# Carleton Finite Fields eSeminar page

## Abstracts (Fall 2021)

Speaker: Tor Helleseth (The Selmer Center, University of Bergen)
Title: The history of the cross correlation between m-sequences: an overview
Abstract: Maximum-length sequences (or m-sequences) of period 2^m-1 are generated by linear feedback shift registers with primitive characteristic polynomials of degree m. These sequences have many important applications in modern communication systems. The most well-known property of m-sequences is their two-level ideal autocorrelation. The first major result on the cross correlation of two different m-sequences of the same period was published by Gold back in January 1968 and the result was used in constructing the famous family of Gold sequences. During more than 50 years the cross correlation between m-sequences of the same period has been intensively studied by many research groups. Many results have been obtained but still many open problems remain in this area. This talk will give an updated survey of the status of the cross correlation of m-sequences as well as some consequences of these results. slides

Speaker: Juliane Capaverde (Universidade Federal de Rio Grande do Sul) and Ariane Masuda (New York City College of Technology)
Title: Redei permutations with the same cycle structure
Abstract: Permutation polynomials over finite fields have been extensively studied over the past decades. Among the major challenges in this area are the questions concerning their cycle structures as they capture relevant properties, both theoretically and practically. In this talk we focus on a family of permutation polynomials, the so called Rédei permutations. Although their cycle structures are known, there are other related questions that can be investigated. For example, when do two Rédei permutations have the same cycle structure? We give a characterization of such pairs, and present explicit families of Rédei permutations with the same cycle structure. We also discuss some results regarding Rédei permutations with a particularly simple cycle structure, consisting of $1$- and $j$-cycles only, when $j$ is $4$ or a prime number. The case $j = 2$ is specially important in some applications. We completely describe Rédei involutions with a prescribed cycle structure, and show that remarkably the only Rédei permutations with a unique cycle structure are the involutions. This is joint work with Virgínia Rodrigues from Universidade Federal do Rio Grande do Sul. slides

Speaker: Svetla Nikova (KU Leuven)
Title: Threshold cryptography against combined physical attacks
Abstract: Recent attacks show that there is a need for protecting implementations jointly against side-channel and fault attacks. Analogously, modern MPC protocols consider active security, i.e. against malicious parties which do not only passively eavesdrop but also actively deviate from the protocol. This provides an opportunity for the field of threshold implementations to evolve with MPC and achieve provable secure implementations against combined passive and active physical attacks. In this talk we will first introduce Threshold Implementations applied to protect various ciphers against SCA and the like with Boolean functions and MPC/SSS. After that we will discuss two recent proposals for combined countermeasures: CAPA and M&M, which both start from passively secure threshold schemes and extend those with information-theoretic MAC tags for protection against active adversaries. While similar in their most basic structure, the two proposals explore very different adversary models and thus employ completely different implementation techniques. CAPA considers the field-probe-and-fault model, which is the embedded analogue of multiple parties jointly computing a function with at least one of the parties honest. Accordingly, CAPA is strongly based on the actively secure MPC protocol SPDZ and inherits its provable security properties in this model. Since this results in very expensive implementations, M&M works in a similar but more realistic adversary model and uses existing building blocks from previous passively secure implementations to build more efficient actively secure threshold cryptography.

Speaker: Alexander Bors (Carleton University)
Title: Cycle types of complete mappings
Abstract: A complete mapping of a finite field $K$ is a bijective function $f:K\rightarrow K$ such that the function $K\rightarrow K, x\mapsto f(x)+x$, is also a bijective. Complete mappings have applications in several areas (combinatorics, cryptography, check-digit systems) and have been studied by various authors. Nonetheless, there are aspects of complete mappings about which little is known yet. An example of this are the cycle types of complete mappings -- the information into how many disjoint cycles of each given length a complete mapping can decompose.

In this talk, I will present results that were achieved recently in collaboration with Qiang Wang (also from Carleton University) and which concern the cycle types of complete mappings in two important classes of functions on finite fields: cyclotomic mappings of first order and an additive analogue thereof which we called coset-wise affine mappings. Our results provide both new examples of cycle types of complete mappings that had never been considered before and new constructions for achieving known cycle types. slides, video

## Previous semesters

### Summer 2021

Full list of Summer 2021 abstracts

Name Title Slides Video
Alex Pott Relaxations of almost perfect nonlinearity slides video
Gary McGuire Linear Fractional Transformations and Irreducible Polynomials over Finite Fields
Emina Soljanin Codes, Graphs and Hyperplanes in Data Access Service slides video

### Winter 2021

Full list of Winter 2021 abstracts

Name Title Slides Video
Daqing Wan Counting solutions of large polynomial systems over finite fields video
Claude Carlet Image sets, nonlinearity and distance to affine functions of delta-uniform functions, and gamma functions of APN functions slides video
Nerdagul Anbar Meidl On nilpotent automorphism groups of functions fields slides video
Markus Grassl Algebraic Quantum Codes: New challenges for classical coding theory? slides video
Ivelisse Rubio On multidimensional periodic arrays slides video
Anne Canteaut Recovering or testing Extended-Affine equivalence slides video
Cathy Swaenepoel Trace of products in finite fields and additive double character sums slides video

### Fall 2020

Full list of Fall 2020 abstracts

Name Title Slides Video
Herivelto Borges Algebraic curves through Fernando Torres' lens slides
Nina Bindel A status update on NIST's post-quantum standardization effort slides video
Jonathan Jedwab Packings of partial difference sets
Ray Perlner The MinRank problem in cryptography and cryptanalysis video
Stephen Cohen Existence theorems for r-primitive elements in finite fields slides video
Daniel Katz Niho's last conjecture slides video

### Summer 2020

Full list of Summer 2020 abstracts

Name Title Slides Video
Anna-Lena Horlemann Invariants of linear rank-metric codes -- and what to do with them slides video
Shuxing Li Intersection distribution of polynomials and its applications slides video
Qi Cheng Discrete logarithms over Kummer and Artin-Schreier extensions slides video
Lucas Reis Character sums estimates over affine spaces applied to existence results in finite fields slides video
Marco Baldi QC-LDPC codes, QC-MDPC codes and their use in post-quantum cryptography slides video
Guillermo Matera The distribution of factorization patterns on nonlinear families of univariate polynomials over a finite field slides video
Alev Topuzoglu On the arithmetic of sequences of permutation polynomials slides
Petr Lisonek Contextual hypergraphs slides
Luciane Quoos Locally recoverable codes slides video
Lilya Budaghyan Optimal cryptographic functions over finite fields slides
Arne Winterhof On the distribution of the Rudin-Shapiro function for finite fields slides video
Felice Manganiello Graphs and finite fields in modern communications slides video
Francisco Rodriguez-Henriquez Parallel strategies for SIDH: Toward computing SIDH twice as fast slides video