Euler's Phi Function Applied to Large NumbersDate: 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/ |
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