In this work we consider a distributed system formed by a master processor and a collection of n processors (workers) that can execute tasks; worker processors are untrusted and might act maliciously. The master assigns tasks to workers to be executed. Each task returns a binary value, and we want the master to accept only correct values with high probability. Furthermore, we assume that the service provided by the workers is not free; for each task that a worker is assigned, the master is charged with a work-unit. Therefore, considering a single task assigned to several workers, our goal is to have the master computer to accept the correct value of the task with high probability, with the smallest possible amount of work (number of workers the master assigns the task). We explore two ways of bounding the number of faulty processors: (a) we consider a fixed bound f < n/2 on the maximum number of workers that may fail, and (b) a probability p < 1/2 of any processor to be faulty (all processors are faulty with probability p, independently of the rest of processors). Our work demonstrates that it is possible to obtain high probability of correct acceptance with low work. In particular, by considering both mechanisms of bounding the number of malicious workers, we first show lower bounds on the minimum amount of (expected) work required, so that any algorithm accepts the correct value with probability of success 1 - ε, where ε 1 (e.g., 1/n). Then we develop and analyze two algorithms, each using a different decision strategy, and show that both algorithms obtain the same probability of success 1 - ε, and in doing so, they require similar upper bounds on the (expected) work. Furthermore, under certain conditions, these upper bounds are asymptotically optimal with respect to our lower bounds.
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