模幂(模运算中的幂)

给定三个数字x，y和p，计算(x ^y )％p。

例子 ：

Input:  x = 2, y = 3, p = 5
Output: 3
Explanation: 2^3 % 5 = 8 % 5 = 3.

Input:  x = 2, y = 5, p = 13
Output: 6
Explanation: 2^5 % 13 = 32 % 13 = 6.

我们已经讨论了电源的递归和迭代解决方案。

下面讨论迭代解决方案。

C++

/* Iterative Function to calculate (x^y) in O(log y) */
int power(int x, int y)
{
     
    // Initialize answer
    int res = 1;
     
    // Check till the number becomes zero
    while (y)
    {
         
        // If y is odd, multiply x with result
        if (y % 2 == 1)
            res = (res * x);
             
        // y = y/2
        y = y >> 1;
         
        // Change x to x^2
        x = (x * x);
    }
    return res;
}
 
// This code is contributed by yaswanth0412

C

/* Iterative Function to calculate (x^y) in O(log y) */
int power(int x, unsigned int y)
{
    int res = 1;     // Initialize result
  
    while (y > 0)
    {
        // If y is odd, multiply x with result
        if (y & 1)
            res = res*x;
  
        // y must be even now
        y = y>>1; // y = y/2
        x = x*x;  // Change x to x^2
    }
    return res;
}

Java

/* Iterative Function to calculate (x^y) in O(log y) */
static int power(int x, int y)
{
    int res = 1;     // Initialize result
  
    while (y > 0)
    {
       
        // If y is odd, multiply x with result
        if ((y & 1) != 0)
            res = res * x;
  
        // y must be even now
        y = y >> 1; // y = y/2
        x = x * x;  // Change x to x^2
    }
    return res;
}
 
// This code is contributed by Dharanendra L V.

Python3

# Iterative Function to calculate (x^y) in O(log y)
def power(x, y):
     
    # Initialize result
    res = 1   
  
    while (y > 0):
       
        # If y is odd, multiply x with result
        if ((y & 1) != 0):
            res = res * x
  
        # y must be even now
        y = y >> 1 # y = y/2
        x = x * x  # Change x to x^2
         
    return res
 
# This code is contributed by Khushboogoyal499

C#

/* Iterative Function to calculate (x^y) in O(log y) */
static int power(int x, int y)
{
    int res = 1;     // Initialize result
  
    while (y > 0)
    {
        // If y is odd, multiply x with result
        if ((y & 1) != 0)
            res = res * x;
  
        // y must be even now
        y = y >> 1; // y = y/2
        x = x * x;  // Change x to x^2
    }
    return res;
}
 
// This code is contributed by Dharanendra L V.

Javascript

/* Iterative Function to calculate (x^y) in O(log y) */
function power(x, y)
{
    let res = 1;     // Initialize result
 
    while (y > 0)
    {
        // If y is odd, multiply x with result
        if (y & 1)
            res = res*x;
 
        // y must be even now
        y = y>>1; // y = y/2
        x = x*x; // Change x to x^2
    }
    return res;
}
 
// This code is contributed by _saurabh_jaiswal

高效方法：

上述解决方案的问题是，对于较大的n或x值可能会发生溢出。因此，通常以大数为模来评估功率。

以下是基本的模块化属性，用于在模块化算术下有效地计算功率。

(ab) mod p = ( (a mod p) (b mod p) ) mod p 

For example a = 50,  b = 100, p = 13
50  mod 13  = 11
100 mod 13  = 9

(50 * 100) mod 13 = ( (50 mod 13) * (100 mod 13) ) mod 13 
or (5000) mod 13 = ( 11 * 9 ) mod 13
or 8 = 8

下面是基于上述属性的实现。

C++ 14

// Iterative C++ program to compute modular power
#include 
using namespace std;
 
/* Iterative Function to calculate (x^y)%p in O(log y) */
int power(long long 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
  
    if (x == 0) return 0; // In case x is divisible by 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;
}
 
// Driver code
int main()
{
    int x = 2;
    int y = 5;
    int p = 13;
    cout << "Power is " << power(x, y, p);
    return 0;
}
 
// This code is contributed by shubhamsingh10

Java

// Iterative Java program to compute modular power
import java.io.*;
class GFG
{
 
  /* Iterative Function to calculate (x^y) in O(log y) */
  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
 
    if (x == 0)
      return 0; // In case x is divisible by p;
 
    while (y > 0)
    {
 
      // If y is odd, multiply x with result
      if ((y & 1) != 0)
        res = (res * x) % p;
 
      // y must be even now
      y = y >> 1; // y = y/2
      x = (x * x) % p;
    }
    return res;
  }
 
  // Driver Code
  public static void main(String[] args)
  {
    int x = 2;
    int y = 5;
    int p = 13;
    System.out.print("Power is " + power(x, y, p));
  }
}
 
// This code is contributed by Dharanendra L V.

Python3

# Iterative Python3 program
# to compute modular power
 
# Iterative Function to calculate
# (x^y)%p in O(log y)
def power(x, y, p) :
    res = 1     # Initialize result
 
    # Update x if it is more
    # than or equal to p
    x = x % p
     
    if (x == 0) :
        return 0
 
    while (y > 0) :
         
        # If y is odd, multiply
        # x with result
        if ((y & 1) == 1) :
            res = (res * x) % p
 
        # y must be even now
        y = y >> 1      # y = y/2
        x = (x * x) % p
         
    return res
     
 
# Driver Code
 
x = 2; y = 5; p = 13
print("Power is ", power(x, y, p))
 
 
# This code is contributed by Nikita Tiwari.

C＃

using System;
public class GFG
{
 
  /* Iterative Function to calculate (x^y) in O(log y) */
  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
 
    if (x == 0)
      return 0; // In case x is divisible by p;
 
    while (y > 0)
    {
 
      // If y is odd, multiply x with result
      if ((y & 1) != 0)
        res = (res * x) % p;
 
      // y must be even now
      y = y >> 1; // y = y/2
      x = (x * x) % p;
    }
    return res;
  }
 
  // Driver Code
  static public void Main ()
  {
    int x = 2;
    int y = 5;
    int p = 13;
    Console.Write("Power is " + power(x, y, p));
  }
}
 
// This code is contributed by Dharanendra L V.

的PHP

 0)
    {
        // If y is odd, multiply
        // x with result
        if ($y & 1)
            $res = ($res * $x) % $p;
 
        // y must be even now
         
        // y = $y/2
        $y = $y >> 1;
        $x = ($x * $x) % $p;
    }
    return $res;
}
 
// Driver Code
$x = 2;
$y = 5;
$p = 13;
echo "Power is ", power($x, $y, $p);
 
// This code is contributed by aj_36
?>

Java脚本

// Iterative Javascript program to
// compute modular power
 
// Iterative Function to
// calculate (x^y)%p in O(log y)
function power(x, y, p)
{
    // Initialize result
    let res = 1;
 
    // Update x if it is more
    // than or equal to p
    x = x % p;
 
    if (x == 0)
        return 0;
 
    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/2
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}
 
// Driver Code
let x = 2;
let y = 5;
let p = 13;
document.write("Power is " + power(x, y, p));
 
// This code is contributed by _saurabh_jaiswal

输出

Power is 6

上述解决方案的时间复杂度为O(Log y)。