# Matching Modulo Divisors, and a Simple N^1/4 Factoring Algorithm

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-187.pdf

We present an algorithm for the following problem: given x(1),..., x(m), distinct modulo N, find two numbers in the list that are congruent modulo a proper divisor of N. The number of steps required is less than m times a polynomial in log N. This gives a simple proof that N can be factored in expected time O( N^(1/4+ e)) for any e>0.

BibTeX citation:

```@techreport{Bach:CSD-84-187,
Author = {Bach, Eric},
Title = {Matching Modulo Divisors, and a Simple N^1/4 Factoring Algorithm},
Institution = {EECS Department, University of California, Berkeley},
Year = {1984},
Month = {Jun},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/5972.html},
Number = {UCB/CSD-84-187},
Abstract = {We present an algorithm for the following problem: given <i>x(1)</i>,...,<i>x(m)</i>, distinct modulo <i>N</i>, find two numbers in the list that are congruent modulo a proper divisor of <i>N</i>.  The number of steps required is less than <i>m</i> times a polynomial in log<i>N</i>.  This gives a simple proof that <i>N</i> can be factored in expected time <i>O</i>(<i>N</i>^(1/4+<i>e</i>)) for any <i>e</i>>0.}
}
```

EndNote citation:

```%0 Report
%A Bach, Eric
%T Matching Modulo Divisors, and a Simple N^1/4 Factoring Algorithm
%I EECS Department, University of California, Berkeley
%D 1984
%@ UCB/CSD-84-187
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/5972.html
%F Bach:CSD-84-187
```