Example

When computing modulo a polynomial, it is of importance to note in which order the computations are carried out. Taking a clever route can gain a lot of time.

For example, let a R[X]/(X2 + 1) be the equivalence class containing the element

(X3 + 1)27 (X2 + X + 1)35

and suppose that the question is to find a representative of degree at most 1 for a. Evidently, it is lot of work to first work out the product and then find the remainder after division by X2 + 1. A considerable reduction in computational work is achieved by the following method mod X2 + 1:

(X3 + 1)27 (X2 + X + 1)35 = (-X + 1 )27 · (-1 + X + 1)35 =

{(-X + 1)2}13 · {(-X + 1) · X}35 = {(1-2X + X2}13 · {(-X + 1) · X}35 =

(-2X)13 · {(-X + 1) · X}35 = -213 · X48 · (-X + 1) = 213X - 213 .

So a representative of a is 213X - 213.

Verify yourself how the arithmetical rules were used.