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BEGIN:VEVENT
SUMMARY:Anna-Lena Horlemann (University of St. Gallen)
DTSTART:20200902T160000Z
DTEND:20200902T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/1
DESCRIPTION:Title: Invariants of linear rank-metric codes -- and what to do with
them.\nby Anna-Lena Horlemann (University of St. Gallen) as part of C
arleton Finite Fields eSeminar\n\n\nAbstract\nWe show that the sequence of
dimensions of the linear spaces\, generated by a given (finite field) ran
k-metric code together with itself under several applications of a field a
utomorphism\, is an invariant for the whole equivalence class of the code.
The same property is proven for the sequence of dimensions of the interse
ctions of itself under several applications of a field automorphism. These
invariants give rise to easily computable criteria to check if two codes
are inequivalent. With these criteria we can derive bounds on the number o
f equivalence classes of rank-metric codes\, derive new characterizations
of the well-known Gabidulin codes\, and show that certain code constructio
ns actually lead to equivalent codes.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Cohen (University of Glasgow)
DTSTART:20201007T160000Z
DTEND:20201007T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/2
DESCRIPTION:Title: Existence theorems for $r$-primitive elements in finite field
s\nby Stephen Cohen (University of Glasgow) as part of Carleton Finite
Fields eSeminar\n\n\nAbstract\nLet $r|q-1$. An element of $\\mathbb{F}_q
$ is $r$-primitive if it has order $(q-1)/r$. Thus\, a primitive element
is $1$-primitive and an $r$-primitive element is the $r$th power of a pri
mitive element of $\\mathbb{F}_q$. We describe some existence theorems for
general $r$-primitive elements and\, in particular\, analogues for $2$-
primitive elements of the following {\\em complete} existence theorems for
primitive elements. \n\n(Theorem A (1990).) For any $n \\geq 2$ and $a\\
in \\mathbb{F}_q$ (necessarily with $a \\neq 0$ if $n=2$) there exists a p
rimitive $\\alpha \\in \\mathbb{F}_{q^n}$ with trace $a$ over $\\mathbb{
F}_q$\, except when $a=0\, n=3\, q=4$.\n\n(Theorem B (1983).) Every l
ine in $\\mathbb{F}_{q^2}$ contains a primitive element. \n (A line in $\
\mathbb{F}_{q^2}$ is a set of the form $\\{\\beta(\\gamma+a): a \\in \\ma
thbb{F}_q\\}$\, for some nonzero $\\beta \\in \\mathbb{F}_{q^2}\, \\gamma
\\in \n\\mathbb{F}_{q^2} \\setminus \\mathbb{F}_q$.\n\nJoint work with G
iorgos Kapetanakis.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Jedwab (Simon Fraser University)
DTSTART:20201104T170000Z
DTEND:20201104T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/3
DESCRIPTION:Title: Packings of partial difference sets\nby Jonathan Jedwab (
Simon Fraser University) as part of Carleton Finite Fields eSeminar\n\n\nA
bstract\nPartial difference sets are highly structured group subsets that
occur in various guises throughout design theory\, finite geometry\, codin
g theory\, and graph theory. They admit only two possible nontrivial chara
cter sums and so are often studied using character theory. The central que
stion is to determine which groups contain a partial difference set with t
wo specified nontrivial character sums. We consider an apparently more dif
ficult question: which groups contain a large disjoint collection of such
partial difference sets? This leads us to identify a certain subgroup as c
ontaining important structural information about the packing. With this in
sight\, we are able to formulate a recursive construction of packings in a
belian groups of increasing exponent. This allows us to unify and extend n
umerous previous results about partial difference sets using a common fram
ework.\n\nThis is joint work with Shuxing Li\, a 2019-2021 PIMS Postdoctor
al Fellow.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shuxing Li (Simon Fraser University)
DTSTART:20200826T160000Z
DTEND:20200826T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/4
DESCRIPTION:Title: Intersection distribution and its applications\nby Shuxin
g Li (Simon Fraser University) as part of Carleton Finite Fields eSeminar\
n\n\nAbstract\nGiven a polynomial f over finite field Fq\, its intersectio
n distribution concerns the collective behaviour of a collection of polyno
mials {f(x)+cx | c \\in Fq}. Each polynomial f canonically induces a (q+1)
-set S_f in the classical projective plane PG(2\,q) and the intersection d
istribution of f reflects how the point set S_f interacts with the lines i
n PG(2\,q). Motivated by the long-standing open problem of classifying ova
l monomials\, which are over F_2^n having the same intersection distributi
on as x^2\, we consider the next simplest case: classifying all monomials
over Fq having the same intersection distribution as x^3. Some characteriz
ations of such monomials are derived and as a consequence\, a conjectured
complete list of such monomials is proposed. As an application\, we observ
e that every monomial over F_3^n with the same intersection distribution a
s x^3 naturally leads to a Steiner triple system. Interestingly\, new exam
ples of Steiner triple systems\, which are nonisomorphic to the classical
ones\, are obtained. This is joint work with Gohar Kyureghyan and Alexande
r Pott.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qi Cheng (Oklahoma University)
DTSTART:20200819T160000Z
DTEND:20200819T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/5
DESCRIPTION:Title: The discrete logarithm over Kummer and Artin-Schreier extensi
ons\nby Qi Cheng (Oklahoma University) as part of Carleton Finite Fiel
ds eSeminar\n\n\nAbstract\nMany cryptography protocols rely on hard comput
ational number theoretical problems for security. The discrete logarithm p
roblem over finite fields or elliptic curves is one of the most important
candidates\, besides the integer factorization problem. In this talk\, I w
ill first survey several algorithms attacking the discrete logarithms over
finite fields\, starting from generic algorithms and the index calculus.
My discussion will then be focusing on the of quasi-polynomial-time descen
ding\, and its application on the Kummer and Artin-Schreier extensions.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucas Reis (Federal University of Minas Gerais)
DTSTART:20200812T160000Z
DTEND:20200812T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/6
DESCRIPTION:Title: Character sum estimates over affine spaces applied to existen
ce results in finite fields\nby Lucas Reis (Federal University of Mina
s Gerais) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nIn thi
s talk\, we will discuss the problem of estimating the sum of a multiplica
tive character over the elements of an affine space. We present a new non-
trivial bound on such sums\, along with some applications. In particular\,
we provide asymptotically sharp results on the existence of special primi
tive elements in finite fields.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Baldi (Università Polytecnica delle Marche)
DTSTART:20200729T160000Z
DTEND:20200729T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/7
DESCRIPTION:Title: QC-LDPC codes\, QC-MDPC codes and their use in post-quantum c
ryptography\nby Marco Baldi (Università Polytecnica delle Marche) as
part of Carleton Finite Fields eSeminar\n\n\nAbstract\nLow-density parity-
check (LDPC) codes are a family of modern error correcting codes exploitin
g a random-based design and iterative decoding algorithms allowing them to
approach the channel capacity. The structured subclass of LDPC codes char
acterized by quasi-cyclicity (QC)\, named QC-LDPC codes\, is known to achi
eve practically the same performance as general LDPC codes while enabling
more compact representation and easier implementation. The use of QC-LDPC
codes and of their variant known as QC-MDPC codes in the framework of the
McEliece cryptosystem has shown to be an important avenue for overcoming t
he main limitations of the original McEliece cryptosystem based on Goppa c
odes. Using QC-LDPC and QC-MDPC codes in cryptography\, however\, poses so
me new challenges with respect to their classical use for data reliability
. Nevertheless\, variants of the McEliece and Niederreiter cryptosystems b
ased on these codes are now under consideration by NIST within the standar
dization process of new post-quantum cryptographic primitives. The seminar
will recall the basics of QC-LDPC and QC-MDPC codes and then describe the
main cryptographic primitives relying on these codes\, along with some op
en research challenges in this area.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Matera (Universidad Nacional de General Sarmiento)
DTSTART:20200722T160000Z
DTEND:20200722T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/8
DESCRIPTION:Title: The distribution of factorization patterns on nonlinear famil
ies of univariate polynomials over a finite field\nby Guillermo Matera
(Universidad Nacional de General Sarmiento) as part of Carleton Finite Fi
elds eSeminar\n\n\nAbstract\nIn this talk we discuss an estimate on the nu
mber |A_λ| of elements on a nonlinear family A of monic polynomials of Fq
[T] of degree r having a given factorization pattern λ. We show that |A_
λ| = T(λ) q^{r−m} + O(q^{r−m−1/2})\, where T(λ) is the proportion
of elements of the symmetric group of r elements with cycle pattern λ an
d m is the codimension of A. We provide explicit upper bounds for the cons
tants underlying the O-notation in terms of λ and A with "good" behavior.
Finally\, we apply these results to analyze the average-case complexity o
f the classical factorization algorithm restricted to the family A\, showi
ng that it behaves as good as in the general case. This is based on joint
work with Mariana Pérez and Melina Privitelli.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alev Topuzoglu (Sabanci University)
DTSTART:20200715T160000Z
DTEND:20200715T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/9
DESCRIPTION:Title: On the arithmetic of sequences of permutation polynomials
\nby Alev Topuzoglu (Sabanci University) as part of Carleton Finite Fields
eSeminar\n\n\nAbstract\nIn this talk\, we will present recent results on
factorization of a large class of permutation polynomials. We also discuss
sequences and iterations of permutation polynomials. In particular\, we a
ddress various problems concerning number theoretic properties of irreduci
ble factors of terms of such sequences. This is based on joint work with T
ekgul Kalayci and Henning Stichtenoth.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francisco Rodriguez-Henriquez (CINVESTAV-IPN)
DTSTART:20200520T160000Z
DTEND:20200520T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/10
DESCRIPTION:Title: Parallel strategies for SIDH: towards computing SIDH twice a
s fast\nby Francisco Rodriguez-Henriquez (CINVESTAV-IPN) as part of Ca
rleton Finite Fields eSeminar\n\n\nAbstract\nOver the last ten years there
has been an intense research to find hard mathematical problems that woul
d be presumably hard to solve by a quantum attacker and at the same time c
ould be used to build reasonably efficient public-key cryptoschemes. One s
uch proposal is the hardness of finding an isogeny map between two ellipti
c curves. This proposal has spawned a new line of research generally known
as isogeny-based cryptography. One salient feature of all isogeny-based p
rotocols proposed up-to-date is that they require exceptionally short key
sizes. However\, the latency associated to those protocols is higher than
the ones reported by other post-quantum cryptosystem proposals. In this ta
lk we present novel strategies and concrete algorithms for the parallel co
mputation of the Supersingular Isogeny-based Diffie-Hellman key exchange (
SIDH) protocol when executed on multi-core platforms. To our knowledge\, t
he work presented here is the first reported multi-core implementation of
SIDH.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Lisonek (Simon Fraser University)
DTSTART:20200624T160000Z
DTEND:20200624T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/11
DESCRIPTION:Title: Contextual hypergraphs\nby Petr Lisonek (Simon Fraser Un
iversity) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nContex
tuality is one of the features that distinguishes quantum mechanics from c
lassical mechanics. There are several ways to formalize contextuality math
ematically. One such formalization consists of a hypergraph whose vertices
are labelled by Hermitian operators such that\, for each hyperedge\, cert
ain conditions are fulfilled by the operators occurring in it. A contextua
l hypergraph is one that admits such vertex labeling. The goal of our work
is to construct large (preferably infinite) families of contextual hyperg
raphs. Historically\, contextual hypergraphs have been found mostly by com
putational searches and ad-hoc constructions. In our work we aim at comput
er-free\, systematical constructions\, which use combinatorial ingredients
such as difference matrices and finite geometries. Finite fields play a c
entral role in obtaining these ingredients. We use appropriate group actio
ns to ensure that our contextual hypergraphs are vertex-transitive\, which
is recognized as an added value in the quantum mechanics applications. Th
e talk does not require any knowledge of quantum physics. This is joint wo
rk with Stefan Trandafir.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luciane Quoos (Federal University of Rio de Janeiro)
DTSTART:20200617T160000Z
DTEND:20200617T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/12
DESCRIPTION:Title: Locally recoverable codes\nby Luciane Quoos (Federal Uni
versity of Rio de Janeiro) as part of Carleton Finite Fields eSeminar\n\n\
nAbstract\nA Locally Recoverable Code is a code such that the value of any
single coordinate of a codeword can be recovered from the values of a sma
ll subset of other coordinates. When we have $\\delta$ non-overlapping sub
sets of cardinality $r_i$ that can be used to recover the missing coordina
te we say that a linear code $\\cC$ with length $n$\, dimension $k$\, mini
mum distance $d$ has $(r_1\,\\ldots\, r_\\delta)$-locality and denote by
$[n\, k\, d\; r_1\, r_2\,\\dots\, r_\\delta].$ In this talk\, I will prese
nt a new upper bound for the minimum distance of these codes and propose a
construction of $[n\, k\, d\; r_1\, r_2\,\\dots\, r_\\delta]$-codes on fu
nction fields of genus $g \\geq 1$. This is joint work with Daniele Bartol
i and Maria Montanucci.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lilya Budaghyan (University of Bergen)
DTSTART:20200610T160000Z
DTEND:20200610T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/13
DESCRIPTION:Title: Optimal cryptographic functions over finite fields\nby L
ilya Budaghyan (University of Bergen) as part of Carleton Finite Fields eS
eminar\n\n\nAbstract\nFunctions over finite fields are used in cryptograph
y\, in particular in block ciphers. An important condition on these functi
ons is a high resistance to the differential and linear cryptanalyses\, wh
ich are among the main attacks on block ciphers. The functions which posse
ss the best resistance to the differential attack are called almost perfec
t nonlinear (APN). Planar\, bent and almost bent (AB) functions are those
mappings which oppose an optimum resistance to both linear and differentia
l attacks. An interesting fact is that planar\, bent\, APN and AB function
s also define optimal objects in other domains of mathematics and informat
ion theory such as coding theory\, finite geometry\, sequence design\, alg
ebra\, combinatorics\, et al. In this talk we will discuss problems and re
cent advances in construction and analysis of these functions.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arne Winterhof (Austrian Academy of Sciences)
DTSTART:20200603T160000Z
DTEND:20200603T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/14
DESCRIPTION:Title: On the distribution of the Rudin-Shapiro function for finite
fields\nby Arne Winterhof (Austrian Academy of Sciences) as part of C
arleton Finite Fields eSeminar\n\n\nAbstract\nSee https://people.math.carl
eton.ca/~finitefields/Files/Arne_abstract.pdf\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felice Manganiello (Clemson University)
DTSTART:20200527T160000Z
DTEND:20200527T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/15
DESCRIPTION:Title: Graphs and finite fields in modern communications\nby Fe
lice Manganiello (Clemson University) as part of Carleton Finite Fields eS
eminar\n\n\nAbstract\nThe origin of communication is based on the concept
of two users exchanging information with each other over a single channel.
The problem of perfect communication over a channel was modeled by Shanno
n in the late 40s. More modern communication networks are not so restricti
ve though. Most of the networks we use nowadays\, connect multiple parties
and graphs can be exploited to represent these networks. The question we
are going to investigate in this seminar is simple: given a graph represen
ting a network\, what is its capacity\, meaning how much information can b
e sent through it\, and by which communication protocol over a finite fiel
d? This question has been already answered for unicast networks\, meaning
networks between a singe source and a single receiver\, and for multicast
networks\, meaning networks used by a source to communicate simultaneously
to multiple receivers. The capacity of communication for most networks wi
th multiple sources is still an open question. Networks of this type are c
haracterized by interference that is represented by the messages sent by u
ndesired sources. A communication strategy has to be determined in order t
o remove the interference. We will focus our work on multiple unicast netw
orks and look at the effectiveness of a practice known as interference ali
gnment. We will define the concepts of linear capacity region of a network
and discover that the points of this region are in relation with the solu
tions of a system of bilinear of equation. Solving such a system is know t
o be hard in general\, so we will finally find the points of this region t
hat are achievable by means of Gaussian elimination.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Bindel (University of Waterloo)
DTSTART:20201118T170000Z
DTEND:20201118T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/16
DESCRIPTION:Title: A status update on NIST's post-quantum standardization effor
t\nby Nina Bindel (University of Waterloo) as part of Carleton Finite
Fields eSeminar\n\n\nAbstract\nIf a general-purpose quantum computer can b
e built\, it will break most widely-deployed public-key cryptography. To p
repare for this risk\, the cryptographic community is busily designing new
cryptographic systems. Furthermore\, the (US-American) National Institute
for Standards and Technology (NIST) is currently aiming at standardizing
several quantum-safe digital signature and public-key encryption schemes (
PKEs). Recently\, NIST announced the candidates that advance further to th
e third round of evaluation in NIST standardization effort. \n\nThis talk
will first give an update on the current status of the NIST's post-quantum
standardization effort. In particular\, we will explain the timeline of t
he ongoing project\, explain reasons for why certain schemes have been cho
sen to advance to the third round\, and what are important evaluation crit
eria during the next phase. Moreover\, we will explain how the concrete se
curity of the schemes is estimated. As an example we take a closer look at
lattice-based encryption schemes. Interestingly\, most of the submitted P
KEs are not perfectly correct schemes\, i.e.\, sometimes honestly generate
d ciphertexts can not be encrypted correctly. Finding such a decryption fa
ilure poses a security risk which will be explained in the talk as well.\n
\nShort bio:\nNina Bindel is affiliated to the Institute for Quantu
m Computing (IQC) as a post doctoral researcher at the Department of Combi
natorics & Optimization at the University of Waterloo in Waterloo\, Ontari
o\, Canada.\n\nBefore joining the IQC\, she was a post doctoral researcher
in the Cryptography and computer algebra group at TU Darmstadt where she
also received her Ph.D. in September 2018. Nina's research interest is mos
tly in the area of cryptography that is secure even in the presence of qua
ntum computers\, so-called post-quantum cryptography.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herivelto Borges (University of São Paolo (São Carlos))
DTSTART:20201202T170000Z
DTEND:20201202T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/17
DESCRIPTION:Title: Algebraic curves through Fernando Torres’ lens\nby He
rivelto Borges (University of São Paolo (São Carlos)) as part of Carleto
n Finite Fields eSeminar\n\n\nAbstract\nThe mathematical legacy of Ferna
ndo Torres is felt in several notions within the theory of curves over
finite fields. Such notions include Weierstrass points\, Stöhr-Voloch
theory\, maximal curves\, coding theory\, and finite geometry. In this ta
lk\, we will highlight and briefly discuss some of Torres’ outstanding
contributions to our mathematical community.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Katz (CSUN Northridge)
DTSTART:20200923T160000Z
DTEND:20200923T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/18
DESCRIPTION:Title: Niho's Last Conjecture\nby Daniel Katz (CSUN Northridge)
as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nThis talk is co
ncerned with character sums\, called Weil sums of\nbinomials\, that determ
ine the nonlinearity (Walsh spectrum) of a power\npermutation x -> x^d of
a finite field F. These Weil sums also\ndetermine the crosscorrelation s
pectrum for a pair of maximum length\nlinear recursive sequences and the w
eight distribution of a cyclic code.\nIn each case\, the binomial involved
is of the form x^d-cx\, and one\nobtains values of the Walsh spectrum by
computing the various Weil sums\nas the coefficient c runs through F. Ce
rtain exponents d\, known as Niho\nexponents\, have a simple form and can
produce Walsh spectra with very\nfew distinct values. The last conjectur
e in Niho's 1972 thesis states\nthat a particular family of such exponents
produces spectra with at most\nfive distinct values. Niho's own techniq
ues show that one has at most\neight distinct values. Each of the eight
candidate values corresponds\nto a possible number of distinct roots of a
seventh degree polynomial on\na subset of the finite field F called the un
it circle. We use symmetry\narguments to show that it is impossible to h
ave four\, six\, or seven\nroots on the unit circle: this proves Niho's la
st conjecture.\n\nThis is joint work with Tor Helleseth and Chunlei Li.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ray Perlner (NIST)
DTSTART:20201021T160000Z
DTEND:20201021T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/19
DESCRIPTION:Title: The MinRank problem in Cryptography and Cryptanalysis\nb
y Ray Perlner (NIST) as part of Carleton Finite Fields eSeminar\n\n\nAbstr
act\nThe MinRank problem\, which seeks to find a nonzero\, low-rank linear
combination of a given set of matrices\, shows up in the cryptanalysis of
a wide variety of Multivariate and Code Based cryptosystems\, including s
everal candidates in the National Institute of Standards and Technology (N
IST)’s Postquantum Cryptography Standardization Process. These include t
he code based cryptosystems ROLLO and RQC\, (which were eliminated from co
nsideration for standardization after the second round due to recent signi
ficant improvements in the special case of the MinRank problem known as th
e Rank Syndrome Decoding problem)\, as well as the third (current) round P
QC standardization candidates Rainbow and GeMSS. This talk will discuss ho
w the MinRank problem relates to the cryptanalysis of this diverse array
of cryptosystems\, as well as recent developments that have dramatically i
mproved the concrete complexity of solving the MinRank problem\, both in s
pecial cases and in general.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daqing Wan (UC Irvine)
DTSTART:20210203T170000Z
DTEND:20210203T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/20
DESCRIPTION:Title: Counting solutions of large polynomial systems over finite f
ields\nby Daqing Wan (UC Irvine) as part of Carleton Finite Fields eSe
minar\n\n\nAbstract\nA fundamental algorithmic problem in mathematics and
computer science is to efficiently count the solutions of a multivariate p
olynomial system over a finite field\, and over all of its finite extensio
ns. All general algorithms so far are fully exponential in terms of the nu
mber of equations. In a recent joint work with Q. Cheng and M. Rojas\, we
have reduced this exponential dependence to a polynomial dependence on the
number of equations. A key new ingredient is an effective version of the
classical Kronecker theorem which says that set-theoretically any polynomi
al system in n variables can be defined by n+1 equations if the field is n
ot too small.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claude Carlet (University of Bergen\, Norway and University of Par
is 8\, France)
DTSTART:20210217T170000Z
DTEND:20210217T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/21
DESCRIPTION:Title: Image sets\, nonlinearity and distance to affine functions o
f $\\delta$-uniform functions\, and $\\gamma$-functions of APN functions\nby Claude Carlet (University of Bergen\, Norway and University of Pari
s 8\, France) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nWe
revisit and take a closer look at a result of 2017\, showing that the dif
ferential uniformity of any vectorial function is bounded from below by an
expression depending on the size of its image set. We make explicit the r
esulting tight lower bound on the image set size of differentially $\\delt
a$-uniform functions.\nWe improve an upper bound on the nonlinearity of ve
ctorial functions obtained in the same reference and involving their image
set size. We study when the resulting bound is sharper than the covering
radius bound. We obtain as a by-product a lower bound on the Hamming dista
nce between differentially $\\delta$-uniform functions and affine function
s\, which we improve significantly with a second bound. This leads us to s
tudy what can be the maximum Hamming distance between vectorial functions
and affine functions. We provide an upper bound which is slightly sharper
than a bound by Liu\, Mesnager and Chen when $m< n$\, and a second upper
bound\, which is much stronger in the case where $m$ is near $n$.\n\nIn a
second part\, we initiate a study\, when $F$ is a general APN function\, o
f the Boolean function $\\gamma_F$ related to the differential spectrum of
$F$ (which is known to be bent if and only if $F$ is almost bent). We cha
racterize its linear structures and specify nonexistence cases\; we show\,
for $n$ even\, their relation with the bent components of $F$. We charac
terize further in terms of $\\gamma_F$ the fact that a component function
of $F$ is bent and study if the number of bent components can be optimal.
We study more deeply the relation between the Walsh transform of $\\gamma_
F$ and the Walsh transform of $F$. By applying the Titsworth relation to t
he Walsh transform $W_{\\gamma_F}$\, we deduce a very simple new relation
satisfied by $W_F^2$. From this latter relation\, we deduce\, for a sub-cl
ass of APN functions\, a lower bound on the nonlinearity\, which is signif
icantly stronger than $nl(F)>0$ (the only general known bound). This sub-c
lass of APN functions includes all known APN functions. We finally show h
ow the nonlinearities of $\\gamma_F$ and $F$ are related by a simple formu
la.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nurdagul Anbar Meidl (Sabanci University)
DTSTART:20210303T170000Z
DTEND:20210303T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/22
DESCRIPTION:Title: On nilpotent automorphism groups of function fields\nby
Nurdagul Anbar Meidl (Sabanci University) as part of Carleton Finite Field
s eSeminar\n\n\nAbstract\nIn this talk\, we give a new result on the autom
orphisms of a function field of genus $g\\geq 2$ over an algebraically clo
sed field of positive characteristic $p$. More precisely\, we show that th
e order of a nilpotent subgroup $G$ of its automorphism group is bounded b
y $16(g-1)$ when $G$ is not a $p$-group. We observe that if $|G|=16(g-1)$\
, then $(g-1)$ is a power of $2$. Furthermore\, we provide an infinite fam
ily of function fields attaining the bound. \n\nThis is a joint work with
Bur\\c{c}in G\\"{u}ne\\c{s}.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Markus Grassl (ICTQT Gdansk)
DTSTART:20210317T160000Z
DTEND:20210317T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/23
DESCRIPTION:Title: Algebraic Quantum Codes: New challenges for classical coding
theory?\nby Markus Grassl (ICTQT Gdansk) as part of Carleton Finite F
ields eSeminar\n\n\nAbstract\nThe talk will discuss connections between qu
antum error-correcting codes (QECCS) and algebraic coding theory. A quantu
m error-correcting code is a subspace of a complex Hilbert space that allo
ws to protect quantum information against certain errors. Using the so-cal
led stabilizer formalism\, we illustrate how QECCs can be constructed usin
g techniques from algebraic coding theory. We will also present some open
problems in classical coding theory that are motivated by the link to quan
tum error-correcting codes. The talk includes a short introduction to the
relevant concepts of quantum mechanics.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivelisse Rubio (UPR Rio Piedras)
DTSTART:20210331T160000Z
DTEND:20210331T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/24
DESCRIPTION:Title: On Multidimensional Periodic Arrays\nby Ivelisse Rubio (
UPR Rio Piedras) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\
nMultidimensional periodic arrays have applications for encoding data duri
ng digital communication or storage. In many applications the arrays are s
tored in memory\, a burden for environments with limited resources. Hence\
, it is important to provide algebraic constructions for the arrays that a
ssure the desired properties\, are easily implemented and have small use o
f memory. In the case of sequences\, their linear complexity is an import
ant parameter\, especially for applications related to information securit
y. In this talk we describe different algebraic constructions of multidime
nsional arrays\, present a generalization of the concept of linear complex
ity\, and analyze the multidimensional linear complexity of several types
of periodic arrays.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Canteaut (INRIA)
DTSTART:20210414T160000Z
DTEND:20210414T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/25
DESCRIPTION:Title: Recovering or Testing Extended-Affine Equivalence\nby An
ne Canteaut (INRIA) as part of Carleton Finite Fields eSeminar\n\n\nAbstra
ct\nExtended Affine (EA) equivalence is the equivalence relation between\n
two vectorial Boolean functions $F$ and $G$ such that there exist\ntwo aff
ine permutations $A$\, $B$\, and an affine function $C$\nsatisfying $G = A
\\circ F \\circ B + C$. While a priori simple\, it is\nvery difficult in
practice to test whether two functions are\nEA-equivalent. This problem h
as two variants: EA-testing deals with\nfiguring out whether the two funct
ions can be EA-equivalent\, and\nEA-recovery is about recovering the tuple
$(A\,B\,C)$ if it exists.\n\nIn this talk\, we present a new efficient al
gorithm that efficiently\nsolves the EA-recovery problem for quadratic fun
ctions. Though its\nworst-case complexity is obtained when dealing with AP
N functions\,\nit supersedes all previously known algorithms in terms of\n
performance\, even in this case. This approach is based on the\nJacobian m
atrix of the functions\, a tool whose study in this context\ncan be of ind
ependent interest.\n\nIn order to tackle EA-testing efficiently\, the best
approach in\npractice relies on class invariants. We discuss a new invari
ant\nbased on the so-called ortho-derivative which is applicable to\nquadr
atic APN functions\, a specific type of functions that is of\ngreat intere
st\, and of which tens of thousands need to be sorted\ninto distinct EA-cl
asses. Our ortho-derivative-based invariant is\nboth very fast to compute\
, and highly discriminating.\n\nJoint work with Alain Couvreur and Léo Pe
rrin\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cathy Swaenepoel (University of Paris)
DTSTART:20210428T160000Z
DTEND:20210428T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/26
DESCRIPTION:Title: Trace of products in finite fields and additive double chara
cter sums\nby Cathy Swaenepoel (University of Paris) as part of Carlet
on Finite Fields eSeminar\n\n\nAbstract\n\\Let $C$ and $D$ be two subsets
of a finite field $\\F_q$ of characteristic $p$ and let $\\mathrm{Tr}$ be
the absolute trace of $\\F_q$. \n\nIn the first part of this talk\, we wil
l consider some ``interesting'' subsets $A$ of $\\F_p$ (such as singletons
or subgroups of $\\F_p^*$) and give lower bounds on $\\mathrm{card}(C)$ a
nd $\\mathrm{card}(D)$ to ensure that $\\mathrm{Tr}(CD)\\cap A\\neq \\empt
yset$.\nOur method allows us to obtain explicit and optimal results (up to
an absolute constant factor). \nSome estimates lead to interesting combin
atorial\nquestions.\n\nIn the second part which is a joint work with Arne
Winterhof\, we will see that if $D$ has some desirable structure then ther
e is a large subset $U$ of $D$ for which the standard upper bound on the a
dditive double character sum $\\sum_{(c\,u)\\in C \\times U} \\psi(cu)$ ca
n be improved. \nThe proof uses a decomposition theorem of Roche-Newton\,
Shparlinski and Winterhof.\nThis new bound allows us to improve one of the
results presented in the first part of the talk as well as a result of Gy
armati and S\\'ark\\"ozy (provided that one of the involved sets has some
desirable structure).\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gary McGuire (University College Dublin)
DTSTART:20210602T160000Z
DTEND:20210602T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/27
DESCRIPTION:Title: Linear Fractional Transformations and Irreducible Polynomial
s over Finite Fields\nby Gary McGuire (University College Dublin) as
part of Carleton Finite Fields eSeminar\n\n\nAbstract\nWe will discuss pol
ynomials over a finite field where linear fractional\n transformations per
mute the roots. For subgroups G of PGL(2\,q) we will\n demonstrate some co
nnections between factorizations of certain polynomials\n into irreducible
polynomials over Fq\, and the field of G-invariant\n rational functions.
Some unusual patterns in the factorizations are explained by\n this connec
tion.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Pott (Otto-von-Guericke-University Magdeburg)
DTSTART:20210707T160000Z
DTEND:20210707T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/28
DESCRIPTION:Title: Relaxations of almost perfect nonlinearity\nby Alexander
Pott (Otto-von-Guericke-University Magdeburg) as part of Carleton Finite
Fields eSeminar\n\n\nAbstract\n(Note: the abstract here was transcribed by
the organizer\, and originally included references I did not include here
. Please see the original on the seminar webpage for the references) \n\nA
function $f : \\mathbb{F}_2^n → \\mathbb{F}_2^n$ is called \\emph{almos
t perfect nonlinear} (APN) if $f(x + a) + f(x) = b$ for all $a\, b$ has at
most $2$ solutions. One may also formulate this as follows: there is no $
4$-set $\\{x\, y\, z\, w\\} \\in \\mathbb{F}_2^n$ \n\\[ f(x) + f(y) + f(z)
+ f(w) = 0 \\]\nwhich is sometimes called the Rodier condition.\n\nSevera
l relaxations of APN functions have been introduced: a function $f$ is cal
led partially\nAPN if $f(y) + f(z) + f(y + z) \\neq 0$ for all $y\, z \\ne
q 0$\, $y \\neq z$. That means that the APN\nproperty is satisfied for $x
= 0$ only. Another popular relaxation are differentially $4$-uniform\nfun
ctions where $f(x + a) + f(x) = b$ has at most 4 solutions.\n\nIn my talk\
, I will discuss the question about the number of $4$-sets $\\{x\, y\, z\,
w\\} \\in \\mathbb{F}_2^n$ such that $f(x) + f(y) + f(z) + f(w) = 0$ for
certain functions $f \\colon \\mathbb{F}_2^n \\to \\mathbb{F}_2^m$ where
$m \\leq n$.\nThis gives rise to a design theoretic interpretation of the
APN property and can be used\nto show\, in a purely combinatorial way\, th
at partially APN permutations exist for all $n$.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emina Soljanin (Rutgers University)
DTSTART:20210804T160000Z
DTEND:20210804T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/29
DESCRIPTION:Title: Codes\, Graphs\, and Hyperplanes in Data Access Service\
nby Emina Soljanin (Rutgers University) as part of Carleton Finite Fields
eSeminar\n\n\nAbstract\nDistributed computing systems strive to maximize t
he number of concurrent data access requests they can support with fixed r
esources. Replicating data objects according to their relative popularity
and access volume helps achieve this goal. However\, these quantities are
often unpredictable. Erasure-coding has emerged as an efficient and robust
form of redundant storage. In erasure-coded models\, data objects are ele
ments of a finite field\, and each node in the system stores one or more l
inear combinations of data objects. This talk asks 1) which data access ra
tes an erasure-coded system can support and 2) which codes can support a s
pecified region of access rates. We will address these questions by castin
g them into some known and some new combinatorial optimization problems on
graphs. We will explain connections with batch codes. This talk will also
describe how\, instead of a combinatorial\, one can adopt a geometric app
roach to the problem.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Bors (Carleton University)
DTSTART:20210929T160000Z
DTEND:20210929T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/30
DESCRIPTION:Title: Cycle types of complete mappings\nby Alexander Bors (Car
leton University) as part of Carleton Finite Fields eSeminar\n\n\nAbstract
\nA complete mapping of a finite field $K$ is a bijective function $f:K\\r
ightarrow K$ 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 eac
h given length a complete mapping can decompose.\n\nIn this talk\, I will
present results that were achieved recently in collaboration with Qiang Wa
ng (also from Carleton University) and which concern the cycle types of co
mplete mappings in two important classes of functions on finite fields: cy
clotomic 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.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tor Helleseth (University of Bergen)
DTSTART:20211201T170000Z
DTEND:20211201T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/32
DESCRIPTION:Title: The history of the cross correlation between m-sequences: an
overview\nby Tor Helleseth (University of Bergen) as part of Carleton
Finite Fields eSeminar\n\n\nAbstract\nMaximum-length sequences (or m-sequ
ences) of period 2^m-1 are\ngenerated by linear feedback shift registers w
ith primitive\ncharacteristic polynomials of degree m. These sequences hav
e\nmany important applications in modern communication systems.\nThe most
well-known property of m-sequences is their two-level\nideal autocorrelati
on. The first major result on the cross\ncorrelation of two different m-se
quences of the same period\nwas published by Gold back in January 1968 and
the result was\nused in constructing the famous family of Gold sequences.
\nDuring more than 50 years the cross correlation between\nm-sequences of
the same period has been intensively studied\nby many research groups. Man
y results have been obtained but\nstill many open problems remain in this
area. This talk will\ngive an updated survey of the status of the cross co
rrelation\nof m-sequences as well as some consequences of these results.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ariane Masuda/Juliane Capaverde (New York City College of Technolo
gy)
DTSTART:20211020T160000Z
DTEND:20211020T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/33
DESCRIPTION:Title: Redei permutations with the same cycle structure\nby Ari
ane Masuda/Juliane Capaverde (New York City College of Technology) as part
of Carleton Finite Fields eSeminar\n\n\nAbstract\nPermutation polynomials
over finite fields have been extensively\nstudied over the past decades.
Among the major challenges in this\narea are the questions concerning thei
r cycle structures as they capture\nrelevant properties\, both theoretical
ly and practically.\n\nIn this talk we focus on a family of permutation po
lynomials\, the so called Rédei permutations. Although their cycle struct
ures are known\, there are other related questions that can be investigate
d. For example\, when do two Rédei permutations have the same cycle struc
ture? We give a characterization of such pairs\, and present explicit fami
lies\nof Rédei permutations with the same cycle structure. We also discus
s some results regarding Rédei permutations with a particularly simple cy
cle structure\, consisting of $1$- and $j$-cycles only\, when $j$ is $4$ o
r a prime number. The case $j = 2$ is specially important in some applicat
ions. We completely describe Rédei involutions with a prescribed cycle st
ructure\, and show that remarkably the only Rédei permutations with a uni
que cycle structure are the involutions.\n\nThis is joint work with Virgí
nia Rodrigues from Universidade Federal do Rio Grande do Sul.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabio Enrique Brochero Martínez (Federal University of Minas Gera
is)
DTSTART:20211103T160000Z
DTEND:20211103T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/34
DESCRIPTION:Title: The functional graph of some family of functions over finite
fields\nby Fabio Enrique Brochero Martínez (Federal University of Mi
nas Gerais) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nLet
$\\mathbb F_q$ be the finite field with $q=p^s$ elements and $f: \\mathbb
F_q\\to \\mathbb F_q$ be a function. The functional graph of $f$ is the
directed graph $G_f=(\\mathcal V\, \\mathcal E)$\, where $\\mathcal V=\\ma
thbb F_q$ and $\\mathcal E=\\{(x\,f(x))\\mid x\\in\\mathbb F_q\\}$. The ch
aracteristics of functional graphs (number of cycles\, cycle lengths\, pre
-cycle lengths and so on) have been studied for several different maps ove
r finite fields\, due to its applications in cryptography.\n\nIn this pres
entation we will present two independent results: the first one we descr
ibe completely the dynamics of the maps $f(x)=x^{q+1}\\pm x^2$ over the fi
nite field $\\mathbb F_{q^2}$ and in the second we study the functional g
raph of maps of the form $f(x)= x^n h( x^{(q-1)/m})$\, where $h$ satisfies
an special condition.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgos Kapetanakis (University of Thessaly)
DTSTART:20211215T170000Z
DTEND:20211215T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/35
DESCRIPTION:Title: The existence of Fq-primitive points on curves using freenes
s\nby Giorgos Kapetanakis (University of Thessaly) as part of Carleton
Finite Fields eSeminar\n\n\nAbstract\nAn element of a finite cyclic group
of order $Q$\, $C_Q$\, is called\n$r$-free (where $r|Q$)\, if it is not a
$p$-th power of any group element for any prime divisor $p$ of $r$. We in
troduce the set\nof $(r\,n)$-free elements of $C_Q$\, where $n|Q$ and $r|(
Q/n)$\, as the\nelements of the subgroup $C_{Q/n}$ that are $r$-free withi
n $C_{Q/n}$.\nInspired by Vinogradov's expression for the characteristic\n
function of primitive elements of the finite field Fq\, we prove\nan analo
gue for the $(r\,n)$-free elements of $C_Q$ and obtain a\nlower bound for
the number of elements $b$ of Fq\, such that $f(b)$\nis $(r\,n)$-free and
$F(b)$ is $(R\,N)$-free\, where $f$ and $F$ are\npolynomials over Fq.\n\nA
s an application\, we consider the problem of the existence of\npoints of
elliptic curves in Fq^2\, whose coordinates are both\nprimitive and provid
e a complete answer for the curves $y^2=x^3±x$.\n\nThis is joint work wit
h Stephen D. Cohen and Lucas Reis.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetla Petkova-Nikova (KU Leuven)
DTSTART:20211013T160000Z
DTEND:20211013T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/36
DESCRIPTION:Title: Threshold Cryptography against Combined Physical Attacks
\nby Svetla Petkova-Nikova (KU Leuven) as part of Carleton Finite Fields e
Seminar\n\n\nAbstract\nRecent attacks show that there is a need for protec
ting implementations jointly against side-channel and fault attacks. Analo
gously\, modern\nMPC protocols consider active security\, i.e. against mal
icious parties\nwhich do not only passively eavesdrop but also actively de
viate from\nthe protocol. This provides an opportunity for the field of th
reshold implementations to evolve with MPC and achieve provable secure imp
lementations against combined passive and active physical attacks.\n\nIn t
his talk we will first introduce Threshold Implementations applied to\npro
tect various ciphers against SCA and the like with Boolean functions\nand
MPC/SSS. After that we will discuss two recent proposals for combined\ncou
ntermeasures: CAPA and M&M\, which both start from passively secure\nthres
hold schemes and extend those with information-theoretic MAC tags\nfor pro
tection against active adversaries. While similar in their most\nbasic str
ucture\, the two proposals explore very different adversary models\nand th
us employ completely different implementation techniques. CAPA\nconsiders
the field-probe-and-fault model\, which is the embedded analogue\nof multi
ple parties jointly computing a function with at least one of the parties
honest. Accordingly\, CAPA is strongly based on the actively secure MPC pr
otocol SPDZ and inherits its provable security properties in this model. S
ince this results in very expensive implementations\, M&M works in a simil
ar but more realistic adversary model and uses existing building blocks fr
om previous passively secure implementations to build more efficient activ
ely secure threshold cryptography.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhicheng (Jason) Gao (Carleton University)
DTSTART:20220131T170000Z
DTEND:20220131T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/37
DESCRIPTION:Title: Some recent results on counting polynomials over $\\fq$ with
prescribed coefficients using the generating function approach\nby Zh
icheng (Jason) Gao (Carleton University) as part of Carleton Finite Field
s eSeminar\n\n\nAbstract\nCounting/estimating some families of polynomials
over $\\fq$ with\nprescribed coefficients has attracted much attention in
the past\n30 years. Three well-known problems are:\n\n(a) existence of ir
reducible polynomials with prescribed coefficients\;\n\n(b) counting irred
ucible polynomials with prescribed leading and/or ending coefficients\;\n\
n(c) counting polynomials with prescribed leading coefficients and\nwith a
given number of roots in a prescribed set. This is closely\nrelated to th
e distance distribution over Reed-Solomon codes.\n\nMost of the published
results about these problems used the character\napproach and Weil's bound
on character sums. In this talk\, I will\ndescribe the generating functio
n approach which leads to some new\nresults in these areas. The generating
functions use the group\nalgebra defined on the group of equivalence clas
ses of polynomials\nwith prescribed leading and /or ending coefficients.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksandr Tuxanidy (Carleton University)
DTSTART:20220214T170000Z
DTEND:20220214T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/38
DESCRIPTION:Title: Equidistribution estimates for palindromic numbers in residu
e classes and applications\nby Aleksandr Tuxanidy (Carleton University
) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nThis talk conc
erns palindromic integers and discusses newly-derived\naverage equidistrib
ution estimates for these in residue classes\nto large moduli. As an appli
cation of this and well-known facts\nfrom sieve theory\, we obtain the fol
lowing:\n\n(1) In any given base\, there are infinitely many palindromic\n
integers having at most six prime divisors.\n\n(2) The density of the prim
e numbers among the base-b palindromes\nat most X is O(1/log X)\, as expec
ted by randomness heuristics. This\nanswers a problem raised by Banks-Hart
-Sakata (2004)\, later proved\nby Col (2009).\n\nWe also make a few remark
s on some related problems in finite fields.\n\nThis is joint work with D.
Panario and Q. Wang\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Magali Bardet (University of Rouen)
DTSTART:20220307T170000Z
DTEND:20220307T180000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/39
DESCRIPTION:Title: Algebraic decoding of Fqm-linear codes in rank metric\nb
y Magali Bardet (University of Rouen) as part of Carleton Finite Fields eS
eminar\n\n\nAbstract\nRank-metric code-based cryptography relies on the ha
rdness of decoding a random linear code in the rank metric. This fundament
al problem is called the Minrank problem\, and is ubiquitous in rank metri
c (or even Hamming metric) code based cryptography as well as in multivari
ate cryptography. For structured instances arising in the former\, their s
ecurity rely on a more specific problem\, namely the Rank Syndrome Decodin
g problem. There is also a generalization called the Rank Support Learning
problem\, where the attacker has access to several syndromes correspondin
g to errors with the same support. Those problems have various application
s in code-based and multivariate cryptography (KEM and signature schemes)\
, and a precise understanding of the complexity of solving them can help d
esigners to create secure parameters.\n\nIn this talk\, I will present the
three problems and their relations to cryptographic schemes\, their algeb
raic modeling and the recent improvements in the understanding of the comp
lexity of solving those systems using algebraic techniques like Gröbner b
ases computations.\n\nThis gathers joint works with P. Briaud\, M. Bros\,
D. Cabarcas\, P. Gaborit\, V. Neiger\, R. Perlner\, O. Ruatta\, D. Smith-T
one\, J.-P. Tillich\, J. Verbel.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariana Perez (Universidad Nacional de Hurlingham and Conicet)
DTSTART:20220321T160000Z
DTEND:20220321T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/40
DESCRIPTION:Title: Families of diagonal equations over finite fields: estimates
and applications\nby Mariana Perez (Universidad Nacional de Hurlingha
m and Conicet) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nI
n this work\, we study the set of $\\mathbb{F}_q$--rational solutions\, th
at is\, solutions with coordinates in the finite field $\\mathbb{F}_q$ of
$q$ elements\, of certain equations and systems defined by families of di
agonal equations with coefficients in $\\mathbb{F}_q$. In \\cite{1} and \\
cite{2} we obtain explicit estimates and results that guarantee the exist
ence of at least an $\\mathbb{F}_q$--rational solution of these families\,
by studying geometric properties of the varieties that define these equat
ions. The results obtained complement those existing in the literature (se
e \\cite{3}).\nFinally we apply these results to a generalization of Wari
ng's\nproblem and the distribution of solutions of congruences modulo a pr
ime number.\n \n\n \\bibitem{1} M. Pérez and M. Privitelli. Estimates on
the number of rational solutions of variants of diagonal equations over f
inite fields\, Finite Fields and Appl. 68 (2020)\, 30 pp.\n\n \\bibitem{2}
M. Pérez and M. Privitelli. On the number of solutions of systems of cer
tain diagonal equations over finite fields. Journal of Number Theory (2021
). \n\n\\bibitem {3} Gary L. Mullen and D. Panario. Handbook of Finite Fie
lds (1st ed.) . Chapman and Hall/CRC\, 2013.\n \n\\end{thebibliography}\n\
n\nThis talk is based on a joint work with Melina Privitelli.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/40/
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SUMMARY:Alfred Wassermann (University of Beyreuth)
DTSTART:20220411T160000Z
DTEND:20220411T170000Z
DTSTAMP:20260422T212710Z
UID:CarletonFiniteFields/41
DESCRIPTION:Title: Designs in Classical Polar Spaces\nby Alfred Wassermann
(University of Beyreuth) as part of Carleton Finite Fields eSeminar\n\n\nA
bstract\nCombinatorial designs have been studied since the 19th century an
d have\nfamous applications in the design of experiments and in coding the
ory.\n50 years ago\, Cameron\, Delsarte and Ray-Chaudhury introduced the n
otion\nof subspace designs\, also known as q-analogs of designs or designs
over finite fields.\nRoughly speaking\, q-analogs of objects arise from t
heir combinatorial counterparts by\nreplacing subsets by subspaces and car
dinalities by dimensions.\nThe first "true" subspace designs\, i.e. design
s with t > 1\,\nwere presented by Thomas only in 1987.\nA next natural gen
eralization of subspace designs are designs\nin polar spaces. For t=1 thes
e objects are known as spreads.\nFor t>1 the first - non-trivial - such de
signs were found by\nDe Bruyn and Vanhove in 2013\, some more designs appe
ared recently in the\nPhD thesis of Landsdown.\n\nIn this talk we will giv
e an overview on the few known structural results\nfor designs in classica
l polar spaces and present quite a few new parameters\nof existing designs
found by computer search.\n
LOCATION:https://researchseminars.org/talk/CarletonFiniteFields/41/
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