Cryptography Reference
In-Depth Information
A.7.1 Galois cohomology
One finds brief summaries of
Galois cohomology
in Appendix B of Silverman [
505
] and
Chapter 19 of Cassels [
114
]. More detailed references are Serre [
488
] and Cassels and
Fr olich [
116
].
Let
K/
). Unlike most references we
write our Galois groups acting on the left (i.e., as
σ
(
f
) rather than
f
σ
). A 1-cocycle in
the additive group
K
is a function
1
ξ
:
G
k
be Galois (we include
K
= k
). Let
G
=
Gal(K
/
k
→
K
such that
ξ
(
στ
)
=
σ
(
ξ
(
τ
))
+
ξ
(
σ
). A 1-
coboundary in
K
is the function
ξ
(
σ
)
K
. The group of 1-cocycles
modulo 1-coboundaries (the group operation is addition (
ξ
1
+
=
σ
(
γ
)
−
γ
for some
γ
∈
ξ
2
(
τ
)) is
denoted
H
1
(
G,
K). Similarly, for the multiplicative group
K
∗
, a 1-cocycle satisfies
ξ
(
στ
)
ξ
2
)(
τ
)
=
ξ
1
(
τ
)
+
=
σ
(
ξ
(
τ
))
ξ
(
σ
), a 1-coboundary is
σ
(
γ
)
/γ
and the quotient group is denoted
H
1
(
G,K
∗
).
k
be Galois. Then H
1
(Gal(
K/
k
={
}
Theorem A.7.2
Let K/
)
,K
)
0
and (
Hilbert 90
)
H
1
(Gal(
K/
k
)
,K
∗
)
={
1
}
(i.e., both groups are trivial).
Proof
The case of finite extensions
K/
is given in Exercise 20.5 of Cassels [
114
]or
Propositions 1 and 2 of Chapter 10 of [
488
]. For a proof in the infinite case see Propositions
2 and 3 (Sections 2.6 and 2.7) of Chapter 5 of [
116
].
k
A.8 Finite fields
Let
p
be a prime. Denote by
F
p
= Z
/p
Z
the finite field of
p
elements. The multiplicative
F
p
. Recall that
F
p
is a cyclic group. A generator for
F
q
is
group of non-zero elements is
F
q
is
ϕ
(
q
−
called a
primitive root
. The number of primitive roots in
1).
Theorem A.8.1
Let p be a prime and m
F
p
m
having p
m
ele-
ments. All such fields are isomorphic. Every finite field can be represented as
∈ N
. Then there exists a field
F
p
[
x
]
/
(
F
(
x
))
where F
(
x
)
∈ F
p
[
x
]
is a monic irreducible polynomial of degree m; the corresponding
vector space basis
{
1
,x,...,x
m
−
1
}
F
p
m
/
F
p
is called a
polynomial basis
.
for
Proof
See Corollary V.5.7 of [
271
] or Section 20.2 of [
497
].
If
p
is a prime and
q
=
p
m
then
F
p
nm
may be viewed as a degree
n
algebraic extension
of
F
q
.
Theorem A.8.2
Every finite field
F
p
m
has a vector space basis over
F
p
of the form
θ,θ
p
,...,θ
p
m
−
1
{
}
;thisiscalleda
normal basis
.
Proof
See Theorem 2.35 or Theorem 3.73 of [
350
,
351
] or Exercise 20.14 of [
497
](the
latter proof works for extensions of
F
p
, but not for all fields).
We discuss methods to construct a normal basis in Section
2.14.1
.
Theorem A.8.3
Let q be a prime power and m
∈ N
. Then
F
q
m
is an algebraic extension
of
F
q
that is Galois. The Galois group is cyclic of order m and generated by the q-power
Frobenius automorphism π
:
x
x
q
.
→
1
It is also necessary that
ξ
satisfy some topological requirements, but we do not explain these here.
