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Algebraic Geometry Codes for Cross-Subspace Alignment in Private Information Retrieval

Okko Makkonen, David Karpuk, Camilla Hollanti

TL;DR

This paper reframes cross-subspace alignment for private information retrieval (PIR) from colluding servers within a unifying algebraic-geometry coding framework. By interpreting CSA as evaluations of Riemann–Roch spaces on curves, the authors construct explicit schemes over genus-zero (projective line) and genus-one (elliptic) curves, enabling $X$-secure and $T$-private PIR with controllable tradeoffs among field size $q$, file length $L$, total servers $N$, and colluding parameters. The genus-zero construction recovers CSA in a streamlined AG-code setting, while the genus-one scheme exploits more rational points to achieve higher rates for fixed $q$ in certain regimes, albeit with shifts in privacy/security dimensions. The work thus provides a flexible, explicit methodology to balance rate, field size, and collusion requirements, and suggests avenues for extension to coded storage and higher-genus curves with potential practical impact in secure distributed data access.

Abstract

A new framework for interference alignment in secure and private information retrieval (PIR) from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. The general scheme is built on algebraic geometry codes and explicit constructions with replicated storage are given over curves of genus zero and one. 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 in cases where the original ones do not.

Algebraic Geometry Codes for Cross-Subspace Alignment in Private Information Retrieval

TL;DR

This paper reframes cross-subspace alignment for private information retrieval (PIR) from colluding servers within a unifying algebraic-geometry coding framework. By interpreting CSA as evaluations of Riemann–Roch spaces on curves, the authors construct explicit schemes over genus-zero (projective line) and genus-one (elliptic) curves, enabling -secure and -private PIR with controllable tradeoffs among field size , file length , total servers , and colluding parameters. The genus-zero construction recovers CSA in a streamlined AG-code setting, while the genus-one scheme exploits more rational points to achieve higher rates for fixed in certain regimes, albeit with shifts in privacy/security dimensions. The work thus provides a flexible, explicit methodology to balance rate, field size, and collusion requirements, and suggests avenues for extension to coded storage and higher-genus curves with potential practical impact in secure distributed data access.

Abstract

A new framework for interference alignment in secure and private information retrieval (PIR) from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. The general scheme is built on algebraic geometry codes and explicit constructions with replicated storage are given over curves of genus zero and one. 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 in cases where the original ones do not.
Paper Structure (17 sections, 5 theorems, 37 equations, 1 figure)

This paper contains 17 sections, 5 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Framework for $X$-Secure and $T$-Private Information Retrieval
  3. Preliminaries from Algebraic Geometry
  4. Divisors and the Riemann--Roch Theorem
  5. Algebraic Geometry Codes
  6. The Hasse Bound and Maximal Curves
  7. Construction for Genus Zero
  8. Basic Scheme Outline
  9. Explicit Scheme Construction
  10. Interpretation in Terms of Interpolation Polynomials
  11. Construction for Genus One
  12. Interpolation Basis
  13. Scheme Construction
  14. Comparison
  15. An Example
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $G_\ell^\mathrm{priv}$ (resp. $G_\ell^\mathrm{sec}$) be a generator matrix of the code $\varphi(V_\ell^\mathrm{priv}) \subseteq \mathbb{F}_q^N$ (resp. $\varphi(V_\ell^\mathrm{sec})$) and let $\mathcal{I} \subseteq [N]$ be a set such that the columns of $G_\ell^\mathrm{priv}$ (resp. $G_\ell^\math

Figures (1)

  • Figure 1: Comparison for $q = 127$. The genus zero curve corresponds to the original CSA scheme Jia_Sun_Jafar_XSTPIR. The genus one construction is using the curve $\mathcal{X} \colon y^2 = x^3 + x + 33$, which has $150$ rational points and $y$ has $Z = 1$ rational zeros. The genus one scheme has a higher rate when $X = T \geq 26$.

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2: Riemann--Roch Theorem
  • Example 3.1
  • Theorem 3.3