Cryptography Reference
In-Depth Information
Let us briefly review Matsumoto-Imai's scheme (Matsumoto and Imai
1988). L et
q
be a prime power and let
m
,
l
be positive integers. Let
=
GF
(
q
) and
12
m
== Î=
. Let
be a set such that each entity
in this set be an identity
d
i
and
Ψ
Ψ
{
xx
|
[
xx
...
x x
],
,
i
1,2,...,
m
}
mi
¹"¹
. Let a one-way algorithm
exist that
implements an injection from
to
m
.
ddi
j
i
j
τ
=
over
randomly and independently from other entities. The secret key-sharing func-
tion
i
is
τ
Setup:
The key-setup server selects
l
(
m
,
m
) symmetric matrices
Ms
, where
1,...,
l
T
Φω φΓω
=
ω ϒ
Î
()
()where
(6.8)
i
i
for each
d
i
. Furthermore,
(
)
T
is the transpose of
(
) and
i
is an (
l,m
) matrix
defined as
T
T
T
φ
=
Γ
Γ
[
Md
(
) ,...,
Md
(
) ]
(6.9)
i
1
i
l
i
Each unit receives its own
i
from the server. For example, if nodes A and B want to
establish a pairwise cryptographic key between each other, A and B would compute
A
(
y
B
) and
B
(
y
A
), respectively, such that there are l-vectors over
that are equal.
Let there be a maximum of
N
sensor nodes, where each of the
N
nodes have a
unique identity ranging from 1 to
N
. Let each of these identities be represented by an
m
bits (
=
m
log (
N
)
.
2
lm
´
symmetric matrices
Key predistribution:
The key generator generates
(
)
=
over the finite field
GF
(2) and
M
τ
is the master secret. It then
computes
i
for each node
S
i
by computing
τ
τ
Ms
, where
1,...,
l
τ
τ
φ
=
τ
=
, and generates
dM
, where
1,...,
l
i
i
, as shown below:
é
ê
ê
ê ú
1
Φ
Φ
i
2
i
Φ
=
=
where
d
is the
m
-dimensional vector and
i
1,...,
N
(6.10)
ê
ê
ê
ê
ë û
i
i
...
l
i
Φ
Key establishment:
On deployment of the sensor nodes, let nodes
S
i
and
S
J
be two
neighboring nodes and be one among the
N
nodes that would like to establish
a pairwise key. These nodes could learn their identities
d
i
and
d
j
through a broad-
cast mechanism. On discovering the identities, the node
S
i
computes a link key as
, where
K
τ
represents the
th bit of the possible pairwise link
key between the two communicating nodes). Similarly,
S
j
constructs the link key
τ
τ
T
K
=
Φ
d
τ
=
1,...,
l
(
ij
i
j
ij
K
τ
.
In addition, MAC is used for verifying if the same key was generated at both ends.
ji
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