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 100 digits. Heuristically, its complexity for factoring an integer n (consisting of bits) is of the form
(in L-notation), where ln is the logarithm in base e. 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.
Read more about General Number Field Sieve: Number Fields, Method, Improving Polynomial Choice, Implementations
Famous quotes containing the words general, number, field and/or sieve:
“It has been an unchallengeable American doctrine that cranberry sauce, a pink goo with overtones of sugared tomatoes, is a delectable necessity of the Thanksgiving board and that turkey is uneatable without it.... There are some things in every country that you must be born to endure; and another hundred years of general satisfaction with Americans and America could not reconcile this expatriate to cranberry sauce, peanut butter, and drum majorettes.”
—Alistair Cooke (b. 1908)
“In many ways, life becomes simpler [for young adults]. . . . We are expected to solve only a finite number of problems within a limited range of possible solutions. . . . It’s a mental vacation compared with figuring out who we are, what we believe, what we’re going to do with our talents, how we’re going to solve the social problems of the globe . . .and what the perfect way to raise our children will be.”
—Roger Gould (20th century)
“Vigil strange I kept on the field one night;
When you my son and my comrade dropt at my side that day,
One look I but gave which your dear eyes return’d with a look I
shall never forget,
One touch of your hand to mine O boy, reach’d up as you lay on the ground,”
—Walt Whitman (1819–1892)
“It’s like pushing marbles through a sieve. It means the sieve will never be the same again.”
—Before the 1972 Democratic Convention in Miami. As quoted in Crazy Salad, ch. 6, by Nora Ephron (1972)