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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