Cryptography Reference
In-Depth Information
which is easily seen to be a ring. Similarly, when the
R
j
are groups, then this
is a direct sum of groups, which is again a group.
We conclude this section with some more comments on Example A.9, since
we will need these ideas, particularly in Chapter 11 when we discuss error-
correcting codes.
If
α
isarootof
f
(
x
) such that
F
p
[
x
]
(
f
(
x
))
,
F
p
(
α
)
=
F
q
=
then
f
(
x
) is uniquely characterized by the conditions that
f
(
α
) = 0 and
g
(
α
)=0
for some
g
(
x
)
∈
F
p
[
x
]if and only if
f
(
x
) divides
g
(
x
). In this case,
f
is called the
minimal polynomial
of
α
over
F
p
, and is assumed to be monic. In particular,
a polynomial is called
primitive
of degree
n
∈
N
over
F
q
if it is a minimal
polynomial over
F
q
of a primitive element of
F
q
n
.
A.5 Vector Spaces
A
vector space
consists of an additive abelian group
V
and a field
F
together
with an operation called
scalar multiplication
of each element of
V
by each
element of
F
on the left, such that for each
r, s
∈
F
and each
α,β
∈
V
the
following conditions are satisfied:
1.
rα
∈
V
.
2.
r
(
sα
)=(
rs
)
α
.
3. (
r
+
s
)
α
=(
rα
)+(
sα
).
4.
r
(
α
+
β
)=(
rα
)+(
rβ
).
5. 1
F
α
=
α
.
The set of elements of
V
are called
vectors
and the elements of
F
are called
scalars
. The generally accepted abuse of language is to say that
V
is a
vector
space over
F
.If
V
1
is a subset of a vector space
V
that is a vector space in its
own right, then
V
1
is called a
subspace of
V
.
Definition A.39
(
Bases, Dependence, and Finite Generation
)
If
is a subset of a vector space
V
, then the intersection of all subspaces of
V
containing
S
S
is called the
subspace generated by
S
,or
spanned by
S
. If there
is a finite set
S
, and
S
generates
V
, then
V
is said to be
finitely generated
.If
S
=
∅
, then
S
generates the zero vector space. If
S
=
{
m
}
, a singleton set, then
the subspace generated by
S
is said to be the
cyclic subspace generated by
m
.
of a vector space
V
is said to be
linearly independent
provided
that for distinct
s
1
,s
2
,...,s
n
∈
S
A subset
S
, and
r
j
∈
F
for
j
=1
,
2
,...,n
,
n
r
j
s
j
=0
implies that
r
j
=0
for
j
=1
,
2
,...,n.
j
=1
Search WWH ::
Custom Search