Modular arithmetic is the set of operations that can be done when working modulo N, where N is an integer greater than 1.
We can visualize this arithmetic using a clock. Suppose that the number 12 in the clock is replaced by zero. Then when we have to add an hour, we get, for example: 3+1 = 4, 10+1 = 11, 11+1 = 0. If we have to add three hours we get: 5+3 = 8, 11+3 = 2 (as in 11 AM + 3 hours = 2 PM). We can also subtract: 2-3 = 11 and even multiply: 5×4 = 8 (because 5+5+5+5 = 8). This is arithmetic modulo 12 and the set of numbers representing the hours 0, 1, 2, 3,..., 11 is known as Z/12Z.
When making these modular operations, the equal symbol is not used. We use the congruence symbol () instead. Note that two numbers A and B are said to be congruent modulo n if A-B is a multiple of n.
The set Z/nZ of numbers modulo n contains the numbers 0, 1, 2, 3, ..., n-2 and n-1. The following operations are defined:
Let A and B be numbers in Z/nZ. Then the addition is defined as:
The subtraction is defined as:
The multiplication is defined as:
where C is the remainder of the division between A × B and n.
The exponentiation is defined as:
where (B times). An efficient method to perform modular exponentiation is the binary method.
Since reducing modulo takes a lot of time, Montgomery multiplication is used in this context where only one modular reduction is needed.
The inverse is defined as:
where B is the number such that (mod ).
The following algorithm to compute the multiplicative inverse n-1 mod q for n with 0 < n < q, where 0 < n-1 < q is based on the Extended Euclidean algorithm. All variables are integers (some programming languages call them Big Integers).
- Set i = q, h = n, v = 0, and d = 1.
- Set t = i DIV h, where DIV is defined as integer division.
- Set x = h.
- Set h = i - tx.
- Set i = x.
- Set x = d.
- Set d = v - tx.
- Set v = x.
- If h > 0, go to step 2.
- Let n-1 = v mod q.
When the modulus is a power of 2, there is an easier method. Let the modulus be , let the number to be inverted be and let rounded to the next integer. Then the method is:
- Set x = 1.
- Perform k times: Set x = x(2-Nx) mod 2^m
The division is defined as: