Merge pull request #50 from AdityaJ7/modular-expo

Modular exponentiation with Python

Author: tejan-singh

Modular Exponentiation/REAMDE.md

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+## Modular Exponentiation
+
+Modular exponentiation is a very important topic in Discrete Mathematics as well as in Cryptography.
+It's main application is in public-key cryptography.
+It is an effective way of calculating modulus values of extremely large exponents values like 2^500 with ease.
+You can refer to this amazing video to understand Modular exponentiation better:-
+[Modular Exponentiation](https://www.youtube.com/watch?v=EHUgNLN8F1Y)
+
+I have used teo different approaches to solve this problem:
+
+1. iterative
+2. recursive
+
+The time complexity in case of iterative approach is O(log(power)) which is awesome.
+
+The time complexity in case of recursive approach is O(power) which is very nice as well.
+
+## How to use these two programs?
+
+1. For iterative approach type:
+
+python iterative.py
+
+Here is a sample use case:
+
+<p align = "center">
+	<img src="iterative.PNG" alt="iter">
+</p>
+
+2. For recursive approach type:
+
+python recursive.py
+
+Here is a sample use case:
+
+<p align = "center">
+	<img src="recursive.PNG" alt="rec">
+</p>

Modular Exponentiation/iterative.PNG

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+def modular_exponentiation(base, power, modulus_to):
+    """This function uses the iterative approach to
+       perform modular exponentiation.
+
+    Args:
+        base (int): The number used for the base value of the expression.
+        power (int): The number choosed for power value of the expression.
+        modulus_to (int): The number used to take modulus of the expression.
+
+    Returns:
+        result(int): The evaluated value of modular exponentiaion.
+    """
+    result = 1 # Intial modular exponentiation result
+    base %= modulus_to # Performing modulus on the base and assigning it back to the base.
+
+    while (power > 0): # Looping over untill the value of power is more than 0.
+
+        # To perform operation accordingly for odd and even powers.
+        if ((power & 1) == 1): # To extract the lowest bit.
+            result = (result * base) % modulus_to # For even power values we perform this.
+
+        power = power//2 # For odd value of power.
+        base = (base ** 2) % modulus_to # Repeatitive squaring of the base and performing modulus operation
+
+    return result
+
+
+if __name__ == "__main__":
+    base = int(input("Enter the base number: "))
+    power = int(input("Enter the power number: "))
+    modulus_to = int(
+        input("Enter the number which is used to perform modulus operation: "))
+    print(f"The result is {modular_exponentiation(base, power, modulus_to)}")

Modular Exponentiation/iterative.py

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+def modular_exponentiation(base, power, modulus_to):
+    """This function uses the recursive approach to
+       perform modular exponentiation.
+
+    Args:
+        base (int): The number used for the base value of the expression.
+        power (int): The number choosed for power value of the expression.
+        modulus_to (int): The number used to take modulus of the expression.
+
+    Returns:
+        result(int): The evaluated value of modular exponentiaion.
+    """
+
+    # Base Cases for recursion
+    if (base == 0): # This reduces computation if base is 0.
+        return 0
+    if (power == 0): # This reduces computation if power is 0.
+        return 1
+    if (modulus_to == 1): # This reduces computation if modulus_to is 1.
+        return 0
+
+    # Recursive cases for recursion
+    result = 0
+    if (power % 2 == 0): # To check if the power is even
+        result = modular_exponentiation(base, power / 2, modulus_to) # Recursive call if power/2
+        result = (result * result) % modulus_to # Storing the squared result value's mod back into result
+
+    else: # For odd power values
+        result = base % modulus_to # Mod of the base value
+        # In the below line the function has been called recursively and successive mods have been performed
+        # with reducing values of power by a factor of 1
+        result = (result * modular_exponentiation(base, power - 1,
+                                                  modulus_to) % modulus_to) % modulus_to 
+    return ((result + modulus_to) % modulus_to)
+
+
+if __name__ == "__main__":
+    base = int(input("Enter the base number: "))
+    power = int(input("Enter the power number: "))
+    modulus_to = int(
+        input("Enter the number which is used to perform modulus operation: "))
+    print(f"The result is {modular_exponentiation(base, power, modulus_to)}")