Protocols for Secure Communication in Wireless Sensor Networks

2.8. Cryptography for Sensor Networks 51
be intensified later.
2.8.1 Hash Functions
A hash function is a mapping from a set of documents of arbitrary length to a set
of hash values, which have a fixed, small size. A cryptographic hash function
h : A → B has the following properties ([122], ch. 9):
1. For any x ∈ A, the hash value h(x) is easy to compute (in linear time).
2. For any hash value y ∈ B, it is computationally infeasible to find a x ∈ A
for which h(x) = y (one-way property/preimage resistance).
3. For a given x ∈ A, it is computationally infeasible to find a x ′ for which x ′ =
x and h(x ′ ) = h(x) (weak collision resistance/second preimage resistance).
4. It is computationally infeasible to find any two distinct x,x ′ ∈ A for which
h(x) = h(x ′ ) (strong collision resistance/collision resistance).
A hash function produces a small representative (also called fingerprint or
message digest) of an input document of arbitrary size. As the cardinality of
the domain A is greater than that of the range B, the existence of collisions is
unavoidable. However, for practically useful hash functions, it is computationally
infeasible to find such collisions. This allows it, for practical purposes, to
identify the output of a hash function with its original input and use it as a substitute,
for example in the process of creating a digital signature of the original
input.
To an observer who only sees the result y of a computation h(x), without
knowing x, the value y seems “random” in the sense that it is unknown how
the value has been created, and it could as well be drawn randomly from B.
This is a consequence of properties 2 and 3: it is practically infeasible to either
find the original value x or any other x ′ with which y could be reconstructed.
Ideally, each value from B appears with equal probability and thus a uniform
distribution can be assumed.
Practical infeasibility refers to the computational power that would be required
in order to find a preimage or a collision. The number of computational
steps required directly corresponds to the length n of the hash function’s output.
Without additional knowledge, finding a preimage or a weak collision
would require 2 n steps, while finding a strong collision would require 2 n/2
steps (cf. [122]). With a sufficiently large n, often 2 80 steps are considered
sufficiently hard as of today’s available technology, it is impossible to break
the security of a hash function within reasonable time.