Quantum Algorithm for the Discrete Logarithm Problem

Keywords and Synonyms

Logarithms in groups          

Problem Definition

Given positive real numbers ${ a \neq 1, b }$, the logarithm of b to base a is the unique real number s such that ${ b = a^s }$. The notion of the discrete logarithm is an extension of this concept to general groups.

Problem 1 (Discrete logarithm)

Input: Group ${ G, a, b \in G }$ such that ${ b = a^s }$ for some positive integer s.

Output: The smallest positive integer s satisfying ${ b = a^s }$, also known as the discrete logarithm of b to the base a in G.

The usual logarithm corresponds to the discrete logarithm problem over the group of positive reals under multiplication. The most common case of the discrete logarithm problem is when the group ${ G = \mathbb{Z}_p^* }$, the multiplicative group of integers between 1 and ${ p - 1 }$ modulo p, where p is a prime. Another important case is when the group G is the group of points of an elliptic curve over a finite field.

Key Results

The discrete...