# Information Security and Cryptography Research Group

## Towards the Equivalence of Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms

### Ueli Maurer

Advances in Cryptology — CRYPTO '94, Lecture Notes in Computer Science, Springer-Verlag, vol. 839, pp. 271–281, Aug 1994.

Let $G$ be an arbitrary cyclic group with generator $g$ and order $|G|$ with known factorization. $G$ could be the subgroup generated by $g$ within a larger group $H$. Based on an assumption about the existence of smooth numbers in short intervals, we prove that breaking the Diffie-Hellman protocol for $G$ and base $g$ is equivalent to computing discrete logarithms in $G$ to the base $g$ when a certain side information string $S$ of length $2\log|G|$ is given, where $S$ depends only on $|G|$ but not on the definition of $G$ and appears to be of no help for computing discrete logarithms in $G$. If every prime factor $p$ of $|G|$ is such that one of a list of expressions in $p$, including $p-1$ and $p+1$, is smooth for an appropriate smoothness bound, then $S$ can efficiently be constructed and therefore breaking the Diffie-Hellman protocol is equivalent to computing discrete logarithms.

## BibTeX Citation

@inproceedings{Maurer94,
author       = {Ueli Maurer},
title        = {Towards the Equivalence of Breaking the {D}iffie-{H}ellman Protocol and Computing Discrete Logarithms},
editor       = {Yvo Desmedt},
booktitle    = {Advances in Cryptology --- CRYPTO~'94},
pages        = 271--281,
series       = {Lecture Notes in Computer Science},
volume       = 839,
year         = 1994,
month        = 8,
publisher    = {Springer-Verlag},
}