General number field sieve

In mathematics, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 100 digits. Heuristically, its complexity for factoring an integer n (consisting of log n bits) is of the form

(in O and L notations) for a constant c which depends on the complexity measure and on the variant of the algorithm^[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, but this is a minor issue). When the term number field sieve (NFS) is used without qualification, it refers to the general number field sieve.

The principle of the number field sieve (both special and general) can be understood as an extension of the simpler rational sieve. When using the rational sieve to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n; the rarity of these causes the rational sieve to be impractical. The general number field sieve, on the other hand, only requires a search for smooth numbers of order n^1/d, where d is some integer greater than one. Since larger numbers are far less likely to be smooth than smaller numbers, this is the key to the efficiency of the number field sieve. But in order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.

Note that log n is the number of digits in the binary representation of n, that is the size of the input to the algorithm. The (worst-case) running time is therefore super-polynomial in the size of the input. It is an important open problem whether factorization can be done in reasonable time — polynomial time — on a classical computer. On a quantum computer, factorization is a tractable problem using Shor's algorithm.

Number fields

Suppose f is an n-degree polynomial over Q (the rational numbers), and r is a complex root of f. Then, f(r) = 0, which can be rearranged to express rⁿ as a linear combination of powers of r less than n. This equation can be used to reduce away any powers of r ≥ n. 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:

(a+bi)(c+di) = ac + (ad+bc)i + (bd)i² = (ac - bd) + (ad+bc)i.

In general, this leads directly to the algebraic number field Q[r], which can be defined as the set of real numbers given by:

a_n-1r^n-1 + ... + a₁r¹ + a₀r⁰, where a₀,...,a_n-1 in Q.

The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of r ≥ n as described above, yielding a value in the same form. To ensure that this field is actually n-dimensional and does not collapse to an even smaller field, it is sufficient that f is an irreducible polynomial. Similarly, one may define the number field ring Z[r] as the subset of Q[r] where a₀,...,a_n-1 are restricted to be integers.

Method

We choose two polynomials f(x) and g(x) of small degrees d and e, which have integer coefficients, which are irreducible over the rationals, and which, when interpreted mod n, have a common integer root m. An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree d for a polynomial, consider the expansion of n in base m (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.

We consider the number field rings Z[r₁] and Z[r₂], where r₁ and r₂ are complex 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. We seek 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 Gaussian elimination, we can get products of certain r and of the corresponding s to be squares at the same time. We need a slightly stronger condition—that they are norms of squares in our number fields, but we can get that condition by this method too. Each r is a norm of a − r₁b and hence we get that the product of the corresponding factors a − r₁b is a square in Z[r₁], with a "square root" which can be determined (as a product of known factors in Z[r₁])—it will typically be represented as an irrational algebraic number. Similarly, we get that the product of the factors a − r₂b is a square in Z[r₂], with a "square root" which we can also compute.

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 mod 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, we get 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 simple method above of choosing polynomials above based on the expansion of n in base m is suboptimal in many practical situations, leading to the development of better methods.

One such method was suggested by Murphy and Brent ^[2]; 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 ^[3] were achieved by the method of Thorsten Kleinjung ^[4], 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

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. In 2007, Jason Papadopoulos developed a faster implementation of final processing as part of msieve, which is public-domain. msieve, however, can only run on a single SMP computer, whilst CWI's implementation can be distributed among several nodes in a cluster with a sufficiently fast interconnect.

Polynomial selection is normally performed by GPL software written by Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.

• NFSNet
• GGNFS.
• pGNFS
• factor by gnfs
• msieve, which contains excellent final-processing code, a good implementation of the polynomial selection which is very good for smaller numbers, and an implementation of the line sieve.
• kmGNFS

References