Optimal Secure Coded Distributed Computation over all Fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909593811025920 |
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| author | Soto, Pedro |
| author_facet | Soto, Pedro |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Secure Coded Distributed Computation over all Fields Soto, Pedro Information Theory Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Symbolic Computation Algebraic Geometry E.4; I.1.2; E.1 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. |
| title | Optimal Secure Coded Distributed Computation over all Fields |
| topic | Information Theory Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Symbolic Computation Algebraic Geometry E.4; I.1.2; E.1 |
| url | https://arxiv.org/abs/2504.18038 |