Cryptography Reference
In-Depth Information
(ii) (, ,)
R is an abelian group with respect to addition and has 0 as its identity.
(iii) The operation ´ is associative.
, ,
abc
R
,
´´=´´
( ) (
(3.7)
abc
abc
)
(iv) The multiplicative operation ( ´ ) is distributive under addition ( + ).
(v)
, ,
abc
R
´+=´+´
( ) ( ) (
abc
ab ac
)
(3.8)
and
( ) ( ) (
+´=´+´
(3.9)
bc a ba
ca
)
Example for R : The set Z under addition and multiplication is a commutative ring. In
addition, the set
Z
under addition modulo n and multiplication modulo n is a com-
n
mutative ring.
3.2.3 Fields
A field , denoted ( F ,
+
, ´ ), is a commutative ring in which all nonzero elements of the
set have multiplicative inverses. The order of the finite field is defined to be the number
of elements in the field. If F q is a finite field where q = p m ( p = prime number and m =
positive number), then p is defined to be the characteristic of a field.
The characteristic of a field is the least positive integer m such that
i å equals 0.
m
1
1
However, it is 0 if
times
1111
m
+++ +¹

m ³
1 0
for any
1
(3.10)
For example, the set Z under the operations of addition and multiplication is not a
field, as the only nonzero integers with multiplicative inverses are 1 and -1. However,
real numbers ( R ), complex numbers ( C ), and the rational numbers ( Q ) form a field.
3.3 Prime Fields
Let F p be a field where p is a prime number. Then integer modulo p is an integer set
with elements {0,1,2,…,p-1}.
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