# Algorithm C/C++ : Fastest way to compute (2^n)%d with a n and d 32 or 64 bit integers

I am searching for an algorithm that allow me to compute `(2^n)%d` with n and d 32 or 64 bits integers.

The problem is that it's impossible to store `2^n` in memory even with multiprecision libraries, but maybe there exist a trick to compute `(2^n)%d` only using 32 or 64 bits integers.

Thank you very much.

Take a look at the Modular Exponentiation algorithm.

The idea is not to compute `2^n`. Instead, you reduce modulus `d` multiple times while you are powering up. That keeps the number small.

Combine the method with Exponentiation by Squaring, and you can compute `(2^n)%d` in only `O(log(n))` steps.

Here's a small example: `2^130 % 123 = 40`

``````2^1   % 123 = 2
2^2   % 123 = 2^2      % 123    = 4
2^4   % 123 = 4^2      % 123    = 16
2^8   % 123 = 16^2     % 123    = 10
2^16  % 123 = 10^2     % 123    = 100
2^32  % 123 = 100^2    % 123    = 37
2^65  % 123 = 37^2 * 2 % 123    = 32
2^130 % 123 = 32^2     % 123    = 40
``````