Modular arithmetic

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 ($\equiv$) 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:

$A + B \equiv C$ (mod $n$)

where:

$C = A + B$ if $A + B \lt n$
$C = A + B - n$ if $A + B \geq n$

Modular subtraction

The subtraction is defined as:

$A - B \equiv C$ (mod $n$)

where:

$C = A - B$ if $A \geq B$
$C = A - B + n$ if $A \lt B$

Modular multiplication

The multiplication is defined as:

$A * B \equiv C$ (mod $n$)

where C is the remainder of the division between A × B and n.

Modular exponentiation

The exponentiation is defined as:

$A^B \equiv C$ (mod $n$)

where $C = A*A*A*A*...*A$ (B times). An efficient method to perform modular exponentiation is the binary method.

Since reducing modulo $n$ takes a lot of time, Montgomery multiplication is used in this context where only one modular reduction is needed.

Modular inversion

The inverse is defined as:

$A^{-1} \equiv B$ (mod $n$)

where B is the number such that $A * B = 1$ (mod $n$).

This operation is only defined when the numbers A and n are coprime, i.e., when gcd(A, n) = 1.

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).

1. Set i = q, h = n, v = 0, and d = 1.
2. Set t = i DIV h, where DIV is defined as integer division.
3. Set x = h.
4. Set h = i - tx.
5. Set i = x.
6. Set x = d.
7. Set d = v - tx.
8. Set v = x.
9. If h > 0, go to step 2.
10. Let n-1 = v mod q.

When the modulus is a power of 2, there is an easier method. Let the modulus be $2^m$, let the number to be inverted be $N$ and let $k = \log_2 m$ rounded to the next integer. Then the method is:

1. Set x = 1.
2. Perform k times: Set x = x(2-Nx) mod 2^m

Modular division

The division is defined as:

$A / B \equiv C$ (mod $n$)

where:

$C \equiv A * B^{-1}$