Function Field Sieve

Related Concepts

Discrete Logarithm Problem; Generic Attacks Against DLP; Index Calculus Attack; Number Field Sieve for the DLP

Definition

The Function Field Sieve (FFS) is an algorithm originally due to Adleman [1, 2] for solving the Discrete Logarithm Problem in (the multiplicative group of) finite fields of small characteristic.

Background

The Function Field Sieve can be used to compute discrete logarithms in $\textrm{ GF}{({p}^{k})}^{{_\ast}}$ as long as k grows at least as fast as (logp)². Under these conditions, the complexity of the function field sieve with respect to p and k is given by the expression in L-notation

$$ \begin{array}{rcl}{ L}_{{p}^{k}}\left (1/3,{(32/9)}^{1/3}\right )& =& \exp \left ({(32/9)}^{1/3}{(k\log p)}^{1/3}\right. \\ & & \left. {(\log (k\log p))}^{2/3} \cdot (1 + o(1))\right ), \\ \end{array} $$

thereby achieving the best known complexity to date for computing discrete logarithms in such finite fields (in full generality).

The Function Field Sieve is...