How to calculate modulus of large numbers?

Okay, so you want to calculate a^b mod m. First we'll take a naive approach and then see how we can refine it.

First, reduce a mod m. That means, find a number a1 so that 0 <= a1 < m and a = a1 mod m. Then repeatedly in a loop multiply by a1 and reduce again mod m. Thus, in pseudocode:

a1 = a reduced mod m
p = 1
for(int i = 1; i <= b; i++) {
    p *= a1
    p = p reduced mod m
}

By doing this, we avoid numbers larger than m^2. This is the key. The reason we avoid numbers larger than m^2 is because at every step 0 <= p < m and 0 <= a1 < m.

As an example, let's compute 5^55 mod 221. First, 5 is already reduced mod 221.

1. 1 * 5 = 5 mod 221
2. 5 * 5 = 25 mod 221
3. 25 * 5 = 125 mod 221
4. 125 * 5 = 183 mod 221
5. 183 * 5 = 31 mod 221
6. 31 * 5 = 155 mod 221
7. 155 * 5 = 112 mod 221
8. 112 * 5 = 118 mod 221
9. 118 * 5 = 148 mod 221
10. 148 * 5 = 77 mod 221
11. 77 * 5 = 164 mod 221
12. 164 * 5 = 157 mod 221
13. 157 * 5 = 122 mod 221
14. 122 * 5 = 168 mod 221
15. 168 * 5 = 177 mod 221
16. 177 * 5 = 1 mod 221
17. 1 * 5 = 5 mod 221
18. 5 * 5 = 25 mod 221
19. 25 * 5 = 125 mod 221
20. 125 * 5 = 183 mod 221
21. 183 * 5 = 31 mod 221
22. 31 * 5 = 155 mod 221
23. 155 * 5 = 112 mod 221
24. 112 * 5 = 118 mod 221
25. 118 * 5 = 148 mod 221
26. 148 * 5 = 77 mod 221
27. 77 * 5 = 164 mod 221
28. 164 * 5 = 157 mod 221
29. 157 * 5 = 122 mod 221
30. 122 * 5 = 168 mod 221
31. 168 * 5 = 177 mod 221
32. 177 * 5 = 1 mod 221
33. 1 * 5 = 5 mod 221
34. 5 * 5 = 25 mod 221
35. 25 * 5 = 125 mod 221
36. 125 * 5 = 183 mod 221
37. 183 * 5 = 31 mod 221
38. 31 * 5 = 155 mod 221
39. 155 * 5 = 112 mod 221
40. 112 * 5 = 118 mod 221
41. 118 * 5 = 148 mod 221
42. 148 * 5 = 77 mod 221
43. 77 * 5 = 164 mod 221
44. 164 * 5 = 157 mod 221
45. 157 * 5 = 122 mod 221
46. 122 * 5 = 168 mod 221
47. 168 * 5 = 177 mod 221
48. 177 * 5 = 1 mod 221
49. 1 * 5 = 5 mod 221
50. 5 * 5 = 25 mod 221
51. 25 * 5 = 125 mod 221
52. 125 * 5 = 183 mod 221
53. 183 * 5 = 31 mod 221
54. 31 * 5 = 155 mod 221
55. 155 * 5 = 112 mod 221

Therefore, 5^55 = 112 mod 221.

Now, we can improve this by using exponentiation by squaring; this is the famous trick wherein we reduce exponentiation to requiring only log b multiplications instead of b. Note that with the algorithm that I described above, the exponentiation by squaring improvement, you end up with the right-to-left binary method.

a1 = a reduced mod m
p = 1
while (b > 0) {
     if (b is odd) {
         p *= a1
         p = p reduced mod m
     }
     b /= 2
     a1 = (a1 * a1) reduced mod m
}

Thus, since 55 = 110111 in binary

1. 1 * (5^1 mod 221) = 5 mod 221
2. 5 * (5^2 mod 221) = 125 mod 221
3. 125 * (5^4 mod 221) = 112 mod 221
4. 112 * (5^16 mod 221) = 112 mod 221
5. 112 * (5^32 mod 221) = 112 mod 221

Therefore the answer is 5^55 = 112 mod 221. The reason this works is because

55 = 1 + 2 + 4 + 16 + 32

so that

5^55 = 5^(1 + 2 + 4 + 16 + 32) mod 221
     = 5^1 * 5^2 * 5^4 * 5^16 * 5^32 mod 221
     = 5 * 25 * 183 * 1 * 1 mod 221
     = 22875 mod 221
     = 112 mod 221

In the step where we calculate 5^1 mod 221, 5^2 mod 221, etc. we note that 5^(2^k) = 5^(2^(k-1)) * 5^(2^(k-1)) because 2^k = 2^(k-1) + 2^(k-1) so that we can first compute 5^1 and reduce mod 221, then square this and reduce mod 221 to obtain 5^2 mod 221, etc.

The above algorithm formalizes this idea.

To add to Jason's answer:

You can speed the process up (which might be helpful for very large exponents) using the binary expansion of the exponent. First calculate 5, 5^2, 5^4, 5^8 mod 221 - you do this by repeated squaring:

 5^1 = 5(mod 221)
 5^2 = 5^2 (mod 221) = 25(mod 221)
 5^4 = (5^2)^2 = 25^2(mod 221) = 625 (mod 221) = 183(mod221)
 5^8 = (5^4)^2 = 183^2(mod 221) = 33489 (mod 221) = 118(mod 221)
5^16 = (5^8)^2 = 118^2(mod 221) = 13924 (mod 221) = 1(mod 221)
5^32 = (5^16)^2 = 1^2(mod 221) = 1(mod 221)

Now we can write

55 = 1 + 2 + 4 + 16 + 32

so 5^55 = 5^1 * 5^2 * 5^4 * 5^16 * 5^32 
        = 5   * 25  * 625 * 1    * 1 (mod 221)
        = 125 * 625 (mod 221)
        = 125 * 183 (mod 183) - because 625 = 183 (mod 221)
        = 22875 ( mod 221)
        = 112 (mod 221)

You can see how for very large exponents this will be much faster (I believe it's log as opposed to linear in b, but not certain.)

Chinese Remainder Theorem comes to mind as an initial point as 221 = 13 * 17. So, break this down into 2 parts that get combined in the end, one for mod 13 and one for mod 17. Second, I believe there is some proof of a^(p-1) = 1 mod p for all non zero a which also helps reduce your problem as 5^55 becomes 5^3 for the mod 13 case as 13*4=52. If you look under the subject of "Finite Fields" you may find some good results on how to solve this.

EDIT: The reason I mention the factors is that this creates a way to factor zero into non-zero elements as if you tried something like 13^2 * 17^4 mod 221, the answer is zero since 13*17=221. A lot of large numbers aren't going to be prime, though there are ways to find large primes as they are used a lot in cryptography and other areas within Mathematics.

What you're looking for is modular exponentiation, specifically modular binary exponentiation. This wikipedia link has pseudocode.

This is part of code I made for IBAN validation. Feel free to use.

    static void Main(string[] args)
    {
        int modulo = 97;
        string input = Reverse("100020778788920323232343433");
        int result = 0;
        int lastRowValue = 1;

        for (int i = 0; i < input.Length; i++)
        {
            // Calculating the modulus of a large number Wikipedia http://en.wikipedia.org/wiki/International_Bank_Account_Number                                                                        
            if (i > 0)
            {
                lastRowValue = ModuloByDigits(lastRowValue, modulo);
            }
            result += lastRowValue * int.Parse(input[i].ToString());
        }
        result = result % modulo;
        Console.WriteLine(string.Format("Result: {0}", result));            
    }

    public static int ModuloByDigits(int previousValue, int modulo)
    {
        // Calculating the modulus of a large number Wikipedia http://en.wikipedia.org/wiki/International_Bank_Account_Number                        
        return ((previousValue * 10) % modulo);
    }
    public static string Reverse(string input)
    {
        char[] arr = input.ToCharArray();
        Array.Reverse(arr);
        return new string(arr);
    }

This is called modular exponentiation(https://en.wikipedia.org/wiki/Modular_exponentiation).

Let's assume you have the following expression:

19 ^ 3 mod 7

Instead of powering 19 directly you can do the following:

(((19 mod 7) * 19) mod 7) * 19) mod 7

But this can take also a long time due to a lot of sequential multipliations and so you can multiply on squared values:

x mod N -> x ^ 2 mod N -> x ^ 4 mod -> ... x ^ 2 |log y| mod N

Modular exponentiation algorithm makes assumptions that:

x ^ y == (x ^ |y/2|) ^ 2 if y is even
x ^ y == x * ((x ^ |y/2|) ^ 2) if y is odd

And so recursive modular exponentiation algorithm will look like this in java:

/**
* Modular exponentiation algorithm
* @param x Assumption: x >= 0
* @param y Assumption: y >= 0
* @param N Assumption: N > 0
* @return x ^ y mod N
*/
public static long modExp(long x, long y, long N) {
    if(y == 0)
        return 1 % N;

    long z = modExp(x, Math.abs(y/2), N);

    if(y % 2 == 0)
        return (long) ((Math.pow(z, 2)) % N);
    return (long) ((x * Math.pow(z, 2)) % N);
}

Special thanks to @chux for found mistake with incorrect return value in case of y and 0 comparison.