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Verifiable Homomorphic Linear Combinations in Multi-Instance Time-Lock Puzzles

Aydin Abadi

TL;DR

This work addresses the need for scalable, verifiable computation on time-lock puzzles (TLPs) by introducing MH-TLP, a Multi-Instance Verifiable Partially Homomorphic TLP, and its extension MMH-TLP for multiple clients. The approach builds on Tempora-Fusion concepts to enable efficient verifiable homomorphic linear combinations of puzzles, solved sequentially rather than in parallel, without requiring asymmetric-key verification or a trusted third party. The key contributions are formal definitions, detailed constructions, and proofs ensuring privacy, correctness, efficiency, and compactness, with a cost analysis showing linear scaling in the number of clients $n$ and puzzles per client $z$. The practical impact lies in enabling scalable private scheduling, secure federated learning SAaaS, and verifiable multi-party computations in e-voting and auctions, all without compromising verifier simplicity or requiring trusted setups. Overall, MH-TLP and MMH-TLP advance the state of verifiable homomorphic TLPs by merging multi-instance scalability with cross-client verifiable computation, offering a flexible, efficient alternative to prior schemes.

Abstract

Time-Lock Puzzles (TLPs) have been developed to securely transmit sensitive information into the future without relying on a trusted third party. Multi-instance TLP is a scalable variant of TLP that enables a server to efficiently find solutions to different puzzles provided by a client at once. Nevertheless, existing multi-instance TLPs lack support for (verifiable) homomorphic computation. To address this limitation, we introduce the "Multi-Instance partially Homomorphic TLP" (MH-TLP), a multi-instance TLP supporting efficient verifiable homomorphic linear combinations of puzzles belonging to a client. It ensures anyone can verify the correctness of computations and solutions. Building on MH-TLP, we further propose the "Multi-instance Multi-client verifiable partially Homomorphic TLP" (MMH-TLP). It not only supports all the features of MH-TLP but also allows for verifiable homomorphic linear combinations of puzzles from different clients. Our schemes refrain from using asymmetric-key cryptography for verification and, unlike most homomorphic TLPs, do not require a trusted third party. A comprehensive cost analysis demonstrates that our schemes scale linearly with the number of clients and puzzles.

Verifiable Homomorphic Linear Combinations in Multi-Instance Time-Lock Puzzles

TL;DR

This work addresses the need for scalable, verifiable computation on time-lock puzzles (TLPs) by introducing MH-TLP, a Multi-Instance Verifiable Partially Homomorphic TLP, and its extension MMH-TLP for multiple clients. The approach builds on Tempora-Fusion concepts to enable efficient verifiable homomorphic linear combinations of puzzles, solved sequentially rather than in parallel, without requiring asymmetric-key verification or a trusted third party. The key contributions are formal definitions, detailed constructions, and proofs ensuring privacy, correctness, efficiency, and compactness, with a cost analysis showing linear scaling in the number of clients and puzzles per client . The practical impact lies in enabling scalable private scheduling, secure federated learning SAaaS, and verifiable multi-party computations in e-voting and auctions, all without compromising verifier simplicity or requiring trusted setups. Overall, MH-TLP and MMH-TLP advance the state of verifiable homomorphic TLPs by merging multi-instance scalability with cross-client verifiable computation, offering a flexible, efficient alternative to prior schemes.

Abstract

Time-Lock Puzzles (TLPs) have been developed to securely transmit sensitive information into the future without relying on a trusted third party. Multi-instance TLP is a scalable variant of TLP that enables a server to efficiently find solutions to different puzzles provided by a client at once. Nevertheless, existing multi-instance TLPs lack support for (verifiable) homomorphic computation. To address this limitation, we introduce the "Multi-Instance partially Homomorphic TLP" (MH-TLP), a multi-instance TLP supporting efficient verifiable homomorphic linear combinations of puzzles belonging to a client. It ensures anyone can verify the correctness of computations and solutions. Building on MH-TLP, we further propose the "Multi-instance Multi-client verifiable partially Homomorphic TLP" (MMH-TLP). It not only supports all the features of MH-TLP but also allows for verifiable homomorphic linear combinations of puzzles from different clients. Our schemes refrain from using asymmetric-key cryptography for verification and, unlike most homomorphic TLPs, do not require a trusted third party. A comprehensive cost analysis demonstrates that our schemes scale linearly with the number of clients and puzzles.
Paper Structure (60 sections, 9 theorems, 5 equations, 5 figures, 2 tables)

This paper contains 60 sections, 9 theorems, 5 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Limitations of State-of-the-Art TLPs
  3. Our Contributions
  4. Multi-Instance Partially Homomorphic TLP.
  5. Multi-Instance Multi-Client Partially Homomorphic TLP.
  6. Applications
  7. Scalable, Private, and Compliant Scheduled Payments in Online Banking.
  8. Asynchronous Multi-Model Training in Secure Aggregation as a Service (SAaaS).
  9. Sequential Verifiable E-Voting and Sealed-Bid Auction Systems.
  10. Related Work
  11. Multi-instance Time Lock Puzzle
  12. Homomorphic Time-lock Puzzles
  13. Preliminaries
  14. Notations and Assumptions
  15. Pseudorandom Function
  16. ...and 45 more sections

Key Result

theorem thmcountertheorem

Let $\bm{\pi}(x)$ be a polynomial of degree $n$ with a random root $\beta$, and $\{(x_{ 1},\pi_{ 1}),\ldots,$$(x_{ l},\pi_{ l})\}$ be point-value representation of $\bm{\pi}(x)$, where $l>n$, ${p}\xspace$ denotes a large prime number, $\log_{ 2}({p}\xspace)=\lambda'$ is the security parameter, $\bm{

Figures (5)

  • Figure 1: The $\mathsf{Exp}_{ \textnormal{prv}}^{\mathcal{A}\xspace}$ experiment.
  • Figure 2: The $\mathsf{Exp}_{ \textnormal{val}}^{\mathcal{A}\xspace}(1^{\lambda}, z)$ experiment.
  • Figure 3: $\text{MH-TLP}$ Workflow Outline.
  • Figure 4: $\text{MMH-TLP}$ Workflow Overview.
  • Figure 5: Enhanced Oblivious Linear function Evaluation ($\mathtt{OLE}\xspace^{ +}$) GhoshN19.

Theorems & Definitions (24)

  • theorem thmcountertheorem: Unforgeable Encrypted Polynomial with a Hidden Root
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Syntax
  • definition thmcounterdefinition: Privacy
  • definition thmcounterdefinition: Solution-Validity
  • definition thmcounterdefinition: Completeness
  • definition thmcounterdefinition: Efficiency
  • definition thmcounterdefinition: Compactness
  • definition thmcounterdefinition: Security
  • ...and 14 more