Optimal Secure Coded Distributed Computation over all Fields

TL;DR

The paper tackles secure, fault-tolerant coded distributed computation over finite fields by extending characteristic-zero constructions to all fields using algebraic-geometry–based evaluation codes and the Hadamard-Shur product. It develops a unified framework—including log-additive codes, curve-degree evaluation codes, and vector extensions—to achieve near-optimal recovery thresholds for matrix and tensor computations, with explicit bounds that depend on the tensor rank, the matrix multiplication exponent $\\omega$, and the genus $g$ of the chosen function field. Key contributions include the dual HS code construction for matrix multiplication, a general tensor computation strategy, and a security-enhanced scheme that accommodates colluding servers while quantifying the trade-offs in recovery thresholds. These results enable efficient and secure coded computation over finite fields, improving the practicality of distributed linear algebra in resource-constrained or adversarial environments.

Abstract

We construct optimal secure coded distributed schemes that extend the known optimal constructions over fields of characteristic 0 to all fields. A serendipitous result is that we can encode \emph{all} functions over finite fields with a recovery threshold proportional to the complexity (tensor rank or multiplicative); this is due to the well-known result that all functions over a finite field can be represented as multivariate polynomials (or symmetric tensors). We get that a tensor of order $\ell$ (or a multivariate polynomial of degree $\ell$) can be computed in the faulty network of $N$ nodes setting within a factor of $\ell$ and an additive term depending on the genus of a code with $N$ rational points and distance covering the number of faulty servers; in particular, we present a coding scheme for general matrix multiplication of two $m \times m $ matrices with a recovery threshold of $2 m^{ω} -1+g$ where $ω$ is the exponent of matrix multiplication which is optimal for coding schemes using AG codes. Moreover, we give sufficient conditions for which the Hadamard-Shur product of general linear codes gives a similar recovery threshold, which we call \textit{log-additive codes}. Finally, we show that evaluation codes with a \textit{curve degree} function (first defined in [Ben-Sasson et al. (STOC '13)]) that have well-behaved zero sets are log-additive.

Theorems & Definitions (23)

• Definition 1
• Remark 1
• Lemma 1
• Definition 2
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Definition 3
• Lemma 5