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Organizing Committee
Virginia Tech
Virginia Tech
Virginia Tech
Universidade Federal do Rio de Janeiro
NIST
University of Zurich
Abstract
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
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
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Workshop Schedule
Monday, August 18, 2025
8:30 - 8:45 AM EDT
Check In
11th Floor Collaborative Space
8:45 - 8:55 AM EDT
Welcome - ICERM Director
Welcome - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
8:55 - 9:00 AM EDT
Organizer Welcome
Welcome - 11th Floor Lecture Hall
Jason LeGrow, Virginia Tech
Hiram López, Virginia Tech
Gretchen Matthews, Virginia Tech
Luciane Quoos, Universidade Federal do Rio de Janeiro
Angela Robinson, NIST
Joachim Rosenthal, University of Zurich
9:00 - 9:50 AM EDT
Network Coding
Tutorial - 11th Floor Lecture Hall
Felice Manganiello, Clemson University
Abstract
Network coding in communication is a technique where intermediate nodes in a network combine incoming data packets using algebraic operations before forwarding them. This differs from traditional routing, where packets are simply relayed without modification. By mixing data, network coding can improve efficiency, reduce latency, and enhance resilience to packet loss and network congestion. This course introduces the fundamental principles of network coding, including its theoretical foundations and practical applications in modern communication systems. Participants will explore key concepts such as linear network coding, multicast scenarios, and in general coding over networks.
9:50 - 10:05 AM EDT
Coffee Break
11th Floor Collaborative Space
10:05 - 10:55 AM EDT
Around Reed-Muller codes
Tutorial - 11th Floor Lecture Hall
Alexander Barg, University of Maryland
Abstract
(I) Reed-Muller (RM) codes form a classic code family that continues to inspire new results and a seemingly inexhaustible range of problems. In this minicourse we discuss several problems inspired by recent works. The first of them concerns storage codes on graphs. A storage code is a set of assignments of bits to the vertices of a graph such that every vertex satisfies a parity check defined by its neighbors. Assuming that the graph is triangle-free (we will explain this assumption), what is the largest-size storage code on it? It was recently shown that there exist graphs on n vertices for which such codes attain the maximum possible size, close to 2^n (arXiv:2212.12117). It is still an open question whether RM codes give rise to similar constructions.
(II) RM codes can also be defined from cosets of the subgroups of the additive group Z_2^m. This construction can be extended to other finite groups of a similar kind, known as Coxeter groups. These groups are not too mysterious: for instance, the permutation group is one of them. It is of interest to explore properties of such Coxeter codes (arXiv:2502.14746). For instance, can we extend the famous |u|u+v| construction? If so, can one construct some analogs of recent decoders of RM codes? More challenging: RM codes achieve the Shannon capacity of the binary erasure channel. Do other Coxeter codes share this property? Also challenging: prove the distance value of Coxeter codes (we have a good idea of the answer).
10:55 - 11:10 AM EDT
Coffee Break
11th Floor Collaborative Space
11:10 AM - 12:00 PM EDT
Research Talk
11th Floor Lecture Hall
Gretchen Matthews, Virginia Tech
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 3:30 PM EDT
Computer Lab
Seminar - 11th Floor Lecture Hall
Rodrigo San José, Virginia Tech
3:30 - 3:45 PM EDT
Coffee Break
11th Floor Collaborative Space
3:45 - 4:30 PM EDT
Computer Lab Continued
Seminar - 11th Floor Lecture Hall
Rodrigo San José, Virginia Tech
4:30 - 5:00 PM EDT
Closing Remarks
11th Floor Lecture Hall
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, August 19, 2025
9:00 - 10:15 AM EDT
q-Matroids and their Codes
Tutorial - 11th Floor Lecture Hall
Eimear Byrne, University College Dublin
Abstract
In this course I will introduce some generalisations of matroids and Hamming metric codes, called q-matroids and rank-metric codes, respectively. These generalisations are examples of q-analogues, where a combinatorial structure defined in relation to a set is instead associated with a vector space. There are many connections between codes and matroids and similarly many connections between rank-metric codes and q-matroids. We will look at different equivalent ways of defining a q-matroid and see how these compare with their classical counterparts. We will also introduce some q-matroid invariants and see how these give invariants of rank-metric codes.
10:15 - 10:45 AM EDT
Coffee Break
11th Floor Collaborative Space
10:45 AM - 12:00 PM EDT
Multivariate Cryptography and MinRank Attacks
Tutorial - 11th Floor Lecture Hall
Ryann Cartor, Clemson University
Abstract
Multivariate public key cryptography is one of the main families of post-quantum cryptography, offering fast and compact schemes built from systems of polynomial equations over finite fields. In this mini-course, we’ll begin with an introduction to past and present multivariate cryptosystems. From there, we’ll explore the MinRank problem, a central tool in cryptanalysis that has been used to break several high-profile schemes in both the multivariate and code-based regimes. This course aims to make these ideas accessible to participants with a basic background in algebra, while offering a window into current research directions in post-quantum cryptography.
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 3:00 PM EDT
Computer Lab
Seminar - 11th Floor Lecture Hall
Rodrigo San José, Virginia Tech
3:00 - 3:15 PM EDT
Coffee Break
11th Floor Collaborative Space
3:15 - 4:30 PM EDT
Group Work
Assigned working locations
4:30 - 5:00 PM EDT
Closing Remarks
11th Floor Lecture Hall
Wednesday, August 20, 2025
9:00 - 9:50 AM EDT
Network Coding
Tutorial - 11th Floor Lecture Hall
Felice Manganiello, Clemson University
Abstract
Network coding in communication is a technique where intermediate nodes in a network combine incoming data packets using algebraic operations before forwarding them. This differs from traditional routing, where packets are simply relayed without modification. By mixing data, network coding can improve efficiency, reduce latency, and enhance resilience to packet loss and network congestion. This course introduces the fundamental principles of network coding, including its theoretical foundations and practical applications in modern communication systems. Participants will explore key concepts such as linear network coding, multicast scenarios, and in general coding over networks.
9:50 - 10:05 AM EDT
Coffee Break
11th Floor Collaborative Space
10:05 - 10:55 AM EDT
Around Reed-Muller codes
Tutorial - 11th Floor Lecture Hall
Alexander Barg, University of Maryland
Abstract
(I) Reed-Muller (RM) codes form a classic code family that continues to inspire new results and a seemingly inexhaustible range of problems. In this minicourse we discuss several problems inspired by recent works. The first of them concerns storage codes on graphs. A storage code is a set of assignments of bits to the vertices of a graph such that every vertex satisfies a parity check defined by its neighbors. Assuming that the graph is triangle-free (we will explain this assumption), what is the largest-size storage code on it? It was recently shown that there exist graphs on n vertices for which such codes attain the maximum possible size, close to 2^n (arXiv:2212.12117). It is still an open question whether RM codes give rise to similar constructions. (II) RM codes can also be defined from cosets of the subgroups of the additive group Z_2^m. This construction can be extended to other finite groups of a similar kind, known as Coxeter groups. These groups are not too mysterious: for instance, the permutation group is one of them. It is of interest to explore properties of such Coxeter codes (arXiv:2502.14746). For instance, can we extend the famous |u|u+v| construction? If so, can one construct some analogs of recent decoders of RM codes? More challenging: RM codes achieve the Shannon capacity of the binary erasure channel. Do other Coxeter codes share this property? Also challenging: prove the distance value of Coxeter codes (we have a good idea of the answer).
10:55 - 11:10 AM EDT
Coffee Break
11th Floor Lecture Hall
11:10 AM - 12:00 PM EDT
12:00 - 12:05 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:05 - 1:30 PM EDT
Lunch/Free Time
1:30 - 3:30 PM EDT
Group Work
Assigned working locations
3:30 - 3:45 PM EDT
Coffee Break
11th Floor Collaborative Space
3:45 - 4:30 PM EDT
Career Panel
Panel Discussion - Virtual
Session Chair
Angela Robinson, NIST
Panelist
Victoria de Quehen, Infosec Global
Panelist
Dustin Moody, National Institute of Standards and Technology
Panelist
Aidan Murphy, Johns Hopkins University
Panelist
Anderson Nascimento, Visa Research
4:30 - 5:00 PM EDT
Closing Remarks
11th Floor Lecture Hall
Thursday, August 21, 2025
9:00 - 10:15 AM EDT
q-Matroids and their Codes
Tutorial - 11th Floor Lecture Hall
Eimear Byrne, University College Dublin
Abstract
In this course I will introduce some generalisations of matroids and Hamming metric codes, called q-matroids and rank-metric codes, respectively. These generalisations are examples of q-analogues, where a combinatorial structure defined in relation to a set is instead associated with a vector space. There are many connections between codes and matroids and similarly many connections between rank-metric codes and q-matroids. We will look at different equivalent ways of defining a q-matroid and see how these compare with their classical counterparts. We will also introduce some q-matroid invariants and see how these give invariants of rank-metric codes.
10:15 - 10:45 AM EDT
Coffee Break
11th Floor Collaborative Space
10:45 AM - 12:00 PM EDT
Multivariate Cryptography and MinRank Attacks
Tutorial - 11th Floor Lecture Hall
Ryann Cartor, Clemson University
Abstract
Multivariate public key cryptography is one of the main families of post-quantum cryptography, offering fast and compact schemes built from systems of polynomial equations over finite fields. In this mini-course, we’ll begin with an introduction to past and present multivariate cryptosystems. From there, we’ll explore the MinRank problem, a central tool in cryptanalysis that has been used to break several high-profile schemes in both the multivariate and code-based regimes. This course aims to make these ideas accessible to participants with a basic background in algebra, while offering a window into current research directions in post-quantum cryptography.
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 3:00 PM EDT
Computer Lab
Seminar - 11th Floor Lecture Hall
Eduardo Camps, Virginia Tech
3:00 - 3:15 PM EDT
Coffee Break
11th Floor Collaborative Space
3:15 - 4:30 PM EDT
Group Work
Assigned working locations
4:30 - 5:00 PM EDT
Closing Remarks
11th Floor Lecture Hall
Friday, August 22, 2025
9:00 - 9:50 AM EDT
Network Coding
Tutorial - 11th Floor Lecture Hall
Felice Manganiello, Clemson University
Abstract
Network coding in communication is a technique where intermediate nodes in a network combine incoming data packets using algebraic operations before forwarding them. This differs from traditional routing, where packets are simply relayed without modification. By mixing data, network coding can improve efficiency, reduce latency, and enhance resilience to packet loss and network congestion. This course introduces the fundamental principles of network coding, including its theoretical foundations and practical applications in modern communication systems. Participants will explore key concepts such as linear network coding, multicast scenarios, and in general coding over networks.
9:50 - 10:05 AM EDT
Coffee Break
11th Floor Collaborative Space
10:05 - 10:55 AM EDT
Around Reed-Muller codes
Tutorial - 11th Floor Lecture Hall
Alexander Barg, University of Maryland
Abstract
(I) Reed-Muller (RM) codes form a classic code family that continues to inspire new results and a seemingly inexhaustible range of problems. In this minicourse we discuss several problems inspired by recent works. The first of them concerns storage codes on graphs. A storage code is a set of assignments of bits to the vertices of a graph such that every vertex satisfies a parity check defined by its neighbors. Assuming that the graph is triangle-free (we will explain this assumption), what is the largest-size storage code on it? It was recently shown that there exist graphs on n vertices for which such codes attain the maximum possible size, close to 2^n (arXiv:2212.12117). It is still an open question whether RM codes give rise to similar constructions. (II) RM codes can also be defined from cosets of the subgroups of the additive group Z_2^m. This construction can be extended to other finite groups of a similar kind, known as Coxeter groups. These groups are not too mysterious: for instance, the permutation group is one of them. It is of interest to explore properties of such Coxeter codes (arXiv:2502.14746). For instance, can we extend the famous |u|u+v| construction? If so, can one construct some analogs of recent decoders of RM codes? More challenging: RM codes achieve the Shannon capacity of the binary erasure channel. Do other Coxeter codes share this property? Also challenging: prove the distance value of Coxeter codes (we have a good idea of the answer).
10:55 - 11:10 AM EDT
Coffee Break
11th Floor Collaborative Space
11:10 AM - 12:00 PM EDT
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 3:30 PM EDT
Computer Lab
Seminar - 11th Floor Lecture Hall
Eduardo Camps, Virginia Tech
3:30 - 3:45 PM EDT
Coffee Break
11th Floor Collaborative Space
3:45 - 4:15 PM EDT
Group Work
Assigned working locations
4:20 - 4:25 PM EDT
Stabilizer Codes over Rings
Lightning Talks - 11th Floor Lecture Hall
Nart Shalqini, Virginia Tech
4:25 - 4:30 PM EDT
Quantum Walks and Monte Carlo for the Galton Box Problem
Lightning Talks - 11th Floor Lecture Hall
Krystal Maughan, University of Vermont
4:30 - 4:35 PM EDT
Goppa codes from a Singer cycle
Lightning Talks - 11th Floor Lecture Hall
Valentino Smaldore, Università degli Studi di Padova
4:35 - 4:40 PM EDT
Introduction to Gabidulin Codes and the Overbeck Distinguisher
Lightning Talks - 11th Floor Lecture Hall
Samuel Orr, Clemson University
4:40 - 4:45 PM EDT
Cyclic AG Code Equivalence
Lightning Talks - 11th Floor Lecture Hall
Olivia Del Guercio, Rice University
4:45 - 4:50 PM EDT
Subspace profiles over finite fields
Lightning Talks - 11th Floor Lecture Hall
Samrith Ram, Indraprastha Institute of Information Technology Delhi
4:50 - 4:55 PM EDT
Curves over finite fields and Projective Reed-Muller Code
Lightning Talks - 11th Floor Lecture Hall
Xu Zhuang, UC Irvine
4:55 - 5:00 PM EDT
Shor's Algorithm
Lightning Talks - 11th Floor Lecture Hall
José Guzmán Vega, Instituto Politécnico Nacional
Request Reimbursement
This section is for general purposes only and does not indicate that all attendees receive funding. Please refer to your personalized invitation to review your offer.
- ORCID iD
- As this program is funded by the National Science Foundation (NSF), ICERM is required to collect your ORCID iD if you are receiving funding to attend this program. Be sure to add your ORCID iD to your Cube profile as soon as possible to avoid delaying your reimbursement.
- Acceptable Costs
- 1 roundtrip between your home institute and ICERM
- Flights on U.S. or E.U. airlines – economy class to either Providence airport (PVD) or Boston airport (BOS)
- Ground Transportation to and from airports and ICERM.
- Unacceptable Costs
- Flights on non-U.S. or non-E.U. airlines
- Flights on U.K. airlines
- Seats in economy plus, business class, or first class
- Change ticket fees of any kind
- Multi-use bus passes
- Meals or incidentals
- Advance Approval Required
- Personal car travel to ICERM from outside New England
- Multiple-destination plane ticket; does not include layovers to reach ICERM
- Arriving or departing from ICERM more than a day before or day after the program
- Multiple trips to ICERM
- Rental car to/from ICERM
- Flights on a Swiss, Japanese, or Australian airline
- Arriving or departing from airport other than PVD/BOS or home institution’s local airport
- 2 one-way plane tickets to create a roundtrip (often purchased from Expedia, Orbitz, etc.)
- Travel Maximum Contributions
- New England: $350
- Other contiguous US: $850
- Asia & Oceania: $2,000
- All other locations: $1,500
- Note these rates were updated in Spring 2023 and supersede any prior invitation rates.
- Reimbursement Requests
Request Reimbursement with Cube
Refer to the back of your ID badge for more information. Checklists are available at the front desk and in the Reimbursement section of Cube.
- Reimbursement Tips
- Scanned original receipts are required for all expenses
- Airfare receipt must show full itinerary and payment
- ICERM does not offer per diem or meal reimbursement
- Allowable mileage is reimbursed at prevailing IRS Business Rate and trip documented via Google Maps PDF
- Keep all documentation until you receive your reimbursement!
- Reimbursement Timing
6–8 weeks after all documentation is sent to ICERM. All reimbursements are reviewed by multiple Brown University offices.
- Reimbursement Deadline
Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. No submissions are accepted after six months.