How to compute (A^B)%M when A,B,M are big numbers?

Question

For example, if A = 864927518, B = 1462579282 ,M = 193773611, how to compute (A^B)%M?

Is there a simple way?

possible duplicate of [Calculating pow(a,b) mod n](http://stackoverflow.com/questions/8496182/calculating-powa-b-mod-n) — phuclv, Sep 22 '15 at 09:20
[Modular exponentiation implementation in Python 3](http://stackoverflow.com/q/18804958/995714), [How did Python implement the built-in function pow()?](http://stackoverflow.com/a/10539256/995714) — phuclv, Sep 22 '15 at 09:23

score 7 · Accepted Answer · answered Sep 22 '15 at 09:01

7

Yes: use modular exponentiation. Python's built in pow function allows you to do this with its optional third argument:

>>> pow(A, B, M)
2767533

answered Sep 22 '15 at 09:01

Alex Riley

169,130
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score 2 · Answer 2 · answered Sep 22 '15 at 09:00

2

3>> pow(864927518, 1462579282, 193773611)
2767533

answered Sep 22 '15 at 09:00

Ignacio Vazquez-Abrams

776,304
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1,341
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How does it run so fast? It's near instantaneous – Tim Sep 22 '15 at 09:02
@TimCastelijns: By using a well-known equivalent equation. – Ignacio Vazquez-Abrams Sep 22 '15 at 09:03

score 1 · Answer 3 · answered Sep 22 '15 at 09:09

I shall blithely assume that you are using ^ as the exponentiation operator here (rather than as XOR). You can either use the built-in pow or you can code your own:

import math

def expmod(i, j, m):
  """Compute (i ** j) % m"""

  acc = 1
  mask = 2 ** int(math.ceil(math.log(j, 2)))
  while mask > 0:
    acc = acc * acc
    if mask & j:
      acc = acc * i
      j = j ^ mask
    mask = mask // 2
    acc = acc % m

  return acc

How to compute (A^B)%M when A,B,M are big numbers?

3 Answers3