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• W2911104 abstract "In traditional security systems, for each task, we either trust an agent or we don’t. If we trust an agent, we allow this agent full access to this particular task. This agent can usually allow his trusted sub-agents the same access, etc. If a trust management system only uses “trust” and “no trust” options, then a person should trust everyone in this potentially long chain. The problem is that trust is rarely a complete trust, there is a certain probability of distrust. So, when the chain becomes long, the probability of a security leak increases. It is desirable to keep track of trust probabilities, so that we should only delegate to agents whose trust is above a certain threshold. In this paper, we present efficient algorithms for handling such probabilities. I. PROBABILISTIC APPROACH IS NEEDED A. Traditional Approach to Trust Management: Brief Idea In traditional security systems (see, e.g., [23]), for each task, we either trust an agent or we don’t. If we trust an agent, we allow this agent full access to this particular task. For example, I trust a bank, where I keep my money, to handle my account. This agent can usually allow his trusted sub-agents the same access, etc. For example, the bank can outsource some money operations to another company and trust this company to handle its accounts. Since I trust the bank, and bank trusts the company, I therefore have to trust the company that handles my account. B. Traditional Approach to Trust Management: Main Problem The problem with the traditional approach is that trust is rarely a complete trust. For example, I trust a bank, where I keep my money, to handle my account. I know that there have been cases when banks cheated on clients, but overall, my trust is pretty high, say p = 99.9% (i.e., distrust is d = 0.1%). The bank, in its turn, trusts a company in India – to which this bank has outsourced to handle my account – with a certain high probability. That company trusts its own employees, etc. If a trust management system only uses “trust” and “no trust” options, then I should trust everyone in this potentially long chain. However, when the chain becomes long, the probability of a security leak increases, and the resulting probability of distrust may get much higher than my original 0.1%. C. What Needs to Be Done It is desirable to keep track of trust probabilities, so that we should only delegate to agents whose trust is above a certain threshold p. II. PROBABILISTIC APPROACH: IDEA To implement the above idea, we can write down the rules describing who directly trusts whom and with what probability. The objective of the resulting Qualitative Trust Management System is, given such rules and the two agents f and s, to determine the probability pt(f, s) with which the first agent f should trust the second agent s. Such systems exist: • a system in which, crudely speaking, all direct trusts are assumed to be of the same probability is described in [7]; • a more complex system in which direct trusts may have different probabilities is described in [12]. In this talk, we will describe new efficient algorithms for computing the corresponding probabilities pt(f, s). III. INDEPENDENT CASE: TOWARDS PRECISE FORMULATION OF THE PROBLEM Let us formulate the problem in precise terms. A. Input Data: Formal Description We have a finite set A; it elements are called agents. For some pairs (a, b) of agents, we are given a number p0(a, b) > 0 from the interval (0, 1]. This number is called a probability with which agent a directly trusts agent b. B. Desired Output: Informal Description Informally, our objective is to describe, for given two agents f and s, the probability pt(f, s) with which the agent f trusts the agent s. C. Graphs: A Natural Description of Input Data From the mathematical viewpoint, it is reasonable to describe this input as a directed graph G = (A,E), in which: • the agents are vertices, and • an edge (a, b) ∈ E connects those pairs of vertices a and b for which we know the probability of direct trust. D. Independence: Reasonable Assumption All we know is the probabilities of direct trust. Since we have no information on the dependence between different direct trust links, it makes sense to assume that the corresponding events are independent; see, e.g., [26]. Under this independence assumption, we can formulate the problem in precise terms. E. Desired Output: Formal Description We have a graph G = (A,E), in which there is a probability p0(a, b) assigned to every edge. We can now describe a random subgraph (A,Er) (Er ⊆ E) of the original graph as follows: • for every edge (a, b) ∈ E, the probability that this edge is present in Er is equal to p0(a, b); • for two different edges, the events describing their presence is Er are statistically independent. In other words, for each edge (a, b) ∈ E: • this edge belongs to Er with probability p0(a, b), and • this edge does not belong to Er with probability 1 − p0(a, b). Since edges are statistically independent, we can provide an explicit formula for the probability p(E′) that the resulting random graph Er coincides with a given subgraph E′ ⊆ E: p(E′) =   ∏ (a,b)∈E′ p0(a, b)   ·   ∏ (a,b)6∈E′ (1− p0(a, b))   . (1) These values p(E′) form a probability distribution on the set of all subsets E′ ⊆ E. Based on this probability distribution, we can then determine the desired probability pt(f, s) as the probability that in the random graph Er, there is a directed path from f to s. If we denote the existence of such a path by f Er → s, we can then describe the desired probability pt(f, s) as follows: pt(f, s) = ∑ E′:f E′ →s p(E′). (2) IV. ALGORITHM FOR THE INDEPENDENT CASE A. Monte-Carlo Simulation: Main Idea Since the edges are assumed to be statistically independent, we can use the following Monte-Carlo simulation algorithm to generate a random graph Er: We loop over all edges (a, b) ∈ E, and for each edge (a, b), we keep it in Er with probability p0(a, b). Specifically, for each edge, we do the following: • we run a standard random number generator that generates numbers uniformly distributed on the interval [0, 1]; as a result, we get a value ξ ∈ [0, 1]; • then, we compare ξ with p0(a, b): – if ξ ≤ p0(a, b), we keep the edge (a, b) in Er; – otherwise, we delete the edge (a, b) from the graph Er. Since the random variable ξ is uniformly distributed on the interval [0, 1], the probability that ξ belongs to any interval [x, y] is equal to the width y−x of this interval. In particular, the probability that ξ ≤ p0(a, b), i.e., that ξ belongs to the interval [0, p0(a, b)], is equal to p0(a, b). As a result of this process, we get different subgraphs E′ with probability that is described by the formula (1). Thus, the desired probability pt(f, s) is equal to the probability that in thus generated random graph Er, there is a directed path from f to s. We can therefore estimate this probability pt(f, s) as the frequency of this event. In other words: • we select a number of iteration N ; • then, N times, we generate the corresponding random graph Er and check whether in this graph, there is a path from f and s; – there are known algorithms for efficiently checking the existence of such a path; see, e.g., [6]; • we then estimate pt(f, s) as the ratio f def = M/N , where M is the number of generated graphs in which f and s were connected. It is well known that after N simulations, the frequency f = M/N estimates the desired probability p with the accuracy" @default.
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• W2911104 date "2004-01-01" @default.
• W2911104 modified "2023-09-26" @default.
• W2911104 title "Probabilistic Approach to Trust: Ideas, Algorithms, And Simulations" @default.
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