How long would it take my i-7 processor to factorise a 1024 bits number (consisting of just 2 prime factors)

Question

We're examining the RSA algorithm and would like to know how much time it would take an intel i-7 core (@ 2.50 gHz) to factorise the RSA-public key.

we wrote a piece of java for this, and I don't know how effective it is

public static String factorise(long l)
{
    double a = Math.floor(Math.sqrt(l));
    while(l/a != Math.round(l/a))
    {
        a--;
    }
    return (long)a + ", " + (long)(l/a);    
}

With a number around 2^45 it took the PC approximately 33 milliseconds. In theory, how long would it take to factorise a number around 2^1024?

Thanks in advance :)

Answer 1

Your algorithm is O(2^n), where n is the number of bits in the original number l. (that means that a single bit more will double the runtime, because twice as many numbers a must be checked - on average)

If 45 bits took 33 ms, then 1024 bits will take approx. 2^1024 / 2^45 * 33ms = 5.34654 * 10^285 years.

This of course assumes, that the 1024bit code is exactly as efficient as your code for long numbers (64bit?). Which is a bold statement, considering that 10^285 years is more than enough time to switch to the General number field sieve and scratch a few million years of that time...

---

In 2009 the 768 bit number rsa-768 was cracked using about 1000 cores and 2 years of calculations. Assuming they used the General number field sieve (a very fair assumption) it would take them 7481 years to crack a 1024 bit number using the same hardware.

Or using only your i7 with this algorithm: about 3 million years. Still a long time.... ;)