Brief Announcement: Breaking the + 1 Barrier: Executing Payment Transactions in Parallel with Less than + 1 Validations.

We consider the problem of validating payment transactions in an asynchronous system in which up to f validators are subject to Byzantine failures under the control of an adaptive adversary. It was shown that this problem can be solved without consensus by using byzantine quorum systems - requiring full-quorum validation, i.e. at least 2 f + 1 validations per transaction. We show that it is possible to validate transactions in parallel with less than f validations per transaction if each transaction spends no more that a small fraction of an initial balance. Our solution relies on ( k 1 , k 2 )-quorum systems, a novel class of quorum systems that we introduce in this paper. Under an adaptive adversary, these systems can be used to allow k 1 transactions to be validated and prevent more than k ′ 2 > k 2 transactions from being validated, the difference k′ 2 − k 2 being dependent on the quorum system's validation slack , which we define in this paper. Using ( k 1 , k 2 )-quorum systems, a payer can execute multiple partial spending transactions to spend a portion of its initial balance with less than full quorum validation - less than f validations per transaction - then reclaim any remaining funds using one fully validated transaction, which we call a settlement transaction.