Dr. Dobb's is part of the Informa Tech Division of Informa PLC

This site is operated by a business or businesses owned by Informa PLC and all copyright resides with them. Informa PLC's registered office is 5 Howick Place, London SW1P 1WG. Registered in England and Wales. Number 8860726.

Channels ▼


The Block Cipher Square Algorithm

Source Code Accompanies This Article. Download It Now.

Dr. Dobb's Journal October 1997: The Galois Field

Dr. Dobb's Journal October 1997

The Galois Field GF(28)

Two byte values can be combined in many different ways to result in a third value. In general, a rule to combine two values to obtain a third value is called a "composition law." Some obvious examples of composition laws that are applicable to byte values are modular addition and multiplication, and the bitwise logical OR, AND, and XOR operations. The combination of a set and two composition laws is called an "algebra." If the laws obey certain rules, one of the laws can be seen as "additive" and the other as "multiplicative." Mathematicians have developed powerful tools to derive and prove nonobvious properties for these algebras. A particularly relevant property of such an algebra is the ability to solve x from the equation a+b[dot] x=0 for any value of a and any nonzero value of b. An algebra exhibiting these properties is called a field.

Finite Field

A field with a finite number of elements is called a "finite field" or "Galois Field" (GF()), after the mathematician Evariste Galois.

The most obvious candidates for the two composition laws on bytes are addition modulo 256 and multiplication modulo 256. Unfortunately, the resulting algebra is not a field. For instance, the equation 1+2 [dot] x=0 has no solution.

Still, 28 is a prime power and it is therefore possible to define two composition laws such that the resulting algebra is the field GF(28). For the additive law we take the bitwise XOR. The multiplicative is more involved, and is constructed as follows: The bits in the bytes are considered as coefficients of polynomials of degree smaller than 8 and the composition law is the multiplication of polynomials modulo some irreducible polynomial (with no other divisors than 1 and itself, similar to prime number). In practice, the multiplication can be easily implemented using table lookup methods.

-- J.D., L.K., and V.R.

Copyright © 1997, Dr. Dobb's Journal

Related Reading

More Insights

Currently we allow the following HTML tags in comments:

Single tags

These tags can be used alone and don't need an ending tag.

<br> Defines a single line break

<hr> Defines a horizontal line

Matching tags

These require an ending tag - e.g. <i>italic text</i>

<a> Defines an anchor

<b> Defines bold text

<big> Defines big text

<blockquote> Defines a long quotation

<caption> Defines a table caption

<cite> Defines a citation

<code> Defines computer code text

<em> Defines emphasized text

<fieldset> Defines a border around elements in a form

<h1> This is heading 1

<h2> This is heading 2

<h3> This is heading 3

<h4> This is heading 4

<h5> This is heading 5

<h6> This is heading 6

<i> Defines italic text

<p> Defines a paragraph

<pre> Defines preformatted text

<q> Defines a short quotation

<samp> Defines sample computer code text

<small> Defines small text

<span> Defines a section in a document

<s> Defines strikethrough text

<strike> Defines strikethrough text

<strong> Defines strong text

<sub> Defines subscripted text

<sup> Defines superscripted text

<u> Defines underlined text

Dr. Dobb's encourages readers to engage in spirited, healthy debate, including taking us to task. However, Dr. Dobb's moderates all comments posted to our site, and reserves the right to modify or remove any content that it determines to be derogatory, offensive, inflammatory, vulgar, irrelevant/off-topic, racist or obvious marketing or spam. Dr. Dobb's further reserves the right to disable the profile of any commenter participating in said activities.

Disqus Tips To upload an avatar photo, first complete your Disqus profile. | View the list of supported HTML tags you can use to style comments. | Please read our commenting policy.