Given four integers A, B, K, M. The task is to find (A + iB)K % M which is a complex number too. A + iB represents a complex number. Examples:
Input : A = 2, B = 3, K = 4, M = 5 Output: 1 + i*0 Input : A = 7, B = 3, K = 10, M = 97 Output: 25 + i*29
Prerequisite: Modular Exponentiation Approach: An efficient approach is similar to the modular exponentiation of a single number. Here, instead of a single we have two number A, B. So, pass a pair of integers as a parameter to the function instead of a single number. Below is the implementation of the above approach :
#include <bits/stdc++.h>
using namespace std;
// Function to multiply two complex numbers modulo M
pair<int, int> Multiply (pair<int, int> p, pair<int, int> q,
int M)
{
// Multiplication of two complex numbers is
// (a + ib)(c + id) = (ac - bd) + i(ad + bc)
int x = ((p.first * q.first) % M - (p.second *
q.second) % M + M) % M;
int y = ((p.first * q.second) % M + (p.second *
q.first) % M) %M;
// Return the multiplied value
return {x, y};
}
// Function to calculate the complex modular exponentiation
pair<int, int> compPow(pair<int, int> complex, int k, int M)
{
// Here, res is initialised to (1 + i0)
pair<int, int> res = { 1, 0 };
while (k > 0)
{
// If k is odd
if (k & 1)
{
// Multiply 'complex' with 'res'
res = Multiply(res, complex, M);
}
// Make complex as complex*complex
complex = Multiply(complex, complex, M);
// Make k as k/2
k = k >> 1;
}
//Return the required answer
return res;
}
// Driver code
int main()
{
int A = 7, B = 3, k = 10, M = 97;
// Function call
pair<int, int> ans = compPow({A, B}, k, M);
cout << ans.first << " + i" << ans.second;
return 0;
}
// Java implementation of the approach
import java.util.*;
class GFG
{
static class pair
{
int first, second;
public pair(int first, int second)
{
this.first = first;
this.second = second;
}
}
// Function to multiply two complex numbers modulo M
static pair Multiply (pair p, pair q, int M)
{
// Multiplication of two complex numbers is
// (a + ib)(c + id) = (ac - bd) + i(ad + bc)
int x = ((p.first * q.first) % M -
(p.second * q.second) % M + M) % M;
int y = ((p.first * q.second) % M +
(p.second * q.first) % M) % M;
// Return the multiplied value
return new pair(x, y);
}
// Function to calculate the
// complex modular exponentiation
static pair compPow(pair complex, int k, int M)
{
// Here, res is initialised to (1 + i0)
pair res = new pair(1, 0 );
while (k > 0)
{
// If k is odd
if (k % 2 == 1)
{
// Multiply 'complex' with 'res'
res = Multiply(res, complex, M);
}
// Make complex as complex*complex
complex = Multiply(complex, complex, M);
// Make k as k/2
k = k >> 1;
}
// Return the required answer
return res;
}
// Driver code
public static void main(String[] args)
{
int A = 7, B = 3, k = 10, M = 97;
// Function call
pair ans = compPow(new pair(A, B), k, M);
System.out.println(ans.first + " + i" +
ans.second);
}
}
// This code is contributed by PrinciRaj1992
# Python3 implementation of the approach
# Function to multiply two complex numbers modulo M
def Multiply (p, q, M):
# Multiplication of two complex numbers is
# (a + ib)(c + id) = (ac - bd) + i(ad + bc)
x = ((p[0] * q[0]) % M - \
(p[1] * q[1]) % M + M) % M
y = ((p[0] * q[1]) % M + \
(p[1] * q[0]) % M) %M
# Return the multiplied value
return [x, y]
# Function to calculate the
# complex modular exponentiation
def compPow(complex, k, M):
# Here, res is initialised to (1 + i0)
res = [1, 0]
while (k > 0):
# If k is odd
if (k & 1):
# Multiply 'complex' with 'res'
res = Multiply(res, complex, M)
# Make complex as complex*complex
complex = Multiply(complex, complex, M)
# Make k as k/2
k = k >> 1
# Return the required answer
return res
# Driver code
if __name__ == '__main__':
A = 7
B = 3
k = 10
M = 97
# Function call
ans = compPow([A, B], k, M)
print(ans[0], "+ i", end = "")
print(ans[1])
# This code is contributed by
# Surendra_Gangwar
// C# implementation of the approach
using System;
class GFG
{
public class pair
{
public int first, second;
public pair(int first, int second)
{
this.first = first;
this.second = second;
}
}
// Function to multiply two complex numbers modulo M
static pair Multiply (pair p, pair q, int M)
{
// Multiplication of two complex numbers is
// (a + ib)(c + id) = (ac - bd) + i(ad + bc)
int x = ((p.first * q.first) % M -
(p.second * q.second) % M + M) % M;
int y = ((p.first * q.second) % M +
(p.second * q.first) % M) % M;
// Return the multiplied value
return new pair(x, y);
}
// Function to calculate the
// complex modular exponentiation
static pair compPow(pair complex, int k, int M)
{
// Here, res is initialised to (1 + i0)
pair res = new pair(1, 0 );
while (k > 0)
{
// If k is odd
if (k % 2 == 1)
{
// Multiply 'complex' with 'res'
res = Multiply(res, complex, M);
}
// Make complex as complex*complex
complex = Multiply(complex, complex, M);
// Make k as k/2
k = k >> 1;
}
// Return the required answer
return res;
}
// Driver code
public static void Main(String[] args)
{
int A = 7, B = 3, k = 10, M = 97;
// Function call
pair ans = compPow(new pair(A, B), k, M);
Console.WriteLine(ans.first + " + i" +
ans.second);
}
}
// This code is contributed by 29AjayKumar
function pair(first, second) {
this.first = first;
this.second = second;
}
function multiply(p, q, M) {
// Multiplication of two complex numbers is
// (a + ib)(c + id) = (ac - bd) + i(ad + bc)
let x = ((p.first * q.first) % M - (p.second * q.second) % M + M) % M;
let y = ((p.first * q.second) % M + (p.second * q.first) % M) % M;
return new pair(x, y);
}
function compPow(complex, k, M) {
let res = new pair(1, 0);
while (k > 0) {
if (k % 2 === 1) {
res = multiply(res, complex, M);
}
complex = multiply(complex, complex, M);
k = k >> 1;
}
return res;
}
let A = 7, B = 3, k = 10, M = 97;
let ans = compPow(new pair(A, B), k, M);
console.log(ans.first + " + i" + ans.second);
// This code is contributed by abn95knd1.
Output:
25 + i29
Time complexity: O(log k).
Auxiliary Space: O(1)
