Why is not known whether integer factorization can be done in polynomial time knowing how to do primality tests efficiently?

Problem Detail: 

First of all, I have just started studying computer science by myself and maybe I just need some clarification of what "polynomial time" means regarding the time complexity of an algorithm and references to study it well.

As I have understood it, whether integer factorization can be done in polynomial time is still an open question and, as this article in wikipedia (https://en.wikipedia.org/wiki/Integer_factorization) puts it,

When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort by several researchers concluded in 2009, factoring a 232-digit number (RSA-768), utilizing hundreds of machines took two years and the researchers estimated that a 1024-bit RSA modulus would take about a thousand times as long.

So, trying to see that for myself, I have written a very naive code in MATLAB checking it with prime numbers up to 15 digits; the reasoning being that if I can check if a number is prime fast, I can easily modify the code to give me the factorization fast.

The time it takes the code to check if a number is prime doesn't grow exponentially with the input.

function[]=prime(n) tic f=floor(sqrt(n)); for i=2:f     if rem(n,i)~=0         b=0;     else         b=1;         disp(i)         break     end end if b==0     disp('prime') else     disp('not prime') end toc end 

And so I go back to the question in the title. What is wrong with my reasoning?

Asked By : kdow

Answered By : Tom van der Zanden

Since your algorithm is "fast", why did you only try it with a 15-digit number and not with a 232-digit one? There's serious money to be made if you indeed have a "fast" algorithm.

Your algorithm takes time (if we count "div" as taking constant time) proportional to $\sqrt{n}$. A $d$-digit number can be as large as $10^d$, so your algorithm takes time proportional to $\sqrt{10^d} \approx 3.16^d$, i.e. exponential in $d$, the number of digits. That is by no means "fast" and grows very quickly as the numbers get larger.

It is polynomial with respect to the value of $n$, but not with respect to the size of $n$. This behavior is called pseudopolynomiality.

The "fast" prime testing algorithms use much more sophisticated approaches which can not be modified (easily) to also give a factorization. They just report yes/no whether the number is prime. The AKS primality test uses time proportional to $d^6$.

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Question Source : http://cs.stackexchange.com/questions/50767