The Star Product of Uniformly Random Codes
Johan V. Dinesen, Ragnar Freij-Hollanti, Camilla Hollanti, Benjamin Jany, Alberto Ravagnani

TL;DR
This paper analyzes the expected dimension of the star product of two random linear codes, providing bounds and asymptotic results with implications for cryptography, quantum error correction, and distributed computation.
Contribution
It establishes a novel connection between star products and bilinear forms, offering new bounds and asymptotic behavior for random codes' star product dimensions.
Findings
Expected dimension reaches maximum asymptotically in field size and code dimensions
Provides a lower bound on the star product dimension for random codes
Implications for cryptography, quantum error correction, and distributed matrix multiplication
Abstract
We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We achieve this by establishing a correspondence between the star product and the evaluation of bilinear forms, which we use to provide a lower bound on the expected star product dimension. We show that asymptotically in both the field size q and the dimensions of the two codes, the expected dimension reaches its maximum. Lastly, we discuss some implications related to private information retrieval, secure distributed matrix multiplication, quantum error correction, and the potential for exploiting the results in cryptanalysis.
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Taxonomy
TopicsCryptography and Data Security · Coding theory and cryptography · Complexity and Algorithms in Graphs
