Siguiente: Representation of Galois fields Arriba: Multiplication in Finite Fields Anterior: Introduction

Ring multiplication

The set $[\![0,2^n-1]\!]$ consists of the collection of remainders modulus $2^n$. It has naturally the ring multiplication operation $(i,j)\mapsto (i j) \bmod 2^n$. These operations are displayed in tables 1-5 for $n\leq 5$. In each display, table (a) contains the numeric values, and, as is usual when displaying matrices, its rows are numbered from up to bottom; the corresponding graphical table (b) displays the same operation in a scale of gray: black color corresponds to the lowest value $0$ while white color corresponds to the greatest value $2^n-1$; this time the rows are numbered from bottom to up.

Table: Ring multiplication table in the modulus ring $[\![0,2^2-1]\!]$.

(a)	(b)

Table: Ring multiplication table in the modulus ring $[\![0,2^3-1]\!]$.

(a)	(b)

Table: Ring multiplication table in the modulus ring $[\![0,2^4-1]\!]$.

(a)

(b)

Table: Ring multiplication table in the modulus ring $[\![0,2^5-1]\!]$ (numerical values).

Table: Ring multiplication table in the modulus ring $[\![0,2^5-1]\!]$ (density plot).

Siguiente: Representation of Galois fields Arriba: Multiplication in Finite Fields Anterior: Introduction