Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Secret Sharing for Secure and Private Information Retrieval: A Construction Using Algebraic Geometry Codes

Okko Makkonen, David Karpuk, Camilla Hollanti

TL;DR

This work reframes secure and private information retrieval as a problem solvable via homomorphic secret sharing built from algebraic-geometry codes. By recasting CSA codes on the projective line as AG codes and extending to hyperelliptic curves, the authors unlock higher PIR rates for fixed field sizes through genus-based increases in rational points. The construction provides flexible tradeoffs among field size, subpacketization, and server counts while maintaining $X$-security and $T$-privacy, and it highlights potential advantages over genus-zero schemes in certain parameter regimes. The results open avenues to apply AG codes to PIR and related distributed computations, with implications for broader interference alignment and secure computation tasks.

Abstract

Private information retrieval (PIR) considers the problem of retrieving a data item from a database or distributed storage system without disclosing any information about which data item was retrieved. Secure PIR complements this problem by further requiring the contents of the data to be kept secure. Privacy and security can be achieved by adding suitable noise to the queries and data using methods from secret sharing. In this paper, a new framework for homomorphic secret sharing in secure and private information retrieval from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. We utilize this framework to give a secure PIR construction using algebraic geometry codes over hyperelliptic curves of arbitrary genus. It is shown that the proposed scheme offers interesting tradeoffs between the field size, file size, number of colluding servers, and the total number of servers. When the field size is fixed, this translates in some cases to higher retrieval rates than those of the original scheme. In addition, the new schemes exist also for some parameters where the original ones do not.

Secret Sharing for Secure and Private Information Retrieval: A Construction Using Algebraic Geometry Codes

TL;DR

This work reframes secure and private information retrieval as a problem solvable via homomorphic secret sharing built from algebraic-geometry codes. By recasting CSA codes on the projective line as AG codes and extending to hyperelliptic curves, the authors unlock higher PIR rates for fixed field sizes through genus-based increases in rational points. The construction provides flexible tradeoffs among field size, subpacketization, and server counts while maintaining -security and -privacy, and it highlights potential advantages over genus-zero schemes in certain parameter regimes. The results open avenues to apply AG codes to PIR and related distributed computations, with implications for broader interference alignment and secure computation tasks.

Abstract

Private information retrieval (PIR) considers the problem of retrieving a data item from a database or distributed storage system without disclosing any information about which data item was retrieved. Secure PIR complements this problem by further requiring the contents of the data to be kept secure. Privacy and security can be achieved by adding suitable noise to the queries and data using methods from secret sharing. In this paper, a new framework for homomorphic secret sharing in secure and private information retrieval from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. We utilize this framework to give a secure PIR construction using algebraic geometry codes over hyperelliptic curves of arbitrary genus. It is shown that the proposed scheme offers interesting tradeoffs between the field size, file size, number of colluding servers, and the total number of servers. When the field size is fixed, this translates in some cases to higher retrieval rates than those of the original scheme. In addition, the new schemes exist also for some parameters where the original ones do not.
Paper Structure (22 sections, 6 theorems, 56 equations, 1 figure)

This paper contains 22 sections, 6 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisors and the Riemann--Roch Theorem
  4. Hyperelliptic Curves
  5. Coding Theory Preliminaries
  6. Generalized Reed--Solomon Codes and Star Products
  7. Algebraic Geometry Codes
  8. Secret Sharing
  9. Linear Secret Sharing
  10. Homomorphic Secret Sharing
  11. Secure and Private Information Retrieval
  12. Construction Using the Projective Line
  13. Construction by Polynomials
  14. Geometric Interpretation
  15. Construction Using Hyperelliptic Curves
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $\mathcal{X}$ be a curve of genus $g$ over $\mathbb{F}_q$, and let $D, D'$ be divisors on $\mathcal{X}$.

Figures (1)

  • Figure 1: Comparison between the maximal achievable rate of the CSA construction of Jia_Sun_Jafar_XSTPIR over the projective line (\ref{['thm:projective_line_construction']}) and the construction over hyperelliptic curves (\ref{['thm:AG_construction']}) over a fixed field. The constructions are over $\mathbb{F}_{2^8}$ with primitive element $\alpha$ satisfying $\alpha^8 + \alpha^4 + \alpha^3 + \alpha^2 + 1 = 0$ (left) and over $\mathbb{F}_{61}$ (right).

Theorems & Definitions (12)

  • Theorem 2.1
  • Example 2.2
  • Proposition 3.1
  • Example 3.2: Shamir secret sharing
  • Example 3.3: Chen--Cramer secret sharing
  • Example 3.4
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 6.1
  • Example 6.2
  • ...and 2 more