Related Articles
Modular Exponentiation in Python
• Difficulty Level : Medium
• Last Updated : 23 Nov, 2020

Given three numbers x, y and p, compute (x^y) % p

Examples:

```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.
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

The problem with above solutions is, overflow may occur for large value of n or x. Therefore, power is generally evaluated under modulo of a large number.

Naive multiplication is O(n) with a very low constant factor with %m.
Pow function calculates in O(log n) time in python but it takes a lot of time when numbers are large enough if you first calculate the value of xy and then mod it with p to get (xy) % p evaluated.

 `# Simple python code that first calls pow() ``# then applies % operator.``a ``=` `2``b ``=` `100``p ``=` `(``int``)(``1e9``+``7``)`` ` `# pow function used with %``d ``=` `pow``(a, b) ``%` `p``print` `(d)`

Output:

```976371285
```

While computing with large numbers modulo, the (%) operator takes a lot of time, so a Fast Modular Exponentiation is used. Python has pow(x, e, m) to get the modulo calculated which takes a lot less time. [Please refer Python Docs for details]

 `# Fast python code that first calls pow() ``# then applies % operator``a ``=` `2``b ``=` `100``p ``=` `(``int``)(``1e9``+``7``)`` ` `# Using direct fast method to compute ``# (a ^ b) % p.``d ``=` `pow``(a, b, p)``print` `(d)`

Output:

```976371285
```

The fast modular exponentiation algorithm has been explained more briefly in link

Attention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.

To begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course. And to begin with your Machine Learning Journey, join the Machine Learning – Basic Level Course

My Personal Notes arrow_drop_up