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BEGIN:VEVENT
SUMMARY:Anna-Lena Horlemann (University of St. Gallen)
DTSTART:20200902T160000Z
DTEND:20200902T170000Z
DTSTAMP:20260404T110745Z
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 pa
rt of Carleton Finite Fields eSeminar\n\n\nAbstract\nWe show that the sequ
ence of dimensions of the linear spaces\, generated by a given (finite fie
ld) rank-metric code together with itself under several applications of a
field automorphism\, is an invariant for the whole equivalence class of th
e code. The same property is proven for the sequence of dimensions of the
intersections 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 n
umber of equivalence classes of rank-metric codes\, derive new characteriz
ations of the well-known Gabidulin codes\, and show that certain code cons
tructions actually lead to equivalent codes.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Cohen (University of Glasgow)
DTSTART:20201007T160000Z
DTEND:20201007T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/2
DESCRIPTION:Title: Existence theorems for $r$-primitive elements in finit
e fields\nby Stephen Cohen (University of Glasgow) as part of Carleton
Finite Fields eSeminar\n\n\nAbstract\nLet $r|q-1$. An element of $\\math
bb{F}_q$ 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 o
f a primitive element of $\\mathbb{F}_q$. We describe some existence theor
ems for general $r$-primitive elements and\, in particular\, analogues f
or $2$-primitive elements of the following {\\em complete} existence theor
ems for primitive elements. \n\n(Theorem A (1990).) For any $n \\geq 2$ a
nd $a\\in \\mathbb{F}_q$ (necessarily with $a \\neq 0$ if $n=2$) there exi
sts a primitive $\\alpha \\in \\mathbb{F}_{q^n}$ with trace $a$ over $\\
mathbb{F}_q$\, except when $a=0\, n=3\, q=4$.\n\n(Theorem B (1983).)
Every line in $\\mathbb{F}_{q^2}$ contains a primitive element. \n (A lin
e in $\\mathbb{F}_{q^2}$ is a set of the form $\\{\\beta(\\gamma+a): a \\
in \\mathbb{F}_q\\}$\, for some nonzero $\\beta \\in \\mathbb{F}_{q^2}\,
\\gamma \\in \n\\mathbb{F}_{q^2} \\setminus \\mathbb{F}_q$.\n\nJoint work
with Giorgos Kapetanakis.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Jedwab (Simon Fraser University)
DTSTART:20201104T170000Z
DTEND:20201104T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/3
DESCRIPTION:Title: Packings of partial difference sets\nby Jonathan J
edwab (Simon Fraser University) as part of Carleton Finite Fields eSeminar
\n\n\nAbstract\nPartial difference sets are highly structured group subset
s that occur in various guises throughout design theory\, finite geometry\
, coding theory\, and graph theory. They admit only two possible nontrivia
l character sums and so are often studied using character theory. The cent
ral question is to determine which groups contain a partial difference set
with two specified nontrivial character sums. We consider an apparently m
ore difficult question: which groups contain a large disjoint collection o
f such partial difference sets? This leads us to identify a certain subgro
up as containing important structural information about the packing. With
this insight\, we are able to formulate a recursive construction of packin
gs in abelian groups of increasing exponent. This allows us to unify and e
xtend numerous previous results about partial difference sets using a comm
on framework.\n\nThis is joint work with Shuxing Li\, a 2019-2021 PIMS Pos
tdoctoral Fellow.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shuxing Li (Simon Fraser University)
DTSTART:20200826T160000Z
DTEND:20200826T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/4
DESCRIPTION:Title: Intersection distribution and its applications\nby
Shuxing Li (Simon Fraser University) as part of Carleton Finite Fields eS
eminar\n\n\nAbstract\nGiven a polynomial f over finite field Fq\, its inte
rsection distribution concerns the collective behaviour of a collection of
polynomials {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 interse
ction distribution of f reflects how the point set S_f interacts with the
lines in PG(2\,q). Motivated by the long-standing open problem of classify
ing oval monomials\, which are over F_2^n having the same intersection dis
tribution as x^2\, we consider the next simplest case: classifying all mon
omials over Fq having the same intersection distribution as x^3. Some char
acterizations of such monomials are derived and as a consequence\, a conje
ctured complete list of such monomials is proposed. As an application\, we
observe that every monomial over F_3^n with the same intersection distrib
ution as x^3 naturally leads to a Steiner triple system. Interestingly\, n
ew examples of Steiner triple systems\, which are nonisomorphic to the cla
ssical ones\, are obtained. This is joint work with Gohar Kyureghyan and A
lexander Pott.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qi Cheng (Oklahoma University)
DTSTART:20200819T160000Z
DTEND:20200819T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/5
DESCRIPTION:Title: The discrete logarithm over Kummer and Artin-Schreier
extensions\nby Qi Cheng (Oklahoma University) as part of Carleton Fini
te Fields eSeminar\n\n\nAbstract\nMany cryptography protocols rely on hard
computational number theoretical problems for security. The discrete loga
rithm problem over finite fields or elliptic curves is one of the most imp
ortant candidates\, besides the integer factorization problem. In this tal
k\, I will first survey several algorithms attacking the discrete logarith
ms over finite fields\, starting from generic algorithms and the index cal
culus. My discussion will then be focusing on the of quasi-polynomial-time
descending\, and its application on the Kummer and Artin-Schreier extensi
ons.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucas Reis (Federal University of Minas Gerais)
DTSTART:20200812T160000Z
DTEND:20200812T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/6
DESCRIPTION:Title: Character sum estimates over affine spaces applied to
existence results in finite fields\nby Lucas Reis (Federal University
of Minas Gerais) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\
nIn this talk\, we will discuss the problem of estimating the sum of a mul
tiplicative character over the elements of an affine space. We present a n
ew non-trivial bound on such sums\, along with some applications. In parti
cular\, we provide asymptotically sharp results on the existence of specia
l primitive elements in finite fields.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Baldi (Università Polytecnica delle Marche)
DTSTART:20200729T160000Z
DTEND:20200729T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/7
DESCRIPTION:Title: QC-LDPC codes\, QC-MDPC codes and their use in post-qu
antum cryptography\nby Marco Baldi (Università Polytecnica delle Marc
he) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nLow-density
parity-check (LDPC) codes are a family of modern error correcting codes ex
ploiting a random-based design and iterative decoding algorithms allowing
them to approach the channel capacity. The structured subclass of LDPC cod
es characterized by quasi-cyclicity (QC)\, named QC-LDPC codes\, is known
to achieve practically the same performance as general LDPC codes while en
abling more compact representation and easier implementation. The use of Q
C-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 overc
oming the main limitations of the original McEliece cryptosystem based on
Goppa codes. Using QC-LDPC and QC-MDPC codes in cryptography\, however\, p
oses some new challenges with respect to their classical use for data reli
ability. Nevertheless\, variants of the McEliece and Niederreiter cryptosy
stems based on these codes are now under consideration by NIST within the
standardization process of new post-quantum cryptographic primitives. The
seminar will recall the basics of QC-LDPC and QC-MDPC codes and then descr
ibe the main cryptographic primitives relying on these codes\, along with
some open research challenges in this area.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Matera (Universidad Nacional de General Sarmiento)
DTSTART:20200722T160000Z
DTEND:20200722T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/8
DESCRIPTION:Title: The distribution of factorization patterns on nonlinea
r families of univariate polynomials over a finite field\nby Guillermo
Matera (Universidad Nacional de General Sarmiento) as part of Carleton Fi
nite Fields eSeminar\n\n\nAbstract\nIn this talk we discuss an estimate on
the number |A_λ| of elements on a nonlinear family A of monic polynomial
s of Fq[T] of degree r having a given factorization pattern λ. We show th
at |A_λ| = T(λ) q^{r−m} + O(q^{r−m−1/2})\, where T(λ) is the prop
ortion of elements of the symmetric group of r elements with cycle pattern
λ and m is the codimension of A. We provide explicit upper bounds for th
e constants underlying the O-notation in terms of λ and A with "good" beh
avior. Finally\, we apply these results to analyze the average-case comple
xity of the classical factorization algorithm restricted to the family A\,
showing 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://stable.researchseminars.org/talk/CarletonFiniteFields/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alev Topuzoglu (Sabanci University)
DTSTART:20200715T160000Z
DTEND:20200715T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/9
DESCRIPTION:Title: On the arithmetic of sequences of permutation polynomi
als\nby Alev Topuzoglu (Sabanci University) as part of Carleton Finite
Fields eSeminar\n\n\nAbstract\nIn this talk\, we will present recent resu
lts on factorization of a large class of permutation polynomials. We also
discuss sequences and iterations of permutation polynomials. In particular
\, we address various problems concerning number theoretic properties of i
rreducible factors of terms of such sequences. This is based on joint work
with Tekgul Kalayci and Henning Stichtenoth.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francisco Rodriguez-Henriquez (CINVESTAV-IPN)
DTSTART:20200520T160000Z
DTEND:20200520T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/10
DESCRIPTION:Title: Parallel strategies for SIDH: towards computing SIDH
twice as fast\nby Francisco Rodriguez-Henriquez (CINVESTAV-IPN) as par
t of Carleton Finite Fields eSeminar\n\n\nAbstract\nOver the last ten year
s there has been an intense research to find hard mathematical problems th
at would be presumably hard to solve by a quantum attacker and at the same
time could be used to build reasonably efficient public-key cryptoschemes
. One such proposal is the hardness of finding an isogeny map between two
elliptic curves. This proposal has spawned a new line of research generall
y known as isogeny-based cryptography. One salient feature of all isogeny-
based protocols proposed up-to-date is that they require exceptionally sho
rt key sizes. However\, the latency associated to those protocols is highe
r than the ones reported by other post-quantum cryptosystem proposals. In
this talk we present novel strategies and concrete algorithms for the para
llel computation of the Supersingular Isogeny-based Diffie-Hellman key exc
hange (SIDH) protocol when executed on multi-core platforms. To our knowle
dge\, the work presented here is the first reported multi-core implementat
ion of SIDH.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Lisonek (Simon Fraser University)
DTSTART:20200624T160000Z
DTEND:20200624T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/11
DESCRIPTION:Title: Contextual hypergraphs\nby Petr Lisonek (Simon Fr
aser University) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\
nContextuality is one of the features that distinguishes quantum mechanics
from classical mechanics. There are several ways to formalize contextuali
ty mathematically. One such formalization consists of a hypergraph whose v
ertices are labelled by Hermitian operators such that\, for each hyperedge
\, certain conditions are fulfilled by the operators occurring in it. A co
ntextual hypergraph is one that admits such vertex labeling. The goal of o
ur work is to construct large (preferably infinite) families of contextual
hypergraphs. Historically\, contextual hypergraphs have been found mostly
by computational searches and ad-hoc constructions. In our work we aim at
computer-free\, systematical constructions\, which use combinatorial ingr
edients such as difference matrices and finite geometries. Finite fields p
lay a central role in obtaining these ingredients. We use appropriate grou
p actions to ensure that our contextual hypergraphs are vertex-transitive\
, which is recognized as an added value in the quantum mechanics applicati
ons. The talk does not require any knowledge of quantum physics. This is j
oint work with Stefan Trandafir.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luciane Quoos (Federal University of Rio de Janeiro)
DTSTART:20200617T160000Z
DTEND:20200617T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/12
DESCRIPTION:Title: Locally recoverable codes\nby Luciane Quoos (Fede
ral University of Rio de Janeiro) as part of Carleton Finite Fields eSemin
ar\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 o
f a small subset of other coordinates. When we have $\\delta$ non-overlapp
ing subsets of cardinality $r_i$ that can be used to recover the missing c
oordinate we say that a linear code $\\cC$ with length $n$\, dimension $k$
\, minimum distance $d$ has $(r_1\,\\ldots\, r_\\delta)$-locality and den
ote by $[n\, k\, d\; r_1\, r_2\,\\dots\, r_\\delta].$ In this talk\, I wil
l present a new upper bound for the minimum distance of these codes and pr
opose a construction of $[n\, k\, d\; r_1\, r_2\,\\dots\, r_\\delta]$-code
s on function fields of genus $g \\geq 1$. This is joint work with Daniele
Bartoli and Maria Montanucci.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lilya Budaghyan (University of Bergen)
DTSTART:20200610T160000Z
DTEND:20200610T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/13
DESCRIPTION:Title: Optimal cryptographic functions over finite fields\nby Lilya Budaghyan (University of Bergen) as part of Carleton Finite Fi
elds eSeminar\n\n\nAbstract\nFunctions over finite fields are used in cryp
tography\, in particular in block ciphers. An important condition on these
functions is a high resistance to the differential and linear cryptanalys
es\, which are among the main attacks on block ciphers. The functions whic
h possess the best resistance to the differential attack are called almost
perfect nonlinear (APN). Planar\, bent and almost bent (AB) functions are
those mappings which oppose an optimum resistance to both linear and diff
erential attacks. An interesting fact is that planar\, bent\, APN and AB f
unctions also define optimal objects in other domains of mathematics and i
nformation theory such as coding theory\, finite geometry\, sequence desig
n\, algebra\, combinatorics\, et al. In this talk we will discuss problems
and recent advances in construction and analysis of these functions.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arne Winterhof (Austrian Academy of Sciences)
DTSTART:20200603T160000Z
DTEND:20200603T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/14
DESCRIPTION:Title: On the distribution of the Rudin-Shapiro function for
finite fields\nby Arne Winterhof (Austrian Academy of Sciences) as pa
rt of Carleton Finite Fields eSeminar\n\n\nAbstract\nSee https://people.ma
th.carleton.ca/~finitefields/Files/Arne_abstract.pdf\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felice Manganiello (Clemson University)
DTSTART:20200527T160000Z
DTEND:20200527T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/15
DESCRIPTION:Title: Graphs and finite fields in modern communications
\nby Felice Manganiello (Clemson University) as part of Carleton Finite Fi
elds eSeminar\n\n\nAbstract\nThe origin of communication is based on the c
oncept of two users exchanging information with each other over a single c
hannel. The problem of perfect communication over a channel was modeled by
Shannon in the late 40s. More modern communication networks are not so re
strictive though. Most of the networks we use nowadays\, connect multiple
parties and graphs can be exploited to represent these networks. The quest
ion we are going to investigate in this seminar is simple: given a graph r
epresenting a network\, what is its capacity\, meaning how much informatio
n can be sent through it\, and by which communication protocol over a fini
te field? This question has been already answered for unicast networks\, m
eaning networks between a singe source and a single receiver\, and for mul
ticast networks\, meaning networks used by a source to communicate simulta
neously to multiple receivers. The capacity of communication for most netw
orks with multiple sources is still an open question. Networks of this typ
e are characterized by interference that is represented by the messages se
nt by undesired sources. A communication strategy has to be determined in
order to remove the interference. We will focus our work on multiple unica
st networks and look at the effectiveness of a practice known as interfere
nce alignment. We will define the concepts of linear capacity region of a
network and discover that the points of this region are in relation with t
he solutions of a system of bilinear of equation. Solving such a system is
know to be hard in general\, so we will finally find the points of this r
egion that are achievable by means of Gaussian elimination.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Bindel (University of Waterloo)
DTSTART:20201118T170000Z
DTEND:20201118T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/16
DESCRIPTION:Title: A status update on NIST's post-quantum standardizatio
n effort\nby Nina Bindel (University of Waterloo) as part of Carleton
Finite Fields eSeminar\n\n\nAbstract\nIf a general-purpose quantum compute
r can be built\, it will break most widely-deployed public-key cryptograph
y. To prepare for this risk\, the cryptographic community is busily design
ing new cryptographic systems. Furthermore\, the (US-American) National In
stitute for Standards and Technology (NIST) is currently aiming at standar
dizing several quantum-safe digital signature and public-key encryption sc
hemes (PKEs). Recently\, NIST announced the candidates that advance furthe
r to the third round of evaluation in NIST standardization effort. \n\nThi
s talk will first give an update on the current status of the NIST's post-
quantum standardization effort. In particular\, we will explain the timeli
ne of the ongoing project\, explain reasons for why certain schemes have b
een chosen to advance to the third round\, and what are important evaluati
on criteria during the next phase. Moreover\, we will explain how the conc
rete security of the schemes is estimated. As an example we take a closer
look at lattice-based encryption schemes. Interestingly\, most of the subm
itted PKEs are not perfectly correct schemes\, i.e.\, sometimes honestly g
enerated ciphertexts can not be encrypted correctly. Finding such a decryp
tion failure poses a security risk which will be explained in the talk as
well.\n\nShort bio:\nNina Bindel is affiliated to the Institute for
Quantum Computing (IQC) as a post doctoral researcher at the Department o
f Combinatorics & Optimization at the University of Waterloo in Waterloo\,
Ontario\, Canada.\n\nBefore joining the IQC\, she was a post doctoral res
earcher in the Cryptography and computer algebra group at TU Darmstadt whe
re she also received her Ph.D. in September 2018. Nina's research interest
is mostly in the area of cryptography that is secure even in the presence
of quantum computers\, so-called post-quantum cryptography.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herivelto Borges (University of São Paolo (São Carlos))
DTSTART:20201202T170000Z
DTEND:20201202T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/17
DESCRIPTION:Title: Algebraic curves through Fernando Torres’ lens
\nby Herivelto Borges (University of São Paolo (São Carlos)) as part of
Carleton Finite Fields eSeminar\n\n\nAbstract\nThe mathematical legacy o
f Fernando Torres is felt in several notions within the theory of curv
es over finite fields. Such notions include Weierstrass points\, Stöhr-
Voloch theory\, maximal curves\, coding theory\, and finite geometry. In
this talk\, we will highlight and briefly discuss some of Torres’ outs
tanding contributions to our mathematical community.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Katz (CSUN Northridge)
DTSTART:20200923T160000Z
DTEND:20200923T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/18
DESCRIPTION:Title: Niho's Last Conjecture\nby Daniel Katz (CSUN Nort
hridge) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nThis tal
k is concerned with character sums\, called Weil sums of\nbinomials\, that
determine the nonlinearity (Walsh spectrum) of a power\npermutation x ->
x^d of a finite field F. These Weil sums also\ndetermine the crosscorrel
ation spectrum for a pair of maximum length\nlinear recursive sequences an
d the weight distribution of a cyclic code.\nIn each case\, the binomial i
nvolved is of the form x^d-cx\, and one\nobtains values of the Walsh spect
rum by computing the various Weil sums\nas the coefficient c runs through
F. Certain exponents d\, known as Niho\nexponents\, have a simple form a
nd can produce Walsh spectra with very\nfew distinct values. The last co
njecture in Niho's 1972 thesis states\nthat a particular family of such ex
ponents produces spectra with at most\nfive distinct values. Niho's own
techniques show that one has at most\neight distinct values. Each of the
eight candidate values corresponds\nto a possible number of distinct root
s of a seventh degree polynomial on\na subset of the finite field F called
the unit circle. We use symmetry\narguments to show that it is impossib
le to have four\, six\, or seven\nroots on the unit circle: this proves Ni
ho's last conjecture.\n\nThis is joint work with Tor Helleseth and Chunlei
Li.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ray Perlner (NIST)
DTSTART:20201021T160000Z
DTEND:20201021T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/19
DESCRIPTION:Title: The MinRank problem in Cryptography and Cryptanalysis
\nby Ray Perlner (NIST) as part of Carleton Finite Fields eSeminar\n\n
\nAbstract\nThe MinRank problem\, which seeks to find a nonzero\, low-rank
linear combination of a given set of matrices\, shows up in the cryptanal
ysis of a wide variety of Multivariate and Code Based cryptosystems\, incl
uding several candidates in the National Institute of Standards and Techno
logy (NIST)’s Postquantum Cryptography Standardization Process. These in
clude the code based cryptosystems ROLLO and RQC\, (which were eliminated
from consideration for standardization after the second round due to recen
t significant improvements in the special case of the MinRank problem know
n as the Rank Syndrome Decoding problem)\, as well as the third (current)
round PQC standardization candidates Rainbow and GeMSS. This talk will dis
cuss how the MinRank problem relates to the cryptanalysis of this diverse
array of cryptosystems\, as well as recent developments that have dramati
cally improved the concrete complexity of solving the MinRank problem\, bo
th in special cases and in general.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daqing Wan (UC Irvine)
DTSTART:20210203T170000Z
DTEND:20210203T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/20
DESCRIPTION:Title: Counting solutions of large polynomial systems over f
inite fields\nby Daqing Wan (UC Irvine) as part of Carleton Finite Fie
lds eSeminar\n\n\nAbstract\nA fundamental algorithmic problem in mathemati
cs and computer science is to efficiently count the solutions of a multiva
riate polynomial system over a finite field\, and over all of its finite e
xtensions. All general algorithms so far are fully exponential in terms of
the number of equations. In a recent joint work with Q. Cheng and M. Roja
s\, 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 p
olynomial system in n variables can be defined by n+1 equations if the fie
ld is not too small.\n
LOCATION:https://stable.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:20260404T110745Z
UID:CarletonFiniteFields/21
DESCRIPTION:Title: Image sets\, nonlinearity and distance to affine func
tions of $\\delta$-uniform functions\, and $\\gamma$-functions of APN func
tions\nby Claude Carlet (University of Bergen\, Norway and University
of Paris 8\, France) as part of Carleton Finite Fields eSeminar\n\n\nAbstr
act\nWe revisit and take a closer look at a result of 2017\, showing that
the differential uniformity of any vectorial function is bounded from belo
w by an expression depending on the size of its image set. We make explici
t the resulting tight lower bound on the image set size of differentially
$\\delta$-uniform functions.\nWe improve an upper bound on the nonlinearit
y of vectorial functions obtained in the same reference and involving thei
r image set size. We study when the resulting bound is sharper than the co
vering radius bound. We obtain as a by-product a lower bound on the Hammin
g distance between differentially $\\delta$-uniform functions and affine f
unctions\, which we improve significantly with a second bound. This leads
us to study what can be the maximum Hamming distance between vectorial fu
nctions 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 funct
ion\, of the Boolean function $\\gamma_F$ related to the differential spec
trum of $F$ (which is known to be bent if and only if $F$ is almost bent).
We characterize its linear structures and specify nonexistence cases\; we
show\, for $n$ even\, their relation with the bent components of $F$. We
characterize further in terms of $\\gamma_F$ the fact that a component fu
nction of $F$ is bent and study if the number of bent components can be op
timal. We study more deeply the relation between the Walsh transform of $\
\gamma_F$ and the Walsh transform of $F$. By applying the Titsworth relati
on to the Walsh transform $W_{\\gamma_F}$\, we deduce a very simple new re
lation satisfied by $W_F^2$. From this latter relation\, we deduce\, for a
sub-class of APN functions\, a lower bound on the nonlinearity\, which is
significantly stronger than $nl(F)>0$ (the only general known bound). Thi
s sub-class of APN functions includes all known APN functions. We finally
show how the nonlinearities of $\\gamma_F$ and $F$ are related by a simpl
e formula.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nurdagul Anbar Meidl (Sabanci University)
DTSTART:20210303T170000Z
DTEND:20210303T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/22
DESCRIPTION:Title: On nilpotent automorphism groups of function fields\nby Nurdagul Anbar Meidl (Sabanci University) as part of Carleton Finit
e Fields eSeminar\n\n\nAbstract\nIn this talk\, we give a new result on th
e automorphisms of a function field of genus $g\\geq 2$ over an algebraica
lly closed field of positive characteristic $p$. More precisely\, we show
that the order of a nilpotent subgroup $G$ of its automorphism group is bo
unded by $16(g-1)$ 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 infin
ite family of function fields attaining the bound. \n\nThis is a joint wor
k with Bur\\c{c}in G\\"{u}ne\\c{s}.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Markus Grassl (ICTQT Gdansk)
DTSTART:20210317T160000Z
DTEND:20210317T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/23
DESCRIPTION:Title: Algebraic Quantum Codes: New challenges for classical
coding theory?\nby Markus Grassl (ICTQT Gdansk) as part of Carleton F
inite Fields eSeminar\n\n\nAbstract\nThe talk will discuss connections bet
ween quantum error-correcting codes (QECCS) and algebraic coding theory. A
quantum error-correcting code is a subspace of a complex Hilbert space th
at allows to protect quantum information against certain errors. Using the
so-called stabilizer formalism\, we illustrate how QECCs can be construct
ed using techniques from algebraic coding theory. We will also present som
e open problems in classical coding theory that are motivated by the link
to quantum error-correcting codes. The talk includes a short introduction
to the relevant concepts of quantum mechanics.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivelisse Rubio (UPR Rio Piedras)
DTSTART:20210331T160000Z
DTEND:20210331T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/24
DESCRIPTION:Title: On Multidimensional Periodic Arrays\nby Ivelisse
Rubio (UPR Rio Piedras) as part of Carleton Finite Fields eSeminar\n\n\nAb
stract\nMultidimensional periodic arrays have applications for encoding da
ta during digital communication or storage. In many applications the array
s are stored in memory\, a burden for environments with limited resources.
Hence\, it is important to provide algebraic constructions for the arrays
that assure the desired properties\, are easily implemented and have smal
l use of memory. In the case of sequences\, their linear complexity is an
important parameter\, especially for applications related to information
security. In this talk we describe different algebraic constructions of mu
ltidimensional arrays\, present a generalization of the concept of linear
complexity\, and analyze the multidimensional linear complexity of several
types of periodic arrays.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Canteaut (INRIA)
DTSTART:20210414T160000Z
DTEND:20210414T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/25
DESCRIPTION:Title: Recovering or Testing Extended-Affine Equivalence
\nby Anne Canteaut (INRIA) as part of Carleton Finite Fields eSeminar\n\n\
nAbstract\nExtended Affine (EA) equivalence is the equivalence relation be
tween\ntwo vectorial Boolean functions $F$ and $G$ such that there exist\n
two affine permutations $A$\, $B$\, and an affine function $C$\nsatisfying
$G = A \\circ F \\circ B + C$. While a priori simple\, it is\nvery diffic
ult in practice to test whether two functions are\nEA-equivalent. This pr
oblem has two variants: EA-testing deals with\nfiguring out whether the tw
o functions can be EA-equivalent\, and\nEA-recovery is about recovering th
e tuple $(A\,B\,C)$ if it exists.\n\nIn this talk\, we present a new effic
ient algorithm that efficiently\nsolves the EA-recovery problem for quadra
tic functions. Though its\nworst-case complexity is obtained when dealing
with APN functions\,\nit supersedes all previously known algorithms in ter
ms of\nperformance\, even in this case. This approach is based on the\nJac
obian matrix of the functions\, a tool whose study in this context\ncan be
of independent interest.\n\nIn order to tackle EA-testing efficiently\, t
he best approach in\npractice relies on class invariants. We discuss a new
invariant\nbased on the so-called ortho-derivative which is applicable to
\nquadratic APN functions\, a specific type of functions that is of\ngreat
interest\, and of which tens of thousands need to be sorted\ninto distinc
t EA-classes. Our ortho-derivative-based invariant is\nboth very fast to c
ompute\, and highly discriminating.\n\nJoint work with Alain Couvreur and
Léo Perrin\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cathy Swaenepoel (University of Paris)
DTSTART:20210428T160000Z
DTEND:20210428T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/26
DESCRIPTION:Title: Trace of products in finite fields and additive doubl
e character sums\nby Cathy Swaenepoel (University of Paris) as part of
Carleton Finite Fields eSeminar\n\n\nAbstract\n\\Let $C$ and $D$ be two s
ubsets of a finite field $\\F_q$ of characteristic $p$ and let $\\mathrm{T
r}$ be the absolute trace of $\\F_q$. \n\nIn the first part of this talk\,
we will consider some ``interesting'' subsets $A$ of $\\F_p$ (such as sin
gletons or subgroups of $\\F_p^*$) and give lower bounds on $\\mathrm{card
}(C)$ and $\\mathrm{card}(D)$ to ensure that $\\mathrm{Tr}(CD)\\cap A\\neq
\\emptyset$.\nOur method allows us to obtain explicit and optimal results
(up to an absolute constant factor). \nSome estimates lead to interesting
combinatorial\nquestions.\n\nIn the second part which is a joint work wit
h Arne Winterhof\, we will see that if $D$ has some desirable structure th
en there is a large subset $U$ of $D$ for which the standard upper bound o
n the additive double character sum $\\sum_{(c\,u)\\in C \\times U} \\psi(
cu)$ can be improved. \nThe proof uses a decomposition theorem of Roche-Ne
wton\, 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 resul
t of Gyarmati and S\\'ark\\"ozy (provided that one of the involved sets ha
s some desirable structure).\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gary McGuire (University College Dublin)
DTSTART:20210602T160000Z
DTEND:20210602T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/27
DESCRIPTION:Title: Linear Fractional Transformations and Irreducible Pol
ynomials over Finite Fields\nby Gary McGuire (University College Dubl
in) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nWe will disc
uss polynomials over a finite field where linear fractional\n transformati
ons permute the roots. For subgroups G of PGL(2\,q) we will\n demonstrate
some connections between factorizations of certain polynomials\n into irre
ducible polynomials over Fq\, and the field of G-invariant\n rational func
tions. Some unusual patterns in the factorizations are explained by\n this
connection.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Pott (Otto-von-Guericke-University Magdeburg)
DTSTART:20210707T160000Z
DTEND:20210707T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/28
DESCRIPTION:Title: Relaxations of almost perfect nonlinearity\nby Al
exander Pott (Otto-von-Guericke-University Magdeburg) as part of Carleton
Finite Fields eSeminar\n\n\nAbstract\n(Note: the abstract here was transcr
ibed by the organizer\, and originally included references I did not inclu
de 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 \\emp
h{almost 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\
nSeveral relaxations of APN functions have been introduced: a function $f$
is called partially\nAPN if $f(y) + f(z) + f(y + z) \\neq 0$ for all $y\,
z \\neq 0$\, $y \\neq z$. That means that the APN\nproperty is satisfied
for $x = 0$ only. Another popular relaxation are differentially $4$-unifo
rm\nfunctions where $f(x + a) + f(x) = b$ has at most 4 solutions.\n\nIn m
y 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 w
ay\, that partially APN permutations exist for all $n$.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emina Soljanin (Rutgers University)
DTSTART:20210804T160000Z
DTEND:20210804T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/29
DESCRIPTION:Title: Codes\, Graphs\, and Hyperplanes in Data Access Servi
ce\nby Emina Soljanin (Rutgers University) as part of Carleton Finite
Fields eSeminar\n\n\nAbstract\nDistributed computing systems strive to max
imize the number of concurrent data access requests they can support with
fixed resources. Replicating data objects according to their relative popu
larity and access volume helps achieve this goal. However\, these quantiti
es are often unpredictable. Erasure-coding has emerged as an efficient and
robust form of redundant storage. In erasure-coded models\, data objects
are elements of a finite field\, and each node in the system stores one or
more linear combinations of data objects. This talk asks 1) which data ac
cess rates an erasure-coded system can support and 2) which codes can supp
ort a specified region of access rates. We will address these questions by
casting them into some known and some new combinatorial optimization prob
lems on graphs. We will explain connections with batch codes. This talk wi
ll also describe how\, instead of a combinatorial\, one can adopt a geomet
ric approach to the problem.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Bors (Carleton University)
DTSTART:20210929T160000Z
DTEND:20210929T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/30
DESCRIPTION:Title: Cycle types of complete mappings\nby Alexander Bo
rs (Carleton University) as part of Carleton Finite Fields eSeminar\n\n\nA
bstract\nA 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 ma
ppings about which little is known yet. An example of this are the cycle t
ypes of complete mappings -- the information into how many disjoint cycles
of each given length a complete mapping can decompose.\n\nIn this talk\,
I will present results that were achieved recently in collaboration with Q
iang Wang (also from Carleton University) and which concern the cycle type
s of complete mappings in two important classes of functions on finite fie
lds: cyclotomic mappings of first order and an additive analogue thereof w
hich we called coset-wise affine mappings. Our results provide both new ex
amples of cycle types of complete mappings that had never been considered
before and new constructions for achieving known cycle types.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tor Helleseth (University of Bergen)
DTSTART:20211201T170000Z
DTEND:20211201T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/32
DESCRIPTION:Title: The history of the cross correlation between m-sequen
ces: an overview\nby Tor Helleseth (University of Bergen) as part of C
arleton Finite Fields eSeminar\n\n\nAbstract\nMaximum-length sequences (or
m-sequences) of period 2^m-1 are\ngenerated by linear feedback shift regi
sters with primitive\ncharacteristic polynomials of degree m. These sequen
ces have\nmany important applications in modern communication systems.\nTh
e most well-known property of m-sequences is their two-level\nideal autoco
rrelation. The first major result on the cross\ncorrelation of two differe
nt m-sequences of the same period\nwas published by Gold back in January 1
968 and the result was\nused in constructing the famous family of Gold seq
uences.\nDuring more than 50 years the cross correlation between\nm-sequen
ces of the same period has been intensively studied\nby many research grou
ps. Many results have been obtained but\nstill many open problems remain i
n this area. This talk will\ngive an updated survey of the status of the c
ross correlation\nof m-sequences as well as some consequences of these res
ults.\n
LOCATION:https://stable.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:20260404T110745Z
UID:CarletonFiniteFields/33
DESCRIPTION:Title: Redei permutations with the same cycle structure\
nby Ariane Masuda/Juliane Capaverde (New York City College of Technology)
as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nPermutation poly
nomials over finite fields have been extensively\nstudied over the past de
cades. Among the major challenges in this\narea are the questions concerni
ng their cycle structures as they capture\nrelevant properties\, both theo
retically and practically.\n\nIn this talk we focus on a family of permuta
tion polynomials\, the so called Rédei permutations. Although their cycle
structures are known\, there are other related questions that can be inve
stigated. For example\, when do two Rédei permutations have the same cycl
e structure? We give a characterization of such pairs\, and present explic
it families\nof Rédei permutations with the same cycle structure. We also
discuss some results regarding Rédei permutations with a particularly si
mple cycle structure\, consisting of $1$- and $j$-cycles only\, when $j$ i
s $4$ or a prime number. The case $j = 2$ is specially important in some a
pplications. We completely describe Rédei involutions with a prescribed c
ycle structure\, and show that remarkably the only Rédei permutations wit
h a unique cycle structure are the involutions.\n\nThis is joint work with
Virgínia Rodrigues from Universidade Federal do Rio Grande do Sul.\n
LOCATION:https://stable.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:20260404T110745Z
UID:CarletonFiniteFields/34
DESCRIPTION:Title: The functional graph of some family of functions over
finite fields\nby Fabio Enrique Brochero Martínez (Federal Universit
y of Minas Gerais) as part of Carleton Finite Fields eSeminar\n\n\nAbstrac
t\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 $\\mathcal
V=\\mathbb F_q$ and $\\mathcal E=\\{(x\,f(x))\\mid x\\in\\mathbb F_q\\}$.
The characteristics of functional graphs (number of cycles\, cycle length
s\, pre-cycle lengths and so on) have been studied for several different m
aps over finite fields\, due to its applications in cryptography.\n\nIn th
is presentation we will present two independent results: the first one we
describe completely the dynamics of the maps $f(x)=x^{q+1}\\pm x^2$ over
the finite field $\\mathbb F_{q^2}$ and in the second we study the funct
ional graph of maps of the form $f(x)= x^n h( x^{(q-1)/m})$\, where $h$ sa
tisfies an special condition.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgos Kapetanakis (University of Thessaly)
DTSTART:20211215T170000Z
DTEND:20211215T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/35
DESCRIPTION:Title: The existence of Fq-primitive points on curves using
freeness\nby Giorgos Kapetanakis (University of Thessaly) as part of C
arleton Finite Fields eSeminar\n\n\nAbstract\nAn element of a finite cycli
c group of order $Q$\, $C_Q$\, is called\n$r$-free (where $r|Q$)\, if it i
s not a $p$-th power of any group element for any prime divisor $p$ of $r$
. We introduce the set\nof $(r\,n)$-free elements of $C_Q$\, where $n|Q$ a
nd $r|(Q/n)$\, as the\nelements of the subgroup $C_{Q/n}$ that are $r$-fre
e within $C_{Q/n}$.\nInspired by Vinogradov's expression for the character
istic\nfunction of primitive elements of the finite field Fq\, we prove\na
n analogue for the $(r\,n)$-free elements of $C_Q$ and obtain a\nlower bou
nd for the number of elements $b$ of Fq\, such that $f(b)$\nis $(r\,n)$-fr
ee and $F(b)$ is $(R\,N)$-free\, where $f$ and $F$ are\npolynomials over F
q.\n\nAs an application\, we consider the problem of the existence of\npoi
nts of elliptic curves in Fq^2\, whose coordinates are both\nprimitive and
provide a complete answer for the curves $y^2=x^3±x$.\n\nThis is joint w
ork with Stephen D. Cohen and Lucas Reis.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetla Petkova-Nikova (KU Leuven)
DTSTART:20211013T160000Z
DTEND:20211013T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/36
DESCRIPTION:Title: Threshold Cryptography against Combined Physical Atta
cks\nby Svetla Petkova-Nikova (KU Leuven) as part of Carleton Finite F
ields eSeminar\n\n\nAbstract\nRecent attacks show that there is a need for
protecting implementations jointly against side-channel and fault attacks
. Analogously\, modern\nMPC protocols consider active security\, i.e. agai
nst malicious parties\nwhich do not only passively eavesdrop but also acti
vely deviate from\nthe protocol. This provides an opportunity for the fiel
d of threshold implementations to evolve with MPC and achieve provable sec
ure implementations against combined passive and active physical attacks.\
n\nIn this talk we will first introduce Threshold Implementations applied
to\nprotect various ciphers against SCA and the like with Boolean function
s\nand MPC/SSS. After that we will discuss two recent proposals for combin
ed\ncountermeasures: CAPA and M&M\, which both start from passively secure
\nthreshold schemes and extend those with information-theoretic MAC tags\n
for protection against active adversaries. While similar in their most\nba
sic structure\, the two proposals explore very different adversary models\
nand thus employ completely different implementation techniques. CAPA\ncon
siders the field-probe-and-fault model\, which is the embedded analogue\no
f multiple parties jointly computing a function with at least one of the p
arties honest. Accordingly\, CAPA is strongly based on the actively secure
MPC protocol SPDZ and inherits its provable security properties in this m
odel. Since this results in very expensive implementations\, M&M works in
a similar but more realistic adversary model and uses existing building bl
ocks from previous passively secure implementations to build more efficien
t actively secure threshold cryptography.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhicheng (Jason) Gao (Carleton University)
DTSTART:20220131T170000Z
DTEND:20220131T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/37
DESCRIPTION:Title: Some recent results on counting polynomials over $\\f
q$ with prescribed coefficients using the generating function approach
\nby Zhicheng (Jason) Gao (Carleton University) as part of Carleton Finit
e Fields eSeminar\n\n\nAbstract\nCounting/estimating some families of poly
nomials over $\\fq$ with\nprescribed coefficients has attracted much atten
tion in the past\n30 years. Three well-known problems are:\n\n(a) existenc
e of irreducible polynomials with prescribed coefficients\;\n\n(b) countin
g irreducible polynomials with prescribed leading and/or ending coefficien
ts\;\n\n(c) counting polynomials with prescribed leading coefficients and\
nwith a given number of roots in a prescribed set. This is closely\nrelate
d to the distance distribution over Reed-Solomon codes.\n\nMost of the pub
lished results about these problems used the character\napproach and Weil'
s bound on character sums. In this talk\, I will\ndescribe the generating
function approach which leads to some new\nresults in these areas. The gen
erating functions use the group\nalgebra defined on the group of equivalen
ce classes of polynomials\nwith prescribed leading and /or ending coeffici
ents.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksandr Tuxanidy (Carleton University)
DTSTART:20220214T170000Z
DTEND:20220214T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/38
DESCRIPTION:Title: Equidistribution estimates for palindromic numbers in
residue classes and applications\nby Aleksandr Tuxanidy (Carleton Uni
versity) as part of Carleton Finite Fields eSeminar\n\n\nAbstract\nThis ta
lk concerns palindromic integers and discusses newly-derived\naverage equi
distribution estimates for these in residue classes\nto large moduli. As a
n application of this and well-known facts\nfrom sieve theory\, we obtain
the following:\n\n(1) In any given base\, there are infinitely many palind
romic\nintegers having at most six prime divisors.\n\n(2) The density of t
he prime numbers among the base-b palindromes\nat most X is O(1/log X)\, a
s expected by randomness heuristics. This\nanswers a problem raised by Ban
ks-Hart-Sakata (2004)\, later proved\nby Col (2009).\n\nWe also make a few
remarks on some related problems in finite fields.\n\nThis is joint work
with D. Panario and Q. Wang\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Magali Bardet (University of Rouen)
DTSTART:20220307T170000Z
DTEND:20220307T180000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/39
DESCRIPTION:Title: Algebraic decoding of Fqm-linear codes in rank metric
\nby Magali Bardet (University of Rouen) as part of Carleton Finite Fi
elds eSeminar\n\n\nAbstract\nRank-metric code-based cryptography relies on
the hardness of decoding a random linear code in the rank metric. This fu
ndamental problem is called the Minrank problem\, and is ubiquitous in ran
k metric (or even Hamming metric) code based cryptography as well as in mu
ltivariate cryptography. For structured instances arising in the former\,
their security rely on a more specific problem\, namely the Rank Syndrome
Decoding problem. There is also a generalization called the Rank Support L
earning problem\, where the attacker has access to several syndromes corre
sponding to errors with the same support. Those problems have various appl
ications in code-based and multivariate cryptography (KEM and signature sc
hemes)\, and a precise understanding of the complexity of solving them can
help designers to create secure parameters.\n\nIn this talk\, I will pres
ent the three problems and their relations to cryptographic schemes\, thei
r algebraic modeling and the recent improvements in the understanding of t
he complexity of solving those systems using algebraic techniques like Gr
öbner bases computations.\n\nThis gathers joint works with P. Briaud\, M.
Bros\, D. Cabarcas\, P. Gaborit\, V. Neiger\, R. Perlner\, O. Ruatta\, D.
Smith-Tone\, J.-P. Tillich\, J. Verbel.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariana Perez (Universidad Nacional de Hurlingham and Conicet)
DTSTART:20220321T160000Z
DTEND:20220321T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/40
DESCRIPTION:Title: Families of diagonal equations over finite fields: es
timates and applications\nby Mariana Perez (Universidad Nacional de Hu
rlingham and Conicet) as part of Carleton Finite Fields eSeminar\n\n\nAbst
ract\nIn this work\, we study the set of $\\mathbb{F}_q$--rational solutio
ns\, that is\, solutions with coordinates in the finite field $\\mathbb{F}
_q$ of $q$ elements\, of certain equations and systems defined by familie
s of diagonal equations with coefficients in $\\mathbb{F}_q$. In \\cite{1}
and \\cite{2} we obtain explicit estimates and results that guarantee th
e existence of at least an $\\mathbb{F}_q$--rational solution of these fam
ilies\, by studying geometric properties of the varieties that define thes
e equations. The results obtained complement those existing in the literat
ure (see \\cite{3}).\nFinally we apply these results to a generalization
of Waring's\nproblem and the distribution of solutions of congruences modu
lo a prime number.\n \n\n \\bibitem{1} M. Pérez and M. Privitelli. Estim
ates on the number of rational solutions of variants of diagonal equations
over finite fields\, Finite Fields and Appl. 68 (2020)\, 30 pp.\n\n \\bib
item{2} M. Pérez and M. Privitelli. On the number of solutions of systems
of certain diagonal equations over finite fields. Journal of Number Theor
y (2021). \n\n\\bibitem {3} Gary L. Mullen and D. Panario. Handbook of Fin
ite Fields (1st ed.) . Chapman and Hall/CRC\, 2013.\n \n\\end{thebibliogra
phy}\n\n\nThis talk is based on a joint work with Melina Privitelli.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/40/
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SUMMARY:Alfred Wassermann (University of Beyreuth)
DTSTART:20220411T160000Z
DTEND:20220411T170000Z
DTSTAMP:20260404T110745Z
UID:CarletonFiniteFields/41
DESCRIPTION:Title: Designs in Classical Polar Spaces\nby Alfred Wass
ermann (University of Beyreuth) as part of Carleton Finite Fields eSeminar
\n\n\nAbstract\nCombinatorial designs have been studied since the 19th cen
tury and have\nfamous applications in the design of experiments and in cod
ing theory.\n50 years ago\, Cameron\, Delsarte and Ray-Chaudhury introduce
d the notion\nof subspace designs\, also known as q-analogs of designs or
designs over finite fields.\nRoughly speaking\, q-analogs of objects arise
from their combinatorial counterparts by\nreplacing subsets by subspaces
and cardinalities by dimensions.\nThe first "true" subspace designs\, i.e.
designs with t > 1\,\nwere presented by Thomas only in 1987.\nA next natu
ral generalization of subspace designs are designs\nin polar spaces. For t
=1 these objects are known as spreads.\nFor t>1 the first - non-trivial -
such designs were found by\nDe Bruyn and Vanhove in 2013\, some more desig
ns appeared recently in the\nPhD thesis of Landsdown.\n\nIn this talk we w
ill give an overview on the few known structural results\nfor designs in c
lassical polar spaces and present quite a few new parameters\nof existing
designs found by computer search.\n
LOCATION:https://stable.researchseminars.org/talk/CarletonFiniteFields/41/
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