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Fractional Payment Transactions: Executing Payment Transactions in Parallel with Less than f+1 Validations

Rida Bazzi, Sara Tucci-Piergiovanni

TL;DR

This paper addresses enabling parallel validation of fractional spending payments in an asynchronous system with up to $f$ Byzantine validators controlled by an adaptive adversary. It introduces $(k_1,k_2)$-Byzantine quorum systems with lower and upper intersection bounds and a validation slack, enabling up to $s_1=k_1$ concurrent payments each spending at most $1/s_2$ of a fund’s balance while ensuring no double spending; a settlement protocol restores unspent funds using larger quorums and secret propagation. Theoretical results include probabilistic safety and progress guarantees with high probability, a simple asynchronous construction, and impossibility results for deterministic or full-balance spending; the approach aims to improve throughput for scalable payments and smart contracts. The work also discusses parameter trade-offs, randomness-based security against adaptive adversaries, and future directions for reducing space complexity and removing inter-party coordination during validation. These insights suggest practical gains in high-throughput, low-latency payment processing under Byzantine fault tolerance.

Abstract

We consider the problem of supporting 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, in the case of a single owner, this problem can be solved without consensus by using byzantine quorum systems (requiring a quorum of $2f+1$ validations per transaction). Nonetheless, the process of validating transactions remains sequential. For example, if one has a balance of ten coins and intends to make separate payments of two coins each to two distinct recipients, both transactions must undergo processing by a common correct validator. On the other hand, these two transactions are non-conflicting as they do not lead to double spending, allowing in principle for parallel validation. In this paper, we show that it is possible to validate payment transactions in parallel with less than $f$ validations per transaction in an asynchronous system, provided that each transaction spends only a small fraction of a balance. Our solution relies on a novel class of probabilistic quorum systems that we introduce in this paper, termed \textit{$(k_1,k_2)$-quorum systems}. In the absence of an adaptive adversary, \textit{$(k_1,k_2)$-quorum systems} can be used to enable concurrent and asynchronous validation of up to $k_1$ transactions while preventing validation of more than $k_2$ transactions. Employing a $(k_1, k_2)$-quorum system, we introduce protocols enabling a payer to validate multiple \textit{fractional spending} transactions in parallel with less than $f+1$ validations per transaction. Subsequently, the payer reclaims any remaining funds through a fully validated transaction, referred to as a \textit{settlement} transaction.

Fractional Payment Transactions: Executing Payment Transactions in Parallel with Less than f+1 Validations

TL;DR

This paper addresses enabling parallel validation of fractional spending payments in an asynchronous system with up to Byzantine validators controlled by an adaptive adversary. It introduces -Byzantine quorum systems with lower and upper intersection bounds and a validation slack, enabling up to concurrent payments each spending at most of a fund’s balance while ensuring no double spending; a settlement protocol restores unspent funds using larger quorums and secret propagation. Theoretical results include probabilistic safety and progress guarantees with high probability, a simple asynchronous construction, and impossibility results for deterministic or full-balance spending; the approach aims to improve throughput for scalable payments and smart contracts. The work also discusses parameter trade-offs, randomness-based security against adaptive adversaries, and future directions for reducing space complexity and removing inter-party coordination during validation. These insights suggest practical gains in high-throughput, low-latency payment processing under Byzantine fault tolerance.

Abstract

We consider the problem of supporting payment transactions in an asynchronous system in which up to validators are subject to Byzantine failures under the control of an adaptive adversary. It was shown that, in the case of a single owner, this problem can be solved without consensus by using byzantine quorum systems (requiring a quorum of validations per transaction). Nonetheless, the process of validating transactions remains sequential. For example, if one has a balance of ten coins and intends to make separate payments of two coins each to two distinct recipients, both transactions must undergo processing by a common correct validator. On the other hand, these two transactions are non-conflicting as they do not lead to double spending, allowing in principle for parallel validation. In this paper, we show that it is possible to validate payment transactions in parallel with less than validations per transaction in an asynchronous system, provided that each transaction spends only a small fraction of a balance. Our solution relies on a novel class of probabilistic quorum systems that we introduce in this paper, termed \textit{-quorum systems}. In the absence of an adaptive adversary, \textit{-quorum systems} can be used to enable concurrent and asynchronous validation of up to transactions while preventing validation of more than transactions. Employing a -quorum system, we introduce protocols enabling a payer to validate multiple \textit{fractional spending} transactions in parallel with less than validations per transaction. Subsequently, the payer reclaims any remaining funds through a fully validated transaction, referred to as a \textit{settlement} transaction.
Paper Structure (28 sections, 19 theorems, 6 equations, 1 figure, 10 algorithms)

This paper contains 28 sections, 19 theorems, 6 equations, 1 figure, 10 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. System Model
  4. The $(s_1,s_2)$-Fractional Spending Problem
  5. Funds, Payments and Settlements
  6. Problem Definition
  7. --Byzantine quorums
  8. Motivation
  9. Quorums definition
  10. --quorums Construction
  11. --quorum systems
  12. Fractional Spending and Settlement Protocols
  13. Protocol Overview
  14. Fractional Spending Protocols
  15. Buyer's Code
  16. ...and 13 more sections

Key Result

lemma 1

An $(m,n)$ uniform balanced quorum system is a $(k_1,k_2)$-quorum system with $\alpha = 1/3$ and $\beta = 2/3$ has $\epsilon = \delta = \negl[n]$ and validation slack = $m/3$, if $p_f+\alpha_1 < 1/3$, where $\alpha_1 = k_1m/n$ and $p_f = f/n$.

Figures (1)

  • Figure 1: Illustration of $(k_1,k2)$-quorums requirements: (a) the size of intersection of $Q$ with up to $k_1$ selected quorums together with previously corrupted servers in $Q$ does not exceed $\alpha |Q|$; (b) the size of the intersection $Q$ with $k_2$ (or more) selected quorums minus any previously corrupted servers in the intersection is at least $\beta |Q|$;

Theorems & Definitions (23)

  • definition 1
  • definition 2
  • definition 3
  • lemma 1
  • definition 4
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • ...and 13 more