The Star Product of Uniformly Random Codes

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.