Discrete logarithm records

The Discrete Logarithm Problem is the problem of finding solutions $x$ to the equation $g^{x}=h$ given elements $g$ and $h$ of a finite cyclic group $G$ . The difficulty of this problem is the basis of several cryptographic systems, including Diffie-Hellman key agreement, ElGamal encryption, ElGamal signatures, the Digital Signature Algorithm, and the Elliptic Curve Cryptography analogs of these. Common choices for $G$ used in these algorithms include the multiplicative group of integers modulo $p$ , the multiplicative group of a finite field, and the group of points on an elliptic curve modulo $p$ or over a finite field.

Integers modulo $p$

On 18 Jun 2005, Antoine Joux and Reynald Lercier announced the computation of a discrete logarithm modulo a 130-digit strong prime in three weeks, using a 1.15 GHz 16-processor HP AlphaServer GS1280 computer and a number field sieve algorithm.^[1]

On 5 Feb 2007 this was superseded by the announcement by Thorsten Kleinjung of the computation of a discrete logarithm modulo a 160-digit safe prime, again using the number field sieve. Most of the computation was done using idle time on various PCs and on a parallel computing cluster.^[2]

^ Antoine Joux, “Discrete logarithms in GF(p) --- 130 digits,” June 18, 2005, [[1]].
^ Thorsten Kleinjung, “Discrete logarithms in GF(p) --- 160 digits,” February 5, 2007, [[2]].

[1] Antoine Joux, “Discrete logarithms in GF(p) --- 130 digits,” June 18, 2005, [[1]].

[2] Thorsten Kleinjung, “Discrete logarithms in GF(p) --- 160 digits,” February 5, 2007, [[2]].

[1]

[2]

Integers modulo p {\displaystyle p}

Integers modulo $p$