Efficient Prime Factorization for large numbers

Question

I've been working on a little problem where I need to compute 18-digit numbers into their respective prime factorization. Everything compiles and it runs just fine, considering that it actually works, but I am looking to reduce the run time of the prime factorization. I have implemented recursion and threading but I think I might need some help in understanding possible algorithms for large number computation.

Every time I run this on the 4 numbers I have pre-made, it takes about 10 seconds. I would like to reduce this to possibly 0.06 seconds if there are any ideas out there.

I noticed a few algorithms like Sieve of Eratosthenes and producing a list of all the prime numbers prior to computing. I'm just wondering if someone could elaborate on it. For instance, I'm having issues understanding how to implement Sieve of Eratosthenes into my program or if it would even be a good idea. Any and all pointers on how to approach this better would be really helpful!

Here is my code:

#include <iostream>
#include <thread>
#include <vector>
#include <chrono>

using namespace std;
using namespace std::chrono;

vector<thread> threads;
vector<long long> inputVector;
bool developer = false; 
vector<unsigned long long> factor_base;
vector<long long> primeVector;

class PrimeNumber
{
    long long initValue;        // the number being prime factored
    vector<long long> factors;  // all of the factor values
public:
    void setInitValue(long long n)
    {
        initValue = n;
    }
    void addToVector(long long m)
    {
        factors.push_back(m);
    }
    void setVector(vector<long long> m)
    {
        factors = m;
    }
    long long getInitValue()
    {
        return initValue;
    }
    vector<long long> getVector()
    {
        return factors;
    }
};

vector<PrimeNumber> primes;

// find primes recursively and have them returned in vectors
vector<long long> getPrimes(long long n, vector<long long> vec)
{
    double sqrt_of_n = sqrt(n);

    for (int i = 2; i <= sqrt_of_n; i++)
    {
        if (n % i == 0) 
        {
            return vec.push_back(i), getPrimes(n / i, vec); //cause recursion
        }
    }

    // pick up the last prime factorization number
    vec.push_back(n);

    //return the finished vector
    return vec;
}

void getUserInput()
{
    long long input = -1;
    cout << "Enter all of the numbers to find their prime factors. Enter 0 to compute" << endl;
    do
    {
        cin >> input;
        if (input == 0)
        {
            break;
        }
        inputVector.push_back(input);
    } while (input != 0);
}

int main() 
{

    vector<long long> temp1;   // empty vector
    vector<long long> result1; // temp vector

    if (developer == false)
    {
        getUserInput();
    }
    else
    {
        cout << "developer mode active" << endl;
        long long a1 = 771895004973090566;
        long long b1 = 788380500764597944;
        long long a2 = 100020000004324000;
        long long b2 = 200023423420000000;
        inputVector.push_back(a1);
        inputVector.push_back(b2);
        inputVector.push_back(b1);
        inputVector.push_back(a2);
    }

    high_resolution_clock::time_point time1 = high_resolution_clock::now();

    // give each thread a number to comput within the recursive function
    for (int i = 0; i < inputVector.size(); i++)
    {   
        PrimeNumber prime;
        prime.setInitValue(inputVector.at(i));
        threads.push_back(thread([&]{
            prime.setVector(result1 = getPrimes(inputVector.at(i), temp1));
            primes.push_back(prime);
        }));
    }

    // allow all of the threads to join back together.
    for (auto& th : threads)
    {
        cout << th.get_id() << endl;
        th.join();
    }

    high_resolution_clock::time_point time2 = high_resolution_clock::now();

    // print all of the information
    for (int i = 0; i < primes.size(); i++)
    {
        vector<long long> temp = primes.at(i).getVector();

        for (int m = 0; m < temp.size(); m++)
        {
            cout << temp.at(m) << " ";
        }
        cout << endl;
    }

    cout << endl;

    // so the running time
    auto duration = duration_cast<microseconds>(time2 - time1).count();

    cout << "Duration: " << (duration / 1000000.0) << endl;

    return 0;
}

Answer 1

Trial division is only suitable for factoring small numbers. For n up to 2^64, you'll need a better algorithm: I recommend starting with wheel factorization to get the small factors, followed by Pollard's rho algorithm to get the rest. Where trial division is O(sqrt(n)), rho is O(sqrt(sqrt(n))), so it's much faster. For 2^64, sqrt(n) = 2^32, but sqrt(sqrt(n)) = 2^16, which is a huge improvement. You should expect to factor your numbers in a few milliseconds, at most.

I don't have C++ code for factoring, but I do have readable Python code. Let me know if you want me to post it. If you want to know more about wheel factorization and the rho algorithm, I have lots of prime number stuff at my blog.