Fractional Spending: VRF&Ring Signatures As Efficient Primitives For Secret Quorums

Digital currencies have emerged as a significant evolution in the financial system, yet they face challenges in distributed settings, particularly regarding double spending. Traditional approaches, such as Bitcoin, use consensus to establish a total order of transactions, ensuring that no more than the currency held by an account is spent in the order. However, consensus protocols are costly, especially when coping with Byzantine faults. It was shown that solving Consensus is not needed to perform currency's transfer, for instance using byzantine quorum systems but validation remains per-account sequential. Recent research also introduced the fractional spending problem, which enables concurrent but non-conflicting transactions i.e., transactions that spend from the same account but cannot lead to a double spending because each is only spending a small fraction of the balance. A solution was proposed based on a new quorum system and specific cryptographic primitives to protect against an adaptive adversary. The quorum system, called (k1, k2)-quorum system, guarantees that at least k1 transactions can be validated concurrently but that no more than k2 can. Employing such quorums, a payer can validate concurrently multiple fractional spending transactions in parallel with high probability. Subsequently, the payer reclaims any remaining sum through a settlement. This paper enhances such solution by integrating different cryptographic primitives, VRF and Ring Signatures, into a similar protocol. But contrarily, these tools ensure quorums to remain secret during settlements, allowing to reduces its communication costs from cubic to quadratic in messages. We also achieve payment transaction with 3 message delays rather then 5. Additionally, we propose a refined formalization of the fractional spending problem, introducing coupons, which simplifies the theoretical framework and proof structure.