Linear Algebra over finite fields is a building block for several applications including data storage, error detection and correction, and public-key cryptography. These applications enable the security and possibility of our daily digital lives. This workshop aims to expose and engage junior researchers in the foundations and applications of linear algebra over finite fields.

Like linear algebra over infinite fields, linear algebra over finite fields considers vector spaces, matrices, and linear transformations, but there are some crucial differences that play a key role in applications. For instance, the intersection of a vector space and its dual may be nontrivial, and its dimension plays a role in entanglement assisted quantum error correcting codes, secret sharing, and protecting against fault injection and side channel attacks. Working over finite fields also facilitates error correction (classical and quantum) and local recovery of data, through linear systems of equations with only finitely many possible solutions. Linear algebra over finite fields plays a key role in code-based cryptography, the basis for families being considered in the final round of the NIST Post-Quantum Cryptography Standardization process. Quantum algorithms lead to vulnerabilities in current public-key cryptosystems which are prompting new studies of how matrices and linear algebra can support new cryptography.

This workshop will feature lectures on various applications of linear algebra over finite fields. Participants will learn about in the interplay between coding theory and cryptography along with mathematics related to a range of applications.

Participants will be assigned a group led by one of the course instructors. Open problems suitable for exploration by non-experts will be shared, and students will be guided in the research by these instructors.

Application Deadline June 9, 2025