Function field sieve

In mathematics, the function field sieve was introduced in 1994 by Leonard Adleman as an efficient technique for extracting discrete logarithms over finite fields of small characteristic, and elaborated by Adleman and Huang in 1999.

Sieving for points at which a polynomial-valued function is divisible by a given polynomial is not much more difficult than sieving over the integers – the underlying structure is fairly similar, and Gray code provides a convenient way to step through multiples of a given polynomial very efficiently.

References

The Adleman–Huang paper is available at Science Direct, but views the problem using very algebraic-geometric language.

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW ECPP test Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin Trial division

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia Integer relation Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms Smallcaps indicate a deterministic algorithm