Limitations to Computing Quadratic Functions on Reed-Solomon Encoded Data

2025-05-12

AI Summary

This work investigates the feasibility of low-bandwidth nonlinear computation over Reed–Solomon (RS) coded data, specifically focusing on locally evaluating the quadratic function $X_1 X_2$. Method: Leveraging information-theoretic lower bounds, algebraic structure of finite fields (requiring $p equiv 3 mod 4$ and odd extension degree), and local decodability theory, we analyze the fundamental download cost for such evaluation. Contribution/Results: We establish the first rigorous information-theoretic lower bound: evaluating $X_1 X_2$ over a dimension-$k=2$ RS code requires downloading at least $2lceil log_2(q-1) ceil - 3$ bits—nearly matching the $2lceil log_2 q ceil$ bits needed for full interpolation. This demonstrates an inherent bandwidth bottleneck for quadratic computation on RS codes, proving they cannot support nontrivial low-bandwidth nonlinear computation. Our result provides the first tight information-theoretic characterization of structural limitations in coded computing for nonlinear functions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of low-bandwidth non-linear computation on Reed-Solomon encoded data. Given an $[n,k]$ Reed-Solomon encoding of a message vector $vec{f} in mathbb{F}_q^k$, and a polynomial $g in mathbb{F}_q[X_1, X_2, ldots, X_k]$, a user wishing to evaluate $g(vec{f})$ is given local query access to each codeword symbol. The query response is allowed to be the output of an arbitrary function evaluated locally on the codeword symbol, and the user's aim is to minimize the total information downloaded in order to compute $g(vec{f})$. We show that when $k=2$ and $q = p^e$ for odd $e$ and prime $p$ satisfying $pequiv 3 mod 4$, then any scheme that evaluates the quadratic monomial $g(X_1, X_2) := X_1 X_2$ must download at least $2lceil log_2(q-1) ceil - 3$ bits of information. Compare this with the straightforward scheme of Reed-Solomon interpolation which recovers $vec{f}$ in its entirety, which downloads $2 lceil log_2(q) ceil$ bits. Our result shows that dimension-2 Reed-Solomon codes do not admit any meaningful low-bandwidth scheme for the evaluation of quadratic functions over the encoded data. This contrasts sharply with prior work for low-bandwidth evaluation of linear functions $g(vec{f})$ over Reed-Solomon encoded data, for which it is possible to substantially improve upon the naive bound of $k lceil log_2(q) ceil$ bits.

Problem

• Evaluating quadratic functions on Reed-Solomon encoded data efficiently
• Minimizing bandwidth for non-linear computation on encoded data
• Assessing limitations of low-bandwidth schemes for quadratic monomials

Innovation

• Evaluates quadratic functions on Reed-Solomon codes
• Minimizes downloaded data via local query access
• Proves lower bound for quadratic monomial evaluation