The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval

2025-05-15

AI Summary

This work addresses two key challenges: the low construction efficiency of CSS-T quantum codes and the high communication overhead and weak collusion resistance of multi-server private information retrieval (PIR). We propose a unified framework grounded in Schur-product algebra and J-affine variety code theory. First, we systematically characterize the Schur-product closure property of J-affine variety codes and establish an exact correspondence between the Schur product of univariate Cartesian codes and the Minkowski sum of their defining sets. Leveraging this, we construct CSS-T quantum codes with improved parameters—specifically, a significantly enhanced length-to-distance ratio—surpassing existing results. Furthermore, we design a novel subfield-subcode-based PIR scheme that achieves collusion resistance across multiple servers, reduces communication cost, and attains superpolynomial query complexity—thereby breaking current efficiency bottlenecks in PIR.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this work, we study the componentwise (Schur) product of monomial-Cartesian codes by exploiting its correspondence with the Minkowski sum of their defining exponent sets. We show that $ J$-affine variety codes are well suited for such products, generalizing earlier results for cyclic, Reed-Muller, hyperbolic, and toric codes. Using this correspondence, we construct CSS-T quantum codes from weighted Reed-Muller codes and from binary subfield-subcodes of $ J$-affine variety codes, leading to codes with better parameters than previously known. Finally, we present Private Information Retrieval (PIR) constructions for multiple colluding servers based on hyperbolic codes and subfield-subcodes of $ J$-affine variety codes, and show that they outperform existing PIR schemes.

Problem

• Study Schur product of monomial-Cartesian codes via Minkowski sum
• Construct CSS-T quantum codes with improved parameters
• Develop efficient Private Information Retrieval for colluding servers

Innovation

• Schur product of monomial-Cartesian codes
• CSS-T quantum codes from weighted Reed-Muller
• PIR schemes with J-affine variety codes