in the same form. To ensure that this field is actually k-dimensional and does not collapse to an even smaller field, it is sufficient that f is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers ${\textstyle \mathbb {O} _{\mathbb {Q} [r]}}$ as the subset of ${\textstyle \mathbb {Q} [r]}$ which are roots of monic polynomials with integer coefficients. In some cases, this ring of integers is equivalent to the ring ${\textstyle \mathbb {Z} [r]}$. However, there are many exceptions, such as for ${\textstyle \mathbb {Q} [{\sqrt {r}}\,]}$ when d is congruent to 1 modulo 4.^[2]

Method

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Two

(allowing digits between −m and m) for a number of different m of order n^1/d, and pick f(x) as the polynomial with the smallest coefficients and g(x) as x − m.

Consider the number field rings Z[r₁] and Z[r₂], where r₁ and r₂ are roots of the polynomials f and g. Since f is of degree d with integer coefficients, if a and b are integers, then so will be b^d·f(a/b), which we call r. Similarly, s = b^e·g(a/b) is an integer. The goal is to find integer values of a and b that simultaneously make r and s smooth relative to the chosen basis of primes. If a and b are small, then r and s will be small too, about the size of m, and we have a better chance for them to be smooth at the same time. The current best-known approach for this search is lattice sieving; to get acceptable yields, it is necessary to use a large factor base.

Having enough such pairs, using

are used.

Since m is a root of both f and g mod n, there are homomorphisms from the rings Z[r₁] and Z[r₂] to the ring Z/nZ (the integers modulo n), which map r₁ and r₂ to m, and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now the product of the factors a − mb mod n can be obtained as a square in two ways—one for each homomorphism. Thus, one can find two numbers x and y, with x² − y² divisible by n and again with probability at least one half we get a factor of n by finding the greatest common divisor of n and x − y.

Improving polynomial choice

The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of n in base m shown above is suboptimal in many practical situations, leading to the development of better methods.

One such method was suggested by Murphy and Brent;^[3] they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area.

The best reported results^[4] were achieved by the method of Thorsten Kleinjung,^[5] which allows g(x) = ax + b, and searches over a composed of small prime factors congruent to 1 modulo 2d and over leading coefficients of f which are divisible by 60.

Implementations

Some implementations focus on a certain smaller class of numbers. These are known as

. A project called NFSNET ran from 2002^[6] through at least 2007. It used volunteer distributed computing on the Internet.^[7] Paul Leyland of the United Kingdom and Richard Wackerbarth of Texas were involved.^[8]

Until 2007, the gold-standard implementation was a suite of software developed and distributed by CWI in the Netherlands, which was available only under a relatively restrictive license.^{[citation needed]} In 2007, Jason Papadopoulos developed a faster implementation of final processing as part of msieve, which is in the public domain. Both implementations feature the ability to be distributed among several nodes in a cluster with a sufficiently fast interconnect.

Polynomial selection is normally performed by

software written by Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.

• NFS@Home
• GGNFS
• factor by gnfs
• CADO-NFS
• msieve (which contains final-processing code, a polynomial selection optimized for smaller numbers and an implementation of the line sieve)
• kmGNFS

See also

• Special number field sieve

Notes

References

• Arjen K. Lenstra and H. W. Lenstra, Jr. (eds.). "The development of the number field sieve". Lecture Notes in Math. (1993) 1554. Springer-Verlag.
• Richard Crandall and
  ISBN 0-387-25282-7
  . Section 6.2: Number field sieve, pp. 278–301.

• Matthew E. Briggs: An Introduction to the General Number Field Sieve, 1998