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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?

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    $\begingroup$ Read up on the difference between polynomial and pseudo-polynomial time. $\endgroup$ – Raphael Dec 15 '15 at 14:12
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    $\begingroup$ I think the key point of confusion is that the time for integer factorization is expressed in terms of the number of digits in the number, not the number itself. Primality is testable in time polynomial in the number of digits; factorization is not known to be solvable in polynomial time in the number of digits. $\endgroup$ – Louis Wasserman Dec 15 '15 at 19:11
  • $\begingroup$ @PålGD How do you know that $exp(15)∼10^7$ means that the algorithm process a 15-digit number fast? And everyone seems to have calculated the running time differently, is there anyway less arbitrary to tackle that question? $\endgroup$ – Alberto Dec 16 '15 at 2:44
  • $\begingroup$ The example here en.wikipedia.org/wiki/Pseudo-polynomial_time is exactly what I was asking. Thanks, @Raphael $\endgroup$ – Alberto Dec 16 '15 at 12:49
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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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  • $\begingroup$ I checked if it grew exponentially with the input number plotting time-number and observing the trend. It seemed logarithmic and it surely wasn't exponential. Also, how can I input a 200-digit number in my computer? $\endgroup$ – Alberto Dec 15 '15 at 19:27
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    $\begingroup$ @Alberto Arbitrary precision integer libraries handle this, and then arithmetic operators are not O(1) but rather O(b) where b is the number of bits in the integer. A 200-digit number will have roughly 650 bits... $\endgroup$ – recursion.ninja Dec 15 '15 at 20:27
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    $\begingroup$ @Alberto don't check how it grows compared to the input number, check how it grows compared to the length of the input number. In this problem, going from 15 digits to 30 digits counts only as twice as much, not as an increase of 1000000000000000 times. $\endgroup$ – Peteris Dec 16 '15 at 9:20
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    $\begingroup$ Seeing the comments below the question, I think you can improve this answer for reference purposes by mentioning the term "pseudo-polynomial". $\endgroup$ – Raphael Dec 16 '15 at 14:38
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The point is that the complexity of an algorithms is measured in the size of the input. For a number $n$ its size, the length of its representation, is $\log n$ bits.

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  • $\begingroup$ So, why Tom van der Zanden says that it goes as $3.14^n$? And Pål GD, in their comment below my question, $exp(15)∼107$? $\endgroup$ – Alberto Dec 15 '15 at 19:33
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    $\begingroup$ 3.16, not 3.14. One is approximately the square root of ten, the other is approximately pi. There were some cultures that thought pi was equal to the square root of ten because of their proximity. $\endgroup$ – David Conrad Dec 15 '15 at 21:56
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There can be different algorithms to solve a problem and there is no reason that all of them should follow some particular structure. It is a common mistake to assume that to check if a number is not a prime an algorithm needs to find a non-trivial factor of the number. This is not true. The AKS algorithm does not find any factors when it reports a number is not prime.

AKS is rather complicated so let's look at the simpler Miller–Rabin algorithm (ignoring the fact that it is a probabilistic algorithm). Let $n$ be the number we want to check if it is prime or not. The case of even numbers is easy so we consider the case of odd $n$. Let $n-1 = d2^s$ where $d$ is an odd number.

The idea is:

$n$ is not prime iff for all $0 < a < n$ and $0 \leq r < s$, $$a^d \not\equiv 1 \ (\text{mod } n)$$ and $$a^{2^rd} \not\equiv -1 \ (\text{mod } n)$$

Assume that we checked these conditions, then we know $n$ is not a prime without finding a nontrivial factor of $n$.

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If one uses N%D to denote the remainder after dividing N by D, and B^E to denote B raised to the power E, then for any integer N and X with no common factors greater than 1, there will exist some integer Y less than N-1 such that (X^Y)%N equals 1. If N is prime, the smallest such integer will be Y=N-2. For most non-prime values of N, however, most values of X will cause (X^(N-2))%N to yield something else.

If one could identify the Y values for which (X^Y)%N equals 1, that information could be used to factor N fairly easily (and likewise knowing the factors one could compute Y values), but there's no known generally-useful way to find such values without knowing or guessing at the factorization of N.

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    $\begingroup$ All of what you wrote may be true (except that N-2 should be N-1) but I don't see how this answers the question that was asked. $\endgroup$ – D.W. Dec 16 '15 at 0:37
  • $\begingroup$ @D.W.: The title of the question notes that primality testing can be done efficiently; I thought it would be helpful to illustrate how a common method of primality testing involves testing whether a given value is a solution to an equation, while factoring would involve finding solutions to that equation. As for N-2 vs N-1, for a prime number N, (X^(N-2))%Y==1 and (X^(N-1))%Y=X, for X in the range 1..N-1 $\endgroup$ – supercat Dec 16 '15 at 2:40

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