Carleton Finite Fields eSeminar - Fall 2020 Abstracts
Fall 2020 Abstracts
Date/Time: December 2, 2020 12:00 Eastern (UTC -4:00)
Speaker: Herivelto Borges (Universidade de Sao Paulo)
Title: Algebraic curves through Fernando Torres' lens
Abstract:
The mathematical legacy of Fernando 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 talk, we will highlight and briefly discuss
some of Torres' outstanding contributions to our mathematical community.
Date/Time: November 18, 2020 12:00 Eastern (UTC -4:00)
Speaker: Nina Bindel (University of Waterloo)
Title: A status update on NIST's post-quantum standardization effort
Abstract:
If a general-purpose quantum computer can be built, it will break most
widely-deployed public-key cryptography. To prepare for this risk,
the cryptographic community is busily designing new cryptographic
systems. Furthermore, the (US-American) National Institut 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 the
third round of evaluation in NIST standardization effort.
This 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 the ongoing project, explain reasons for why certain schemes
have been chosen to advance to the third round, and what are important
evaluation criteria during the next phase. Moreover, we will explain how
the concrete security of the schemes is estimated. As an example we take
a closer look at lattice-based encryption schemes. Interestingly, most
of the submitted PKEs are not perfectly correct schemes, i.e., sometimes
honestly generated ciphertexts can not be encrypted correctly. Finding
such a decryption failure poses a security risk which will be explained
in the talk as well. slides, video
Date/Time: November 4, 2020 12:00 Eastern (UTC -4:00)
Speaker: Jonathan Jedwab (Simon Fraser University)
Title: Packings of partial difference sets
Abstract:
Partial difference sets are highly structured group subsets that occur in
various guises throughout design theory, finite geometry, coding theory, and
graph theory. They admit only two possible nontrivial character sums and so
are often studied using character theory. The central question is to
determine which groups contain a partial difference set with two specified
nontrivial character sums. We consider an apparently more difficult
question: which groups contain a large disjoint collection of such partial
difference sets? This leads us to identify a certain subgroup as containing
important structural information about the packing. With this insight, we
are able to formulate a recursive construction of packings in abelian groups
of increasing exponent. This allows us to unify and extend numerous previous
results about partial difference sets using a common framework.
This is joint work with Shuxing Li, a 2019-2021 PIMS Postdoctoral Fellow.
Date/Time: October 21, 2020
Speaker: Ray Perlner (NIST)
Title: The MinRank problem in Cryptography and Cryptanalysis
Abstract: The 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 several candidates in the National Institute of Standards and Technology (NIST)s Postquantum Cryptography Standardization Process. These include the code based cryptosystems ROLLO and RQC, (which were eliminated from consideration for standardization after the second round due to recent significant improvements in the special case of the MinRank problem known as the Rank Syndrome Decoding problem), as well as the third (current) round PQC standardization candidates Rainbow and GeMSS. This talk will discuss how the MinRank problem relates to the cryptanalysis of this diverse array of cryptosystems, as well as recent developments that have dramatically improved the concrete complexity of solving the MinRank problem, both in special cases and in general. video.
Date/Time: October 7, 2020
Speaker: Stephen Cohen (Aberdeen)
Title: Existence theorems for r-primitive elements in finite fields
Abstract: link, slides, video.
Date/Time: September 23, 2020 12:00 Eastern (UTC -4:00)
Speaker: Daniel Katz (CSUN Northridge)
Title: Niho's Last Conjecture
Abstract:
This talk is concerned with character sums, called Weil sums of binomials,
that determine the nonlinearity (Walsh spectrum) of a power permutation
x -> x^d of a finite field F. These Weil sums also determine the
crosscorrelation spectrum for a pair of maximum length linear recursive
sequences and the weight distribution of a cyclic code. In each case, the
binomial involved is of the form x^d-cx, and one obtains values of the
Walsh spectrum by computing the various Weil sums as the coefficient c
runs through F. Certain exponents d, known as Niho exponents, have a
simple form and can produce Walsh spectra with very few distinct values.
The last conjecture in Niho's 1972 thesis states that a particular family
of such exponents produces spectra with at most five distinct values.
Niho's own techniques show that one has at most eight distinct values.
Each of the eight candidate values corresponds to a possible number of
distinct roots of a seventh degree polynomial on a subset of the finite
field F called the unit circle. We use symmetry arguments to show that
it is impossible to have four, six, or seven roots on the unit circle:
this proves Niho's last conjecture.
This is joint work with Tor Helleseth and Chunlei Li.
slides,
video.