问题
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.
回答1:
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
来源：https://stackoverflow.com/questions/8963686/algorithm-c-c-fastest-way-to-compute-2nd-with-a-n-and-d-32-or-64-bit-int