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 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 (in L-notation), where ln is the natural logarithm. 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). 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 improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. 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 log2 n is the number of bits in the binary representation of n, that is the size of the input to the algorithm, so any element of the order nc 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. Suppose f is a k-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 rk as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of r ≥ 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: In general, this leads directly to the algebraic number field Q, which can be defined as the set of real numbers given by: The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of r ≥ 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 OQ as the subset of Q which are roots of monic polynomials with integer coefficients. In some cases, this ring of integers is equivalent to the ring Z. However, there are many exceptions, such as for Q when d is equal to 1 modulo 4. Two polynomials f(x) and g(x) of small degrees d and e are chosen, 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 n1/d, and pick f(x) as the polynomial with the smallest coefficients and g(x) as x − m. Consider the number field rings Z and Z, 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 bd·f(a/b), which we call r. Similarly, s = be·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.