Special number field sieveIn number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form re ± s, where r and s are small (for instance Mersenne numbers). Heuristically, its complexity for factoring an integer is of the form: in O and L-notations.
General number field sieveIn 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 O and L-notations. 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).
Quadratic sieveThe quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties.
Discrete logarithmIn mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases.
