# Discrete Logarithms and Factoring

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-186.pdf

This note discusses the relationship between the two problems of the title. We present probabilistic polynomial-time reductions that show:
1) To factor n, it suffices to be able to compute discrete logarithms modulo n.
2) To compute a discrete logarithm modulo a prime power p^(e), it suffices to know it mod p.
3) To compute a discrete logarithm modulo any n, it suffices to be able to factor and compute discrete logarithms modulo primes.

To summarize: solving the discrete logarithm problem for a composite modulus is exactly as hard as factoring and solving it modulo primes.

BibTeX citation:

```@techreport{Bach:CSD-84-186,
Author = {Bach, Eric},
Title = {Discrete Logarithms and Factoring},
Institution = {EECS Department, University of California, Berkeley},
Year = {1984},
Month = {Jun},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5973.html},
Number = {UCB/CSD-84-186},
Abstract = {This note discusses the relationship between the two problems of the title.  We present probabilistic polynomial-time reductions that show:  <br />1) To factor <i>n</i>, it suffices to be able to compute discrete logarithms modulo <i>n</i>.  <br />2) To compute a discrete logarithm modulo a prime power <i>p^(e)</i>, it suffices to know it mod <i>p</i>.  <br />3) To compute a discrete logarithm modulo any <i>n</i>, it suffices to be able to factor and compute discrete logarithms modulo primes.  <p>To summarize: solving the discrete logarithm problem for a composite modulus is exactly as hard as factoring and solving it modulo primes.}
}
```

EndNote citation:

```%0 Report
%A Bach, Eric
%T Discrete Logarithms and Factoring
%I EECS Department, University of California, Berkeley
%D 1984
%@ UCB/CSD-84-186
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5973.html
%F Bach:CSD-84-186
```