Find power of power under mod of a prime

Given four numbers A, B, C and M, where M is prime number. Our task is to find ABC (mod M).

Example:

Input  : A = 2, B = 4, C = 3, M = 23
Output : 6
243(mod 23) = 6

A Naive Approach is to calculate res = BC and then calculate Ares % M by modular exponential. The problem of this approach is that we can’t apply directly mod M on BC, so we have to calculate this value without mod M. But if we solve it directly then we will come up with the large value of exponent of A which will definitely overflow in final answer.



An Efficient approach is to reduce the BC to a smaller value by using the Fermat’s Little Theorem, and then apply Modular exponential.

According the Fermat's little
a(M - 1) = 1 (mod M) if M is a prime.

So if we rewrite BC as x*(M-1) + y, then the
task of computing ABC becomes Ax*(M-1) + y
which can be written as Ax*(M-1)*Ay.
From Fermat's little theorem, we know Ax*(M-1) = 1.
So task of computing ABC reduces to computing Ay

What is the value of y?
From BC = x * (M - 1) + y,
y can be written as BC % (M-1)

We can easily use the above theorem such that we can get
A ^ (B ^ C) % M = (A ^ y ) %  M

Now we only need to find two things as:-
1. y = (B ^ C) % (M - 1)
2. Ans = (A ^ y) % M

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to find (a^b) mod m for a large 'a'
#include<bits/stdc++.h>
using namespace std;
  
// Iterative Function to calculate (x^y)%p in O(log y)
unsigned int power(unsigned int x, unsigned int y, 
                                   unsigned int p)
{
    unsigned 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;
}
  
unsigned int Calculate(unsigned int A, unsigned int B,
                       unsigned int C, unsigned int M)
{
    unsigned int res, ans;
  
    // Calculate B ^ C (mod M - 1)
    res = power(B, C, M-1);
  
    // Calculate A ^ res ( mod M )
    ans = power(A, res, M);
  
    return ans;
}
  
// Driver program to run the case
int main()
{   // M must be be a Prime Number
    unsigned int A = 3, B = 9, C = 4, M = 19;
  
    cout << Calculate(A, B, C, M);
  
    return 0;
}

chevron_right






                        filter_none









Java














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// Java program to find (a^b)  


// mod m for a large 'a' 


import java.util.*; 


  


class GFG { 


      


// Iterative Function to  


// calculate (x^y)%p in O(log y) 


static int power(int x, int y, int p) { 


      


    // Initialize result 


    int res = 1;  


  


    // Update x if it is more 


    // than or equal to p 


    x = x % p;  


  


    while (y > 0) { 


          


    // If y is odd, multiply x with result 


    if (y % 2 != 0) 


        res = (res * x) % p; 


  


    // y must be even now 


    y = y >> 1; // y = y/2 


    x = (x * x) % p; 


    } 


    return res; 


} 


  


static int Calculate(int A, int B, int C, int M) { 


    int res, ans; 


  


    // Calculate B ^ C (mod M - 1) 


    res = power(B, C, M - 1); 


  


    // Calculate A ^ res ( mod M ) 


    ans = power(A, res, M); 


  


    return ans; 


} 


  


// Driver code 


public static void main(String[] args) { 


      


    // M must be be a Prime Number 


    int A = 3, B = 9, C = 4, M = 19; 


  


    System.out.print(Calculate(A, B, C, M)); 


} 


} 


  


// This code is contributed by Anant Agarwal. 



















                        chevron_right







                        filter_none









Python














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















# Python program to calculate the ans 


def calculate(A, B, C, M): 


  


    # Calculate B ^ C (mod M - 1) 


    res = pow(B, C, M-1) 


  


    # Calculate A ^ res ( mod M ) 


    ans = pow(A, res, M) 


  


    return ans 


  


# Driver program to run the case 


A = 3


B = 9


C = 4


  


# M must be Prime Number 


M = 19


print( calculate(A, B, C, M) ) 



















                        chevron_right







                        filter_none









C#














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// C# program to find (a^b) mod m for 


// a large 'a' 


using System; 


  


class GFG { 


      


    // Iterative Function to calculate 


    // (x^y)%p in O(log y) 


    static int power(int x, int y, int p) 


    { 


          


        // Initialize result 


        int res = 1;  


      


        // Update x if it is more 


        // than or equal to p 


        x = x % p;  


      


        while (y > 0) { 


              


            // If y is odd, multiply x  


            // with result 


            if (y % 2 != 0) 


                res = (res * x) % p; 


          


            // y must be even now 


            y = y >> 1; // y = y/2 


            x = (x * x) % p; 


        } 


        return res; 


    } 


      


    static int Calculate(int A, int B,  


                          int C, int M) 


    { 


        int res, ans; 


      


        // Calculate B ^ C (mod M - 1) 


        res = power(B, C, M - 1); 


      


        // Calculate A ^ res ( mod M ) 


        ans = power(A, res, M); 


      


        return ans; 


    } 


      


    // Driver code 


    public static void Main() 


    { 


          


        // M must be be a Prime Number 


        int A = 3, B = 9, C = 4, M = 19; 


      


        Console.Write(Calculate(A, B, C, M)); 


    } 


} 


  


// This code is contributed by nitin mittal. 



















                        chevron_right







                        filter_none









PHP














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















<?php 


// PHP program to find  


// (a^b) mod m for a  


// large 'a' 


  


// Iterative Function 


// to calculate (x^y)%p 


// in O(log y) 


function 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; 


} 


  


function Calculate($A, $B, $C, $M) 


{ 


    $res; $ans; 


  


    // Calculate B ^ C 


    // (mod M - 1) 


    $res = power($B, $C, $M - 1); 


  


    // Calculate A ^  


    // res ( mod M ) 


    $ans = power($A, $res, $M); 


  


    return $ans; 


} 


  


// Driver Code 


  


// M must be be  


// a Prime Number 


$A = 3; $B = 9;  


$C = 4; $M = 19; 


  


echo Calculate($A, $B,   


               $C, $M); 


  


// This code is contributed 


// by ajit 


?> 



















                        chevron_right







                        filter_none










Output:


18



Time Complexity: O(log(B) + log(C))

Auxiliary space: O(1)


This article is contributed by  Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.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.


GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details 





    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:

Improved By :  nitin mittal, jit_t