Factorizing a large odd integer, n, into its corresponding prime factors can prove to be a difficult task. A brute approach can be testing all integers less than n until a divisor is found. This proves to be very time consuming as a divisor might be a very large prime itself. Pollard p-1 algorithm is a better approach to find out prime factors of any integer. Using the combined help of Modular Exponentiation and GCD, it is able to calculate all the distinct prime factors in no time.
Algorithm
- Given a number n. Initialize a = 2, i = 2
- Until a factor is returned do a <- (a^i) mod n d <- GCD(a-1, n) if 1 < d < n then return d else i <- i+1
- Other factor, d’ <- n/d
- If d’ is not prime n <- d’ goto 1 else d and d’ are two prime factors.
In the above algorithm, the power of ‘a’ is continuously raised until a factor, ‘d’, of n is obtained. Once d is obtained, another factor, ‘d”, is n/d. If d’ is not prime, the same task is repeated for d’ Examples:
Input : 1403 Output : Prime factors of 1403 are 61 23. Explanation : n = 1403, a = 2, i = 2 1st Iteration: a = (2^2) mod 1403 = 4 d = GCD(3, 1403) = 1 i = 2 + 1 = 3 2nd Iteration: a = (4^3) mod 1403 = 64 d = GCD(63, 1403) = 1 i = 3 + 1 = 4 3rd Iteration: a = (64^4) mod 1403 = 142 d = GCD(141, 1403) = 1 i = 4 + 1 = 5 4th Iteration: a = (142^5) mod 1403 = 794 d = GCD(793, 1403) = 61 Since 1 < d < n, one factor is 61. d' = 1403 / 61 = 23. Input : 2993 Output : Prime factors of 2993 are 41 73.
Below is the implementation.
// C++ code for Pollard p-1 // factorization Method #include <bits/stdc++.h> using namespace std;
// function for // calculating GCD int gcd( int a, int b)
{ if (a == 0)
return b;
return gcd(b % a, a);
} // function for // checking prime bool isPrime( int n)
{ if (n <= 1)
return false ;
if (n == 2 || n == 3)
return true ;
if (n % 2 == 0)
return false ;
for ( int i = 3; i * i <= n; i += 2)
if (n % i == 0)
return false ;
return true ;
} // function to generate // prime factors int pollard( int n)
{ // defining base
long long a = 2;
// defining exponent
int i = 2;
// iterate till a prime factor is obtained
while ( true )
{
// recomputing a as required
a = (( long long ) pow (a, i)) % n;
a += n;
a %= n;
// finding gcd of a-1 and n
// using math function
int d = gcd(a-1,n);
// check if factor obtained
if (d > 1)
{
//return the factor
return d;
break ;
}
// else increase exponent by one
// for next round
i += 1;
}
} // Driver code int main()
{ int n = 1403;
// temporarily storing n
int num = n;
// list for storing prime factors
vector< int > ans;
// iterated till all prime factors
// are obtained
while ( true )
{
// function call
int d = pollard(num);
// add obtained factor to list
ans.push_back(d);
// reduce n
int r = (num/d);
// check for prime
if (isPrime(r))
{
// both prime factors obtained
ans.push_back(r);
break ;
}
// reduced n is not prime, so repeat
else
num = r;
}
// print the result
cout << "Prime factors of " << n << " are " ;
for ( int elem : ans)
cout << elem << " " ;
} // This code is contributed by phasing17 |
// Java code for Pollard p-1 // factorization Method import java.util.*;
class GFG
{ // function for
// calculating GCD
static long gcd( long a, long b)
{
if (a == 0 )
return b;
return gcd(b % a, a);
}
// function for
// checking prime
static boolean isPrime( long n)
{
if (n <= 1 )
return false ;
if (n == 2 || n == 3 )
return true ;
if (n % 2 == 0 )
return false ;
for ( long i = 3 ; i * i <= n; i += 2 )
if (n % i == 0 )
return false ;
return true ;
}
// function to generate
// prime factors
static long pollard( long n)
{
// defining base
long a = 2 ;
// defining exponent
long i = 2 ;
// iterate till a prime factor is obtained
while ( true )
{
// recomputing a as required
a = (( long ) Math.pow(a, i)) % n;
a += n;
a %= n;
// finding gcd of a-1 and n
// using math function
long d = gcd(a- 1 ,n);
// check if factor obtained
if (d > 1 )
{
//return the factor
return d;
}
// else increase exponent by one
// for next round
i += 1 ;
}
}
// Driver code
public static void main(String[] args)
{
long n = 1403 ;
// temporarily storing n
long num = n;
// list for storing prime factors
ArrayList<Long> ans = new ArrayList<Long>();
// iterated till all prime factors
// are obtained
while ( true )
{
// function call
long d = pollard(num);
// add obtained factor to list
ans.add(d);
// reduce n
long r = (num/d);
// check for prime
if (isPrime(r))
{
// both prime factors obtained
ans.add(r);
break ;
}
// reduced n is not prime, so repeat
else
num = r;
}
// prlong the result
System.out.print( "Prime factors of " + n + " are " );
for ( long elem : ans)
System.out.print(elem + " " );
}
} // This code is contributed by phasing17 |
# Python code for Pollard p-1 # factorization Method # importing "math" for # calculating GCD import math
# importing "sympy" for # checking prime import sympy
# function to generate # prime factors def pollard(n):
# defining base
a = 2
# defining exponent
i = 2
# iterate till a prime factor is obtained
while ( True ):
# recomputing a as required
a = (a * * i) % n
# finding gcd of a-1 and n
# using math function
d = math.gcd((a - 1 ), n)
# check if factor obtained
if (d > 1 ):
#return the factor
return d
break
# else increase exponent by one
# for next round
i + = 1
# Driver code n = 1403
# temporarily storing n num = n
# list for storing prime factors ans = []
# iterated till all prime factors # are obtained while ( True ):
# function call
d = pollard(num)
# add obtained factor to list
ans.append(d)
# reduce n
r = int (num / d)
# check for prime using sympy
if (sympy.isprime(r)):
# both prime factors obtained
ans.append(r)
break
# reduced n is not prime, so repeat
else :
num = r
# print the result print ( "Prime factors of" , n, "are" , * ans)
|
// C# code for Pollard p-1 // factorization Method using System;
using System.Collections.Generic;
class GFG
{ // function for
// calculating GCD
static long gcd( long a, long b)
{
if (a == 0)
return b;
return gcd(b % a, a);
}
// function for
// checking prime
static bool isPrime( long n)
{
if (n <= 1)
return false ;
if (n == 2 || n == 3)
return true ;
if (n % 2 == 0)
return false ;
for ( long i = 3; i * i <= n; i += 2)
if (n % i == 0)
return false ;
return true ;
}
// function to generate
// prime factors
static long pollard( long n)
{
// defining base
long a = 2;
// defining exponent
long i = 2;
// iterate till a prime factor is obtained
while ( true )
{
// recomputing a as required
a = (( long ) Math.Pow(a, i)) % n;
a += n;
a %= n;
// finding gcd of a-1 and n
// using math function
long d = gcd(a-1,n);
// check if factor obtained
if (d > 1)
{
//return the factor
return d;
}
// else increase exponent by one
// for next round
i += 1;
}
}
// Driver code
public static void Main( string [] args)
{
long n = 1403;
// temporarily storing n
long num = n;
// list for storing prime factors
List< long > ans = new List< long >();
// iterated till all prime factors
// are obtained
while ( true )
{
// function call
long d = pollard(num);
// add obtained factor to list
ans.Add(d);
// reduce n
long r = (num/d);
// check for prime
if (isPrime(r))
{
// both prime factors obtained
ans.Add(r);
break ;
}
// reduced n is not prime, so repeat
else
num = r;
}
// prlong the result
Console.Write( "Prime factors of " + n + " are " );
foreach ( long elem in ans)
Console.Write(elem + " " );
}
} // This code is contributed by phasing17 |
// JavaScript code for Pollard p-1 // factorization Method // function for // calculating GCD function gcd(x, y)
{ x = Math.abs(x);
y = Math.abs(y);
while (y) {
var t = y;
y = x % y;
x = t;
}
return x;
} // function for // checking prime function isPrime(n)
{ if (n <= 1)
return false ;
if (n == 2 || n == 3)
return true ;
if (n % 2 == 0)
return true ;
for ( var i = 3; i * i <= n; i += 2)
if (n % i == 0)
return false ;
return true ;
} // function to generate // prime factors function pollard(n)
{ // defining base
let a = 2
// defining exponent
let i = 2
// iterate till a prime factor is obtained
while ( true )
{
// recomputing a as required
a = (a**i) % n
// finding gcd of a-1 and n
// using math function
d = gcd((a-1), n)
// check if factor obtained
if (d > 1)
{
//return the factor
return d
break
}
// else increase exponent by one
// for next round
i += 1
}
} // Driver code let n = 1403 // temporarily storing n let num = n // list for storing prime factors let ans = [] // iterated till all prime factors // are obtained while ( true )
{ // function call
let d = pollard(num)
// add obtained factor to list
ans.push(d)
// reduce n
r = Math.floor(num/d)
// check for prime
if (isPrime(r))
{
// both prime factors obtained
ans.push(r)
break
}
// reduced n is not prime, so repeat
else
num = r
} // print the result console.log( "Prime factors of" , n, "are" , ans.join( " " ))
// This code is contributed by phasing17 |
Output:
Prime factors of 1403 are 61 23