Carleton Finite Fields eSeminar - Summer 2021 Abstracts
Abstracts (Fall 2021)
Speaker: Giorgos Kapetanakis (University of Thessaly)
Title: The existence of Fq-primitive points on curves using freeness
Abstract:
An element of a finite cyclic group of order Q, C_Q, is called
r-free (where r|Q), if it is not a p-th power of any group
element for any prime divisor p of r. We introduce the set
of (r,n)-free elements of C_Q, where n|Q and r|(Q/n), as the
elements of the subgroup C_{Q/n} that are r-free within C_{Q/n}.
Inspired by Vinogradov's expression for the characteristic
function of primitive elements of the finite field Fq, we prove
an analogue for the (r,n)-free elements of C_Q and obtain a
lower bound for the number of elements b of Fq, such that f(b)
is (r,n)-free and F(b) is (R,N)-free, where f and F are
polynomials over Fq.
As an application, we consider the problem of the existence of
points of elliptic curves in Fq^2, whose coordinates are both
primitive and provide a complete answer for the curves y^2=x^3±x.
This is joint work with Stephen D. Cohen and Lucas Reis.
slides
,video
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,
video
Speaker: Fabio Enrique Brochero Martinez
Title: The functional graph of some family of functions over finite fields
Abstract: Let Fq be the finite field with q=p^s elements and f: Fq -> Fq be a function. The functional graph of f is the directed graph G_f=(V, E), where V=Fq and E={(x,f(x)) : x in Fq}. The characteristics of functional graphs (number of cycles, cycle lengths, pre-cycle lengths and so n) have been studied for several different maps over finite fields, due to its applications in cryptography.
In this presentation we will present two independent results: the first one we describe completely the dynamics of the maps f(x)=x^{q+1} +/- x^2 over the finite field F{q^2} and in the second we study the functional graph of maps of the form f(x)= x^n h( x^{(q-1)/m}), where h satisfies a special condition.
slides,
video
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.
video
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