Cryptography Reference
In-Depth Information
4.45. for
p
= 2689,
α
= 19,
y
= 2221,
β
= 954, and
γ
= 2154, so Alice sends,
m
= 96 and sig
k
(
m,r
)=(
β,γ
) = (954
,
2154).
4.46. for
p
= 4657,
α
= 15,
y
= 3484,
β
= 284, and
γ
= 2503, so Alice sends,
m
= 1111 and sig
k
(
m,r
)=(
β,γ
) = (284
,
1865).
4.47. for
p
= 8761,
α
= 23,
y
= 5807,
β
= 2973, and
γ
= 2678, so Alice sends,
m
= 2069 and sig
k
(
m,r
)=(
β,γ
) = (2973
,
2678).
In Exercises 4.48 and 4.49, use the elliptic curve given in Example 4.9 on
page 190 to decipher the given cryptogram with the private key
a
, provided
in each case.
4.48.
c
= ((11
,
1)
,
(1
,
3)), and
a
=2.
4.49.
c
= ((11
,
1)
,
(11
,
12)), and
a
=3.
Exercises 4.50-4.56 look at the cryptographic applications of Dickson poly-
nomials, defined below.
The
Dickson polynomial
G.1
4.50.
of the
first kind
of degree
n
∈
N
in the
indeterminant
x
with parameter
a
∈
R
, where
R
is a commutative ring
with identity, is defined by
n
(
D
n
(
x,a
)=
n/
2
j
=0
n
−
i
a
)
j
x
n
−
2
j
.
−
n
−
j
i
n
−
j
n
−
j
is an integer. Then establish that, when
R
=
n
Prove that
F
q
,a
finite field, we may write the polynomials in the form,
D
n
(
x,a
)=
x
+
x
2
4
a
/
2
n
+
x
4
a
/
2
n
x
2
−
−
−
.
Moreover, if
A
=(
x
+
√
x
2
4
a
)
/
2, then
A
2
−
−
Ax
+
a
= 0, and
D
n
A
+
a
A
,a
=
A
n
+
a
A
n
.
(G.1)
(
Hint: You may use
Waring's Formula
, given as follows:
n
(
AB
)
j
(
A
+
B
)
n
−
2
j
,
A
n
+
B
n
=
n/
2
j
=0
−
n
j
1)
j
(
−
n
−
j
j
G.1
In1896,L.E.DicksonhadthesepolynomialsaspartofhisdoctoralthesisattheUniversity
of Chicago. In a paper [244], published in 1973, I. Schur put Dickson's name to these poly-
nomials in his honour, and related these Dickson polynomials to the well-known Chebyshev
polynomials. Fromthe1970stothepresent, thetheoryofDicksonpolynomialshasflourished.
Much of this development is due to W.B. Nobauer and his followers (see [179] and [180], for
instance). In particular, applications to cryptography have come to the fore, and it is this in
which we are interested, albeit many other applications of these polynomials abound.
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