In AES, all operations are performed on 8-bit bytes. In particular, the arithmetic operations of addition, multiplication, and division are performed over the finite field GF(2 power 8).

**FINITE FIELD
ARITHMETIC**

In
AES, all operations are performed on 8-bit bytes. In particular, the arithmetic
operations of addition, multiplication, and division are performed over the
finite field GF(28).
Section 4.7 discusses such operations in some detail. For the reader who has
not studied Chapter 4, and as a quick review for those who have, this section
summarizes the important concepts.

In
essence, a field is a set in which we can do addition, subtraction, multipli-
cation, and division without leaving the set. Division is defined with the
following rule: *a*/*b *= *a*(*b *- 1).

An
example of a finite field (one with a finite number of ele- ments) is the set Z*p *consisting of all
the integers {0, 1, ... , *p *- 1},
where *p *is a prime number and in which arithmetic is carried out
modulo *p*.

Virtually all encryption algorithms, both conventional and public-key, involve arithmetic operations on
integers. If one of the operations used in the algorithm is division, then
we need to work in arithmetic defined
over a field;
this is because
divi- sion requires that each nonzero element have a multiplicative
inverse. For conve- nience and for implementation efficiency, we would also like to work with integers
that fit exactly into a given number of bits,
with no wasted bit patterns. That is, we wish to work with integers in the
range 0 through 2*n *- 1, which fit into an *n*-bit word. Unfortunately, the set of such integers, *Z*2*n*, using modular arithmetic, is not a field. For example,
the integer 2 has no multiplicative inverse in *Z*2*n*, that is, there is no integer
*b*, such
that 2*b *mod 2*n *= 1.

There
is a way of defining a finite field containing 2*n *elements;
such a field is referred to as GF(2*n*).
Consider the set, *S*, of all polynomials
of degree *n *- 1 or less with binary coefficients. Thus, each polynomial
has the form

where
each *ai *takes on the value 0 or 1. There
are a total of 2*n *different polynomials in
*S*. For *n *= 3, the 23 = 8 polynomials in the set are

With the appropriate definition of arithmetic operations, each such set *S *is a finite field. The
definition consists of the following elements.

**1.
**Arithmetic follows the ordinary
rules of polynomial arithmetic using the basic
rules of algebra with the following two refinements.

**2.
**Arithmetic on the coefficients is
performed modulo 2. This is the same
as the XOR operation.

**3.
**If multiplication results
in a polynomial of degree greater than *n *- 1, then the
polynomial is reduced modulo some irreducible polynomial *m*(*x*) of degree *n*.
That is, we divide
by *m*(*x*) and keep the remainder. For a polynomial *f*(*x*), the remainder is expressed
as *r*(*x*) = *f*(*x*) mod *m*(*x*). A polynomial *m*(*x*) is called **irreducible **if and only if *m*(*x*) cannot be expressed as a product
of two poly- nomials, both of degree lower
than that of *m*(*x*).

For example, to construct the finite field
GF(23),
we need to choose an irre- ducible polynomial of degree 3. There are only two
such polynomials: (*x*3 + *x*2 + 1) and (*x*3 + *x *+ 1). Addition is equivalent
to taking the XOR of like terms. Thus,
(*x *+ 1) + *x *= 1.

A
polynomial in GF(2*n*) can be uniquely
represented by its *n *binary coeffi-
cients (*an *- 1*an *- 2 ... *a*0). Therefore, every polynomial in GF(2*n*) can be represented by an
*n*-bit number. Addition is performed by taking the bitwise XOR of the two *n*-bit
elements. There is no simple XOR operation
that will accomplish multiplication in GF(2*n*). However, a reasonably straightforward,
easily implemented, technique is available.
In essence, it can be shown that multiplication of a number in GF(2*n*) by 2
consists of a left shift
followed by a conditional XOR with a constant. Multiplication by larger numbers can be achieved
by repeated application of this rule.

To summarize, AES operates on 8-bit bytes.
Addition of two bytes is defined as the bitwise XOR operation. Multiplication
of two bytes is defined as multiplication
in the finite field GF(28), with the irreducible polynomial2 *m*(*x*) = *x*8 + *x*4 + *x*3 + *x *+ 1. The developers of Rijndael give as their
motivation for selecting this one of the 30 possible irreducible polynomials of
degree 8 that it is the first one on the list given in [LIDL94].

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Cryptography and Network Security Principles and Practice : One Symmetric Ciphers : Advanced Encryption Standard : Finite Field Arithmetic |

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