Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond Yao's Millionaires: Secure Multi-Party Computation of Non-Polynomial Functions

Seyed Reza Hoseini Najarkolaei, Mohammad Mahdi Mojahedian, Mohammad Reza Aref

TL;DR

The proposed scheme, utilizing the binary representation of private inputs to determine the $\max$ without disclosing any private inputs or intermediate results, is the only information-theoretically secure method for comparing private numbers without revealing either the private inputs or any intermediate results.

Abstract

In this paper, we present an unconditionally secure $N$-party comparison scheme based on Shamir secret sharing, utilizing the binary representation of private inputs to determine the $\max$ without disclosing any private inputs or intermediate results. Specifically, each party holds a private number and aims to ascertain the greatest number among the $N$ available private numbers without revealing its input, assuming that there are at most $T < \frac{N}{2}$ honest-but-curious parties. The proposed scheme demonstrates a lower computational complexity compared to existing schemes that can only compare two secret numbers at a time. To the best of our knowledge, our scheme is the only information-theoretically secure method for comparing $N$ private numbers without revealing either the private inputs or any intermediate results. We demonstrate that by modifying the proposed scheme, we can compute other well-known non-polynomial functions of the inputs, including the minimum, median, and rank. Additionally, in the proposed scheme, before the final reveal phase, each party possesses a share of the result, enabling the nodes to compute any polynomial function of the comparison result. We also explore various applications of the proposed comparison scheme, including federated learning.

Beyond Yao's Millionaires: Secure Multi-Party Computation of Non-Polynomial Functions

TL;DR

The proposed scheme, utilizing the binary representation of private inputs to determine the without disclosing any private inputs or intermediate results, is the only information-theoretically secure method for comparing private numbers without revealing either the private inputs or any intermediate results.

Abstract

In this paper, we present an unconditionally secure -party comparison scheme based on Shamir secret sharing, utilizing the binary representation of private inputs to determine the without disclosing any private inputs or intermediate results. Specifically, each party holds a private number and aims to ascertain the greatest number among the available private numbers without revealing its input, assuming that there are at most honest-but-curious parties. The proposed scheme demonstrates a lower computational complexity compared to existing schemes that can only compare two secret numbers at a time. To the best of our knowledge, our scheme is the only information-theoretically secure method for comparing private numbers without revealing either the private inputs or any intermediate results. We demonstrate that by modifying the proposed scheme, we can compute other well-known non-polynomial functions of the inputs, including the minimum, median, and rank. Additionally, in the proposed scheme, before the final reveal phase, each party possesses a share of the result, enabling the nodes to compute any polynomial function of the comparison result. We also explore various applications of the proposed comparison scheme, including federated learning.

Paper Structure

This paper contains 26 sections, 2 theorems, 9 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Works
  4. Garbled Circuit
  5. Quantum-Based Comparison
  6. Homomorphic Encryption
  7. Arithmetic Black-Box
  8. Bit-Wise Operations
  9. System Model and Preliminaries
  10. Shamir Secret Sharing
  11. Random Secret Generation
  12. Partition and $0$-Coded Vectors
  13. Proposed Unconditionally Secure Comparison Scheme
  14. Secure Comparison
  15. Equality Test and Zero Indicator
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that $a$ and $b$ are two numbers with length-$L$ binary representations of $\overline{a_1a_2\dots a_L}$ and $\overline{b_1b_2\dots b_L}$, respectively. We have $a>b$ if and only if vector $\mathbf{v}_a-\mathbf{v}_b^0$ has exactly one $0$ entity.

Figures (4)

  • Figure 1: The SCI Module is a combination of secure comparison and zero indicators. It takes the partition vector of $a$ and the $0$-coded vector of $b$, and outputs $0$ if $a>b$ or $1$ in other cases.
  • Figure 2: SCG is a component that takes partition vectors and $0$-coded vectors of the inputs $a$ and $b$, and produces the partition vector and $0$-coded vector of $\max(a,b)$.
  • Figure 3: Using SCG, nodes can iteratively compare secrets and ultimately collaboratively obtain the partition vector and '0'-coded vector of $\max\left(s^{(1)},s^{(2)},\dots,s^{(N)}\right)$.
  • Figure 4: ESCG receives the partition vector, $0$-coded vector, and index of each of $a$ and $b$, and outputs the partition vector, $0$-coded vector, and index of $\max(a,b)$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Example 1
  • Theorem 2