CS 4363, Spring 2007
Lecture 3 Notes, 17 April 2007
Fields
References:
The Field of Fractions or Rational Numbers:
Start with the positive integers:
1, 2, 3, 4, 5, . . .
Using addition (+), we want to make these into a
commutative group, so for
any integers x, y, and z, the following should be true:
- Associativity: ( x + y ) + z = x + ( y + z )
- Commutativity: x + y = y + x
- Identity: there is a special element 0,
called the identity under addition such that x + 0 = x
- Inverses: for each x, there must be an element
-x such that x + (-x) = 0
In order for the positive integers to become a group using addition,
we need to add in the number 0 and the negative integers:
. . ., -4, -3, -2, -1,
0, 1, 2, 3, 4, 5, . . .
This is the set of integers, call it Z.
Now we want to make the set of non-zero integers into a
group using multiplication (*), with the same rules as above:
- Associativity: ( x * y ) * z = x * ( y * z )
- Commutativity: x * y = y * x
- Identity: there is a special element 1,
called the identity under multiplication such that x * 1 = x
- Inverses: for each x, there must be an element
x-1 such that x * x-1 = 1
In order for this to be a group, we need to add in the reciprocal
of each integer, and the keep adding until we have all fractions.
The resulting set of all fractions is called a field.
These are also called rational numbers, denoted by Q.
Fields have the additional property:
- Distributive Rule: ( x + y ) * z = ( x * z ) + ( y * z )
Finite Fields Zp:
- We can turn the five integers 0, 1,
2, 3, 4 into a finite
commutative group using (x + y) % 5 for addition
(add, divide by 5, take remainder).
- 0 is the additive identity, and the additive inverse of
non-zero x is (5 - x), while
0 is its own inverse.
- Then take the non-zero numbers 1, 2,
3, 4 and use
( x * y ) % 5 for multiplication.
- 1 is the multiplicative identity, and the multiplicative inverse of
non-zero x is a little tricky (obtained from the strong form of
the GCD algorithm).
- In this case 1 * 1 = 1, 2 * 3 = 1, 4 * 4 = 1,
so 1 and 4 are their own inverses,
and 2 and 3 are inverses of one another.
- Altogether this makes these five elements into a
finite field, called Z5.
- In a similar way, we can get a finite field Zp
for any prime number p. (This field is unique.)
- This doesn't work for a non-prime, like 6, say, because
2 * 3 = 0, and so 2 and 3 cannot have
multiplicative inverses.
- The main finite field of this kind that we will use
is Z2 with elements 0 and 1.
The Fields GF(pn):
- For any prime p and any integer n greater
than 1, there is a
unique field with pn elements in it.
- We are interested in the case p = 2 and n = 8, to
get a field with 28 = 256 elements.
The elements of this field can be written using 8 bits,
or 2 hex digits, that is, one byte.
- Now start in at page 119 of
Laws of Cryptography
and on page 2 of
Rewrite of the AES Specs.
Revision date: 2007-04-11.
(Please use ISO
8601, the International Standard.)