Euler's Phi Function Applied to Large Numbers

Date: 08/23/2004 at 12:27:22
From: Aruna
Subject: Finding the phi (Euler's function)  of a large number

Euler's phi function [phi(N)] is defined as the number of numbers less
than N and coprime to N.  I need to find the phi of a large number.  I
understand that if this number is a prime P, then phi(P) = P - 1.
However, the number I have in mind is composite and may be a 10 digit
number.

I want to know if this can be done in polynomial time, and if so, is 
there an algorithm for it?

I realize that one way to do this is to find the prime factorization 
of the number N.  We know that phi(P*Q) = (P-1) * (Q-1), if P and Q 
are primes.  However, factorization itself is a hard problem.

Thank you!

Date: 08/23/2004 at 15:21:38
From: Doctor Vogler
Subject: Re: Finding the phi (Euler's function)  of a large number

Hi Aruna,

Thanks for writing to Dr Math.  The answer to your question is no,
there is no known way to get phi(n) in polynomial time for general
(large) n.  In fact, if n is a product of two primes,

  n = p*q

then finding phi(n) is equivalent to factoring n, because if you can
find phi(n), then

  p + q = n + 1 - phi(n)
  p * q = n,

so p and q are, respectively,

  n + 1 - phi(n) + sqrt((n + 1 - phi(n))^2 - 4n)
  ----------------------------------------------
                        2

and

  n + 1 - phi(n) - sqrt((n + 1 - phi(n))^2 - 4n)
  ----------------------------------------------.
                        2

And those two expressions are very easy to calculate.  So if you could
compute phi(n) in polynomial time, then you could factor n in 
polynomial time, and it is conjectured (but not yet proven) that it is
impossible to factor n in polynomial time.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/