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DDH-based schemes for multi-party Function Secret Sharing

Marc Damie, Florian Hahn, Andreas Peter, Jan Ramon

Abstract

Function Secret Sharing (FSS) schemes enable sharing efficiently secret functions. Schemes dedicated to point functions, referred to as Distributed Point Functions (DPFs), are the center of FSS literature thanks to their numerous applications including private information retrieval, anonymous communications, and machine learning. While two-party DPFs benefit from schemes with logarithmic key sizes, multi-party DPFs have seen limited advancements: $O(\sqrt{N})$ key sizes (with $N$, the function domain size) and/or exponential factors in the key size. We propose a DDH-based technique reducing the key size of existing multi-party schemes. In particular, we build an honest-majority DPF with $O(\sqrt[3]{N})$ key size. Our benchmark highlights key sizes up to $10\times$ smaller (on realistic problem sizes) than state-of-the-art schemes. Finally, we extend our technique to schemes supporting comparison functions.

DDH-based schemes for multi-party Function Secret Sharing

Abstract

Function Secret Sharing (FSS) schemes enable sharing efficiently secret functions. Schemes dedicated to point functions, referred to as Distributed Point Functions (DPFs), are the center of FSS literature thanks to their numerous applications including private information retrieval, anonymous communications, and machine learning. While two-party DPFs benefit from schemes with logarithmic key sizes, multi-party DPFs have seen limited advancements: key sizes (with , the function domain size) and/or exponential factors in the key size. We propose a DDH-based technique reducing the key size of existing multi-party schemes. In particular, we build an honest-majority DPF with key size. Our benchmark highlights key sizes up to smaller (on realistic problem sizes) than state-of-the-art schemes. Finally, we extend our technique to schemes supporting comparison functions.
Paper Structure (40 sections, 1 theorem, 5 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 40 sections, 1 theorem, 5 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. 2/3-party DPF
  4. Multi-party DPF
  5. DCF
  6. Gap in the literature
  7. Our Contributions
  8. Definitions
  9. Threat model
  10. Function Secret Sharing
  11. DDH-based Distributed Point Function
  12. Scheme
  13. Asymptotic key size
  14. Supporting other sub-DPFs
  15. Security
  16. ...and 25 more sections

Key Result

theorem 1

Let $\lambda \in \mathbb{N}$, $N, p \in \mathbb{N}$, then $(\mathop{\mathrm{Gen}}\nolimits_{\mathop{\mathrm{DPF}}\nolimits}, \mathop{\mathrm{Eval}}\nolimits_{\mathop{\mathrm{DPF}}\nolimits}, \mathop{\mathrm{Decode}}\nolimits_{\mathop{\mathrm{DPF}}\nolimits})$ as described in Algorithm alg:dpf-ddh is

Figures (3)

  • Figure 1: High-level structure of our DDH-based DPF
  • Figure 2: DPF key sizes for varying moduli. Parameters: $p=5$ parties, $N=10^6$.
  • Figure 3: Key sizes for varying domain sizes and number of parties.

Theorems & Definitions (6)

  • definition 1
  • definition 2: Correctness boyle_function_2015
  • definition 3: Privacy boyle_function_2022
  • theorem 1
  • proof
  • proof