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Primitive root of a prime number n modulo n

  • Difficulty Level : Hard
  • Last Updated : 24 May, 2021

Given a prime number n, the task is to find its primitive root under modulo n. The primitive root of a prime number n is an integer r between[1, n-1] such that the values of r^x(mod n) where x is in the range[0, n-2] are different. Return -1 if n is a non-prime number.

Examples:  

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Input : 7
Output : Smallest primitive root = 3
Explanation: n = 7
3^0(mod 7) = 1
3^1(mod 7) = 3
3^2(mod 7) = 2
3^3(mod 7) = 6
3^4(mod 7) = 4
3^5(mod 7) = 5

Input : 761
Output : Smallest primitive root = 6 

A simple solution is to try all numbers from 2 to n-1. For every number r, compute values of r^x(mod n) where x is in the range[0, n-2]. If all these values are different, then return r, else continue for the next value of r. If all values of r are tried, return -1.

An efficient solution is based on the below facts. 
If the multiplicative order of a number r modulo n is equal to Euler Totient Function Φ(n) ( note that the Euler Totient Function for a prime n is n-1), then it is a primitive root. 



1- Euler Totient Function phi = n-1 [Assuming n is prime]
1- Find all prime factors of phi.
2- Calculate all powers to be calculated further 
   using (phi/prime-factors) one by one.
3- Check for all numbered for all powers from i=2 
   to n-1 i.e. (i^ powers) modulo n.
4- If it is 1 then 'i' is not a primitive root of n.
5- If it is never 1 then return i;.

Although there can be multiple primitive roots for a prime number, we are only concerned with the smallest one. If you want to find all the roots, then continue the process till p-1 instead of breaking up by finding the first primitive root. 

C++




// C++ program to find primitive root of a
// given number n
#include<bits/stdc++.h>
using namespace std;
 
// Returns true if n is prime
bool isPrime(int n)
{
    // Corner cases
    if (n <= 1)  return false;
    if (n <= 3)  return true;
 
    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n%2 == 0 || n%3 == 0) return false;
 
    for (int i=5; i*i<=n; i=i+6)
        if (n%i == 0 || n%(i+2) == 0)
            return false;
 
    return true;
}
 
/* Iterative Function to calculate (x^n)%p in
   O(logy) */
int power(int x, unsigned int y, int p)
{
    int res = 1;     // Initialize result
 
    x = x % p; // Update x if it is more than or
    // equal to p
 
    while (y > 0)
    {
        // If y is odd, multiply x with result
        if (y & 1)
            res = (res*x) % p;
 
        // y must be even now
        y = y >> 1; // y = y/2
        x = (x*x) % p;
    }
    return res;
}
 
// Utility function to store prime factors of a number
void findPrimefactors(unordered_set<int> &s, int n)
{
    // Print the number of 2s that divide n
    while (n%2 == 0)
    {
        s.insert(2);
        n = n/2;
    }
 
    // n must be odd at this point. So we can skip
    // one element (Note i = i +2)
    for (int i = 3; i <= sqrt(n); i = i+2)
    {
        // While i divides n, print i and divide n
        while (n%i == 0)
        {
            s.insert(i);
            n = n/i;
        }
    }
 
    // This condition is to handle the case when
    // n is a prime number greater than 2
    if (n > 2)
        s.insert(n);
}
 
// Function to find smallest primitive root of n
int findPrimitive(int n)
{
    unordered_set<int> s;
 
    // Check if n is prime or not
    if (isPrime(n)==false)
        return -1;
 
    // Find value of Euler Totient function of n
    // Since n is a prime number, the value of Euler
    // Totient function is n-1 as there are n-1
    // relatively prime numbers.
    int phi = n-1;
 
    // Find prime factors of phi and store in a set
    findPrimefactors(s, phi);
 
    // Check for every number from 2 to phi
    for (int r=2; r<=phi; r++)
    {
        // Iterate through all prime factors of phi.
        // and check if we found a power with value 1
        bool flag = false;
        for (auto it = s.begin(); it != s.end(); it++)
        {
 
            // Check if r^((phi)/primefactors) mod n
            // is 1 or not
            if (power(r, phi/(*it), n) == 1)
            {
                flag = true;
                break;
            }
         }
 
         // If there was no power with value 1.
         if (flag == false)
           return r;
    }
 
    // If no primitive root found
    return -1;
}
 
// Driver code
int main()
{
    int n = 761;
    cout << " Smallest primitive root of " << n
         << " is " << findPrimitive(n);
    return 0;
}

Java




// Java program to find primitive root of a
// given number n
import java.util.*;
 
class GFG
{
 
    // Returns true if n is prime
    static boolean isPrime(int n)
    {
        // Corner cases
        if (n <= 1)
        {
            return false;
        }
        if (n <= 3)
        {
            return true;
        }
 
        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
        {
            return false;
        }
 
        for (int i = 5; i * i <= n; i = i + 6)
        {
            if (n % i == 0 || n % (i + 2) == 0)
            {
                return false;
            }
        }
 
        return true;
    }
 
    /* Iterative Function to calculate (x^n)%p in
    O(logy) */
    static int power(int x, int y, int p)
    {
        int res = 1;     // Initialize result
 
        x = x % p; // Update x if it is more than or
        // equal to p
 
        while (y > 0)
        {
            // If y is odd, multiply x with result
            if (y % 2 == 1)
            {
                res = (res * x) % p;
            }
 
            // y must be even now
            y = y >> 1; // y = y/2
            x = (x * x) % p;
        }
        return res;
    }
 
    // Utility function to store prime factors of a number
    static void findPrimefactors(HashSet<Integer> s, int n)
    {
        // Print the number of 2s that divide n
        while (n % 2 == 0)
        {
            s.add(2);
            n = n / 2;
        }
 
        // n must be odd at this point. So we can skip
        // one element (Note i = i +2)
        for (int i = 3; i <= Math.sqrt(n); i = i + 2)
        {
            // While i divides n, print i and divide n
            while (n % i == 0)
            {
                s.add(i);
                n = n / i;
            }
        }
 
        // This condition is to handle the case when
        // n is a prime number greater than 2
        if (n > 2)
        {
            s.add(n);
        }
    }
 
    // Function to find smallest primitive root of n
    static int findPrimitive(int n)
    {
        HashSet<Integer> s = new HashSet<Integer>();
 
        // Check if n is prime or not
        if (isPrime(n) == false)
        {
            return -1;
        }
 
        // Find value of Euler Totient function of n
        // Since n is a prime number, the value of Euler
        // Totient function is n-1 as there are n-1
        // relatively prime numbers.
        int phi = n - 1;
 
        // Find prime factors of phi and store in a set
        findPrimefactors(s, phi);
 
        // Check for every number from 2 to phi
        for (int r = 2; r <= phi; r++)
        {
            // Iterate through all prime factors of phi.
            // and check if we found a power with value 1
            boolean flag = false;
            for (Integer a : s)
            {
 
                // Check if r^((phi)/primefactors) mod n
                // is 1 or not
                if (power(r, phi / (a), n) == 1)
                {
                    flag = true;
                    break;
                }
            }
 
            // If there was no power with value 1.
            if (flag == false)
            {
                return r;
            }
        }
 
        // If no primitive root found
        return -1;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 761;
        System.out.println(" Smallest primitive root of " + n
                + " is " + findPrimitive(n));
    }
}
 
/* This code contributed by PrinciRaj1992 */

Python3




# Python3 program to find primitive root
# of a given number n
from math import sqrt
 
# Returns True if n is prime
def isPrime( n):
 
    # Corner cases
    if (n <= 1):
        return False
    if (n <= 3):
        return True
 
    # This is checked so that we can skip
    # middle five numbers in below loop
    if (n % 2 == 0 or n % 3 == 0):
        return False
    i = 5
    while(i * i <= n):
        if (n % i == 0 or n % (i + 2) == 0) :
            return False
        i = i + 6
 
    return True
 
""" Iterative Function to calculate (x^n)%p
    in O(logy) */"""
def power( x, y, p):
 
    res = 1 # Initialize result
 
    x = x % p # Update x if it is more
              # than or equal to p
 
    while (y > 0):
 
        # If y is odd, multiply x with result
        if (y & 1):
            res = (res * x) % p
 
        # y must be even now
        y = y >> 1 # y = y/2
        x = (x * x) % p
 
    return res
 
# Utility function to store prime
# factors of a number
def findPrimefactors(s, n) :
 
    # Print the number of 2s that divide n
    while (n % 2 == 0) :
        s.add(2)
        n = n // 2
 
    # n must be odd at this po. So we can 
    # skip one element (Note i = i +2)
    for i in range(3, int(sqrt(n)), 2):
         
        # While i divides n, print i and divide n
        while (n % i == 0) :
 
            s.add(i)
            n = n // i
         
    # This condition is to handle the case
    # when n is a prime number greater than 2
    if (n > 2) :
        s.add(n)
 
# Function to find smallest primitive
# root of n
def findPrimitive( n) :
    s = set()
 
    # Check if n is prime or not
    if (isPrime(n) == False):
        return -1
 
    # Find value of Euler Totient function
    # of n. Since n is a prime number, the
    # value of Euler Totient function is n-1
    # as there are n-1 relatively prime numbers.
    phi = n - 1
 
    # Find prime factors of phi and store in a set
    findPrimefactors(s, phi)
 
    # Check for every number from 2 to phi
    for r in range(2, phi + 1):
 
        # Iterate through all prime factors of phi.
        # and check if we found a power with value 1
        flag = False
        for it in s:
 
            # Check if r^((phi)/primefactors)
            # mod n is 1 or not
            if (power(r, phi // it, n) == 1):
 
                flag = True
                break
             
        # If there was no power with value 1.
        if (flag == False):
            return r
 
    # If no primitive root found
    return -1
 
# Driver Code
n = 761
print("Smallest primitive root of",
         n, "is", findPrimitive(n))
 
# This code is contributed by
# Shubham Singh(SHUBHAMSINGH10)

C#




// C# program to find primitive root of a
// given number n
using System;
using System.Collections.Generic;
 
class GFG
{
 
    // Returns true if n is prime
    static bool isPrime(int n)
    {
        // Corner cases
        if (n <= 1)
        {
            return false;
        }
        if (n <= 3)
        {
            return true;
        }
 
        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
        {
            return false;
        }
 
        for (int i = 5; i * i <= n; i = i + 6)
        {
            if (n % i == 0 || n % (i + 2) == 0)
            {
                return false;
            }
        }
 
        return true;
    }
 
    /* Iterative Function to calculate (x^n)%p in
    O(logy) */
    static int power(int x, int y, int p)
    {
        int res = 1;     // Initialize result
 
        x = x % p; // Update x if it is more than or
        // equal to p
 
        while (y > 0)
        {
            // If y is odd, multiply x with result
            if (y % 2 == 1)
            {
                res = (res * x) % p;
            }
 
            // y must be even now
            y = y >> 1; // y = y/2
            x = (x * x) % p;
        }
        return res;
    }
 
    // Utility function to store prime factors of a number
    static void findPrimefactors(HashSet<int> s, int n)
    {
        // Print the number of 2s that divide n
        while (n % 2 == 0)
        {
            s.Add(2);
            n = n / 2;
        }
 
        // n must be odd at this point. So we can skip
        // one element (Note i = i +2)
        for (int i = 3; i <= Math.Sqrt(n); i = i + 2)
        {
            // While i divides n, print i and divide n
            while (n % i == 0)
            {
                s.Add(i);
                n = n / i;
            }
        }
 
        // This condition is to handle the case when
        // n is a prime number greater than 2
        if (n > 2)
        {
            s.Add(n);
        }
    }
 
    // Function to find smallest primitive root of n
    static int findPrimitive(int n)
    {
        HashSet<int> s = new HashSet<int>();
 
        // Check if n is prime or not
        if (isPrime(n) == false)
        {
            return -1;
        }
 
        // Find value of Euler Totient function of n
        // Since n is a prime number, the value of Euler
        // Totient function is n-1 as there are n-1
        // relatively prime numbers.
        int phi = n - 1;
 
        // Find prime factors of phi and store in a set
        findPrimefactors(s, phi);
 
        // Check for every number from 2 to phi
        for (int r = 2; r <= phi; r++)
        {
            // Iterate through all prime factors of phi.
            // and check if we found a power with value 1
            bool flag = false;
            foreach (int a in s)
            {
 
                // Check if r^((phi)/primefactors) mod n
                // is 1 or not
                if (power(r, phi / (a), n) == 1)
                {
                    flag = true;
                    break;
                }
            }
 
            // If there was no power with value 1.
            if (flag == false)
            {
                return r;
            }
        }
 
        // If no primitive root found
        return -1;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int n = 761;
        Console.WriteLine(" Smallest primitive root of " + n
                + " is " + findPrimitive(n));
    }
}
 
// This code is contributed by Rajput-Ji

Javascript




<script>
// Javascript program to find primitive root of a
// given number n
 
 
// Returns true if n is prime
function isPrime(n) {
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
 
    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
 
    for (let i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;
 
    return true;
}
 
/* Iterative Function to calculate (x^n)%p in
O(logy) */
   
function power(x, y, p) {
    let res = 1;     // Initialize result
 
    x = x % p; // Update x if it is more than or
    // equal to p
 
    while (y > 0) {
        // If y is odd, multiply x with result
        if (y & 1)
            res = (res * x) % p;
 
        // y must be even now
        y = y >> 1; // y = y/2
        x = (x * x) % p;
    }
    return res;
}
 
// Utility function to store prime factors of a number
function findPrimefactors(s, n) {
    // Print the number of 2s that divide n
    while (n % 2 == 0) {
        s.add(2);
        n = n / 2;
    }
 
    // n must be odd at this point. So we can skip
    // one element (Note i = i +2)
    for (let i = 3; i <= Math.sqrt(n); i = i + 2) {
        // While i divides n, print i and divide n
        while (n % i == 0) {
            s.add(i);
            n = n / i;
        }
    }
 
    // This condition is to handle the case when
    // n is a prime number greater than 2
    if (n > 2)
        s.add(n);
}
 
// Function to find smallest primitive root of n
function findPrimitive(n) {
    let s = new Set();
 
    // Check if n is prime or not
    if (isPrime(n) == false)
        return -1;
 
    // Find value of Euler Totient function of n
    // Since n is a prime number, the value of Euler
    // Totient function is n-1 as there are n-1
    // relatively prime numbers.
    let phi = n - 1;
 
    // Find prime factors of phi and store in a set
    findPrimefactors(s, phi);
 
    // Check for every number from 2 to phi
    for (let r = 2; r <= phi; r++) {
        // Iterate through all prime factors of phi.
        // and check if we found a power with value 1
        let flag = false;
        for (let it of s) {
 
            // Check if r^((phi)/primefactors) mod n
            // is 1 or not
            if (power(r, phi / it, n) == 1) {
                flag = true;
                break;
            }
        }   
 
        // If there was no power with value 1.
        if (flag == false)
            return r;
    }
 
    // If no primitive root found
    return -1;
}
 
// Driver code
 
let n = 761;
document.write(" Smallest primitive root of " + n + " is " + findPrimitive(n));
 
// This code is contributed by gfgking
</script>

Output:  

Smallest primitive root of 761 is 6

This article is contributed by Niteesh kumar and Sahil Chhabra (akku). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 




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