Linear Iterations on Ordered Semirings for Trust Metric Computation and Attack Resiliency Evaluation

Within the realm of network security, we interpret the concept of trust as a relation among entities that participate in various protocols. Trust relations are based on evidence created by the previous interactions of entities within a protocol. In this work, we are focusing on the evaluation of trust evidence in Ad Hoc Networks. Because of the dynamic nature of Ad Hoc Networks, trust evidence may be uncertain and incomplete. Also, no pre-established infrastructure can be assumed. The evaluation process is modelled as a path problem on a directed graph, where nodes represent entities, and edges represent trust relations. We develop a novel formulation of trust computation as linear iterations on ordered semirings. Using the theory of semirings, we analyze several key problems on the performance of trust algorithms. We also analyze the resilience to attacks of the resulting schemes. I. TRUST DEFINITION MOTIVATION The notion of trust, in the realm of network security, will for our purposes correspond to a set of relations among entities that participate in a protocol. These relations are based on the evidence generated by the previous interactions of entities within a protocol. In general, if the interactions have been faithful to the protocol, then trust will “accumulate” between these entities. Exactly how trust is computed depends on the particular protocol (application). The application determines the exact semantics of trust, and the entity determines how the trust relation will be used in the ensuing steps of the protocol. Trust influences decisions like access control, choice of public keys, etc. It could be useful as a complement to a Public Key Infrastructure (PKI), where an entity would accept or reject a public key according to the trustworthiness of the entities that vouch for it (i.e. have signed a certificate for it) – this is the idea behind PGP’s Web of Trust [3]. It can also be used for routing decisions: Instead of the shortest path, we could be looking for the most trusted path between two nodes (this has been already proposed in P2P networks [4]). In this work we model and analyze trust schemes for mobile ad hoc networks (MANET). Ad Hoc networks are envisioned to have dynamic, sometimes rapidlychanging, random, multihop topologies which are composed of bandwidth-constrained wireless links. The nodes themselves form the network routing infrastructure in an ad hoc fashion [5]. Based on these characteristics, we are imposing the following constraints on our schemes: First, there is no preestablished infrastructure. The computation process cannot rely on, e.g., a Trusted Third Party. There is no centralized Public Key Infrastructure, Certification Authorities, or Registration Authorities with elevated privileges. Second, evidence is uncertain and incomplete. Uncertain, because it is generated by the users on the fly, without lengthy processes. Incomplete, because in the presence of adversaries we cannot assume that all friendly nodes will be reachable: the malicious users may have rendered a small or big part of the network unreachable. Despite the above, we require that the results are as accurate as possible, yet robust in the presence of attackers. It is desirable to, for instance, identify all allied nodes, but it is even more desirable that no adversary is misidentified as good. In this work we do not assume the existence of any globally trusted entity: on the contrary, everything is up to the individual nodes of the network. They themselves sign certificates for each other’s keys, and they themselves have to judge how much to trust these certificates and, essentially, their issuers. The specification of admissible types of evidence, the generation, distribution, discovery and evaluation of trust evidence are collectively called Trust Establishment. In this work, we are focusing on the evaluation process of trust evidence in Ad-Hoc Networks, i.e. we are focusing on the trust metric itself. In particular, we are not dealing with the collection of evidence from the network, and the accompanying communication and signaling overhead. This issue is important, and obviously needs to be addressed in a complete system. Trust computation is the application of a metric to a body of evidence. This evidence is based on interactions of users within a network, and the result of the computation (“trust”) is a quantitative belief of User i about User j’s behavior (i.e. is User j trustworthy according to User i?). For example, assume we have a wireless network where users are supposed to forward data they receive. An interaction in this setting would play out as follows: User i sends a packet to his neighbor. The neighbor has the choice to either forward the packet (as he is supposed to), or drop it (since he may not want to waste his energy). This choice is observed by User i, and counts either as a Good or as a Bad interaction, respectively. Repeated interactions of this type build up the previously mentioned evidence, which will be called direct opinions. The trust computation will compute indirect opinions, that is, opinions of a user for others with which he has had no previous direct interaction. The idea is to take advantage of the interactions (and thus the direct opinions) that intermediate users have had with each other. II. TRUST AS A PATH PROBLEM We treat the trust computation problem as a generalized shortest path problem on a weighted directed graph G(V,E) (trust graph). The vertices of the graph are the users/entities in the network. A weighted edge from vertex i to vertex j corresponds to the opinion that entity i has about entity j. The weight function is w(i, j) : V ×V −→ S, where S is the opinion space. The set S and the precise semantics of opinions are parameters of the model and can differ according to the application. Assume that User s wants to compute the trustworthiness of User d. So, s will ask the people he knows (i.e. has an opinion about) and they will tell him their opinion about d, or they will ask the users they know, etc., until persons with a direct interaction with d are found. Formally, all the direct information that exists about the destination d, is contained in the weighted, directed edges that point to d. On the other hand, all the direct information that s has about the rest of the network is contained in s’s outgoing edges. In effect, s knows about the rest of the network only through his one-hop trust neighbors. Therefore, all information about d that s can use is contained in the paths from s to d. For example, edges pointing to d (or, in general, directed paths to d) that are not reachable from s are useless, and edges pointing out from s that are not on directed paths from s to d are dead ends. This is the starting observation of this work, and the reason why the subsequent model was chosen: because it fits the pathbased nature of the problem. Along each path, concatenation of opinions occurs: If s has opinion ws1 about 1, and 1 has opinion w12 about 2, then s can form an indirect opinion ws2 about 2 that is a function of ws1 and w12 (denoted by ws1 ⊗w12). If there are multiple paths from s to d, then indirect opinions from each path are aggregated to form the overall opinion of s for d (denoted by tsd = t p1 sd ⊕ t p2 sd ⊕ · · · ⊕ t pn sd , where the pi’s are the paths from s to d.). We have mentioned two operations: one is the concatenation of opinions along a path and the other is the Fig. 1. Concatenation of opinions Fig. 2. Aggregation of opinions aggregation across paths. These operators, along with the carrier set S, form a semiring (S,⊕,⊗): • ⊕ is commutative, associative, with a neutral element 0 © ∈ S. • ⊗ is associative, with a neutral element 1 © ∈ S, and 0 © as an absorbing element. • ⊗ distributes over ⊕. In addition, the ordering relations described in Figures 1 and 2, regarding concatenation of trust along a path and aggregation of trust across paths, above, introduce a partial order over our semiring, and thus the semiring we are considering is an ordered semiring [6]. The semiring property is very desirable because it fits the path-based nature of the problem: many other path problems can be expressed as semiring computations [6]. For example, suppose the edge weights are transmission delays, and we want to compute the least delay path from i to j. The semiring to use is (R+ ∪ {∞},min,+), i.e. ⊕ is min, and ⊗ is +: The total delay of a path is equal to the sum of all constituent edge delays, whereas the shortest path is the one with minimum delay among all paths. Also, 0 © is ∞, and 1 © is 0. On the other hand, if edge weights are link capacities, then the maximum bottleneck capacity path is found by the semiring (R+∪{∞},max,min), with 0 © ≡ 0, 1 © ≡ ∞. Then, the result dij is equal to the maximum rate of traffic that i can send to j along a single link. The transitive closure of a graph uses the Boolean semiring: ({0, 1},∨,∧), where all edge weights are equal to 1. This answers the problem of path existence from i to j, i.e. dij = 1 if and only if there exists an i −→ j path. We now look at the expected trust behavior of the operators. • First, we don’t want B to be able to increase A’s trust in C beyond A’s trust in B. For instance, assume A trusts B only moderately, and B trusts C a lot. Then it makes sense that A’s trust in C is also moderate, since A has only B’s word to count on. In general, concatenation should not increase trust. Note that the total opinion along a path is “limited” by the source’s opinion for the first node in the path. • Second, it is better to have multiple independent opinion paths to the destination. In principle, the more independent information there is, the better decision the source can reach. For example, path independence in Public Key Authentication has been argued in [2]. In order to quantify this case, we require the aggregation operator to increase something about the resulting opinion. However, trust cannot increase. If, say, there are multiple opinions all saying that the destination is untrustworthy, then obviously the source’s aggregate opinion should be along these lines. So, for our own semiring we introduce an extra parameter (or better metric) called confidence which is what increases when we have multiple paths. Thus the setting and formalism we have introduced for trust computation is more along the lines of multicriteria (or multi metrics) computation within an ordered semiring [6].