Modular arithmetic is the branch of arithmetic mathematics related with the “mod” functionality. Basically, modular arithmetic is related with computation of “mod” of expressions. Expressions may have digits and computational symbols of addition, subtraction, multiplication, division or any other. Here we will discuss briefly about all modular arithmetic operations.
It states that, for any pair of integers a and b (b is positive), there exist two unique integers q and r such that:
a = b x q + r where 0 <= r < b
Example: If a = 20, b = 6 then q = 3, r = 2 20 = 6 x 3 + 2
Rule for modular addition is:
(a + b) mod m = ((a mod m) + (b mod m)) mod m
(15 + 17) % 7 = ((15 % 7) + (17 % 7)) % 7 = (1 + 3) % 7 = 4 % 7 = 4
The same rule is to modular subtraction. We don’t require much modular subtraction but it can also be done in the same way.
The Rule for modular multiplication is:
(a x b) mod m = ((a mod m) x (b mod m)) mod m
(12 x 13) % 5 = ((12 % 5) x (13 % 5)) % 5 = (2 x 3) % 5 = 6 % 5 = 1
The modular division is totally different from modular addition, subtraction and multiplication. It also does not exist always.
(a / b) mod m is not equal to ((a mod m) / (b mod m)) mod m.
This is calculated using the following formula:
(a / b) mod m = (a x (inverse of b if exists)) mod m
The modular inverse of a mod m exists only if a and m are relatively prime i.e. gcd(a, m) = 1. Hence, for finding the inverse of an under modulo m, if (a x b) mod m = 1 then b is the modular inverse of a.
Example: a = 5, m = 7 (5 x 3) % 7 = 1 hence, 3 is modulo inverse of 5 under 7.
Finding a^b mod m is the modular exponentiation. There are two approaches for this – recursive and iterative. Example:
a = 5, b = 2, m = 7 (5 ^ 2) % 7 = 25 % 7 = 4
Below are some more important concepts related to Modular Arithmetic
- Euler’s Totient Function
- Compute n! under modulo p
- Wilson’s Theorem
- How to compute mod of a big number?
- Find value of y mod (2 raised to power x)