Lecture Notes in Computer Science 5185

32 I. Agudo, C. Fernandez-Gago, and J. Lopez
Definition 6 (Parallel Trust Function). A parallel trust function is used to calculate
the trust level associated to a set of paths or chains of trust statements. It is defined as,
g : � n
� �� �
∞
n=2 TD×···×TD−→ TD, where T D is a trust domain and
1. g(z1,...,zi−1,zi,zi+1,...,zn)=g(z1,...,zi−1,zi+1,...,zn) if zi = 0
2. g(z)=z
g may verify the following desirable properties:
– Idempotency, g(z,z,...,z)=z.
– Monotony, g(z1,z2,...,zn) ≤ g(z ′ 1 ,z′ 2 ,...,z′ n ) If zi ≤ z ′ i for alli ∈ N.
– Associativity, g(g(z1,z2),z3)=g(z1,z2,z3)=g(z1,g(z2,z3)).
The generator parallel function, ortheparallel operator, ⊕ for the function g, is
defined analogously as the operator ⊙.
Definition 7 (Parallel Operator). A Parallel Operator or Generator Parallel Function
is defined as a function, ⊕ : TD× TD−→ TD, such that a ⊕ 0 = 0 ⊕ a = a
We say that the two operators ⊕ and ⊙ are distributive if (a ⊕ b)⊙c =(a ⊙ c)⊕(b ⊙ c).
In the case where there are no cycles in the trust graph, the set of paths connecting
two any given entities is finite. Then, given a sequential operator ⊙ and a commutative
parallel operator ⊕,i.e.a ⊕ b = b ⊕ a, the associated trust evaluation, � FG,isdefinedas
follows,
Definition 8. Let S s,t
x be the set of all paths of trust statements for task x starting in s and
ending in t. For each path p ∈ S s,t
x represented as s v1 vn
−→··· −→ tletzpbe v1 ⊙···⊙vn,
then � FG(s,t,x) is defined as �
p∈S s,t
x zp.
Given a fixed sequential operator, for any parallel operator that verifies idempotency and
monotony properties then, z∗ = min s,t
p∈Sx zp ≤ �
p∈S s,t
x zp ≤ max s,t
p∈Sx zp = z∗ . Therefore,
the maximum and minimum possible trust values associated to a path from s to t are the
upper and lower bounds for the trust evaluation � FG.
Fortunately, we do not need to compute the trust values of each path in order to
compute those bounds, i.e. z∗ and z∗ . In this case we can use an algorithm, adapted
from the Dijkstra algorithm [8], to find for example, the maximum trust path from a
fixed entity s to any other entity on the system. The minimum trust path can be computed
in an analogous way.
This is a particular case of a trust evaluation where we use the maximum function as
a parallel function. Unfortunately we can not generalize this kind of algorithms for other
combinations of parallel and sequential functions as it heavily relies on the properties
of the max and min functions.
3.2 Performing Trust Computations Using Matrices
Let us first assume the case where there are no cycles in the trust graph. We could model
the trust network as a matrix, A where each element aij represents the trust level that