3 Arithmetic modulo N

Now that we have studied the basic arithmetic operations we can perform computations in the ring $\mathbb {Z}$/N$\mathbb {Z}$. The objects in this ring are cosets of the form a + N$\mathbb {Z}$ for integers b. By the process of division we can write a = d ^. N + r for r between 0 and N - 1. Thus b + N$\mathbb {Z}$ = r + N$\mathbb {Z}$ and so the objects of $\mathbb {Z}$/N$\mathbb {Z}$ can be identified with integers between 0 and N = 1. Thus one way to perform arithmetic operations in this ring is to perform them with integers and then, by division, reduce to this canonical form. However, this is uses more space and time than required.