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

### Eric Bach

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-84-187

June 1984

### http://www.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://www.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://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5972.html %F Bach:CSD-84-187