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Title: Constructions of locally recoverable codes with availability Speaker: Katie Haymaker, Villanova University Time: 2:30pm–3:30pm Place: CMC 130 Sponsor: REU Site Program

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlăduţ to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures. This talk is based on joint work with Beth Malmskog (Colorado College) and Gretchen Matthews (Clemson University).

Title: Quantum estimates on lattice enumeration Speaker: Shi Bai, Florida Atlantic University Time: 4:00pm–5:00pm Place: CMC 130 Sponsor: REU Site Program

Lattice reduction techniques such as BKZ (Block-Korkine-Zolotarev) are critical in assessing the security of lattice-based cryptography. The subroutine in BKZ which finds the shortest vector in a projected sublattice can be instantiated with enumeration algorithms, which can be viewed as a depth-first search on an enumeration tree whose nodes represent a partial assignment of the coefficients.

In this talk, we will discuss some concrete estimates on the cost of quantum lattice enumeration using Montanaro's quantum tree backtracking algorithm. We will show a concrete implementation in the quantum circuit model and how to optimize the circuit depth by parallelizing the components.

Title: The Clifford Hierarchy Speaker: Tefjol Pllaha, University of Nebraska Lincoln Time: 2:30pm–3:30pm Place: CMC 130 Sponsor: REU Site Program

In 1999, Gottesman and Chuang demonstrated that universal quantum computing is possible. As part of their proof, they introduced the Clifford hierarchy, and it has been proven to be a central structure both in theory and practice. It this talk, I will define the hierarchy and discuss its deep connections with group theory, symplectic geometry, and representation theory. Additionally, I will use these connections to discuss some recent insights on the second and third level of the hierarchy.

Title: Rank-Metric Codes: Invariants and Extremality Speaker: Giuseppe Cotardo, Virginia Tech Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: REU Site Program

Tensor rank-metric codes were introduced by Roth in 1991 as a natural generalization of matrix rank-metric codes. They are defined to be subspaces of \(r\)-tensors where the ambient space is endowed with the tensor rank as a distance function.

In this talk, we present the general class of tensor rank-metric codes and we study their invariants that correspond to different families of anticodes. In this framework, an anticode is defined to be a perfect space with some additional properties. A perfect space is one that is spanned by tensors of rank 1. We identify four different classes of tensor anticodes and show how these gives different information on the codes they describe. The use of the anticodes concept is motivated by an interest in capturing structural properties of tensor codes. We define the generalized tensor weights associated with each set of anticodes. We describe the properties of these invariants and we show how different invariants offer different information on the underlying code. We also derive the MacWilliam identities for codes of tensors via the generalized tensor binomial moments. Finally, we introduce new families of extremal tensor codes, namely the i-tensor binomial moment determined and the \(i\)-tensor maximum rank distance codes.

Title: Just how hard are rotations of \(\mathbb{Z}^n\)? Algorithms and cryptography with the simplest lattice Speaker: Huck Bennett, University of Colorado Boulder Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: REU Site Program

A lattice is a regular grid of points in \(n\)-dimensional space. Lattices are classically studied objects in mathematics and in the past few decades, have played a central role in many areas of computer science and especially cryptography.

In this talk, I will discuss algorithms and cryptography on one of the simplest families of lattices: rotations of the integer lattice \(\mathbb{Z}^n\). I will start by presenting a new, faster algorithm for the Shortest Vector Problem (SVP) and Lattice Isomorphism Problem (LIP) on rotations of \(\mathbb{Z}^n\). Previously, no faster algorithm for these problems on rotations of \(\mathbb{Z}^n\) than was known for general lattices. I will then show how to build a secure public-key cryptosystem (similar to one built from LWE) based on the assumption that close relatives of these problems are intractable.

Joint work with Atul Ganju, Pura Peetathawatchai, and Noah Stephens-Davidowitz. Published at EUROCRYPT 2023. Preprint available at https://eprint.iacr.org/2021/1548.

Title: A survey of compositional inverses of permutation polynomials over finite fields Speaker: Qiang (Steven) Wang, Carleton University Time: 2:30pm–3:30pm Place: CMC 130 Sponsor: REU Site Program

Let \(q\) be a prime power and Fq be the finite field with q elements. We call a polynomial \(f(x)\in Fq[x]\) a permutation polynomial (PP) of \(Fq\) when the evaluation map \(f:a\to f(a)\) is a bijection. PPs play important roles in finite field theory and they have broad applications in coding theory, combinatorial designs, and cryptography. Explicitly determining the compositional inverse of a PP is also useful because both a PP and its inverse are required in many applications. For example, during the decryption process in a cryptographic algorithm with Substitution-Permutation Network structure, the compositional inverse of the \(S\)-box plays an essential role. In this talk we survey on the recent results and methods in the study of compositional inverses of PPs over finite fields. In particular, we describe a framework in terms of a commutative diagram which unifies several recent methods in finding the inverses of PPs.