7 research outputs found
The Distribution of Sandpile Groups of Random Graphs with their Pairings
We determine the distribution of the sandpile group (also known as the
Jacobian) of the Erd\H{o}s-R\'{e}nyi random graph along with its
canonical duality pairing as tends to infinity, fully resolving a
conjecture from 2015 due to Clancy, Leake, and Payne and generalizing the
result by Wood on the groups. In particular, we show that a finite abelian
-group equipped with a perfect symmetric pairing appears as the
Sylow -part of the sandpile group and its pairing with frequency inversely
proportional to , where
is the set of automorphisms of preserving the pairing . While this
distribution is related to the Cohen-Lenstra distribution, the two
distributions are not the same on account of the additional algebraic data of
the pairing. The proof utilizes the moment method: we first compute a complete
set of moments for our random variable (the average number of epimorphisms from
our random object to a fixed object in the category of interest) and then show
the moments determine the distribution. To obtain the moments, we prove a
universality result for the moments of cokernels of random symmetric integral
matrices whose dual groups are equipped with symmetric pairings that is strong
enough to handle both the dependence in the diagonal entries and the additional
data of the pairing. We then apply results due to Sawin and Wood to show that
these moments determine a unique distribution.Comment: 43 page
The geometric interpretation of the Tate pairing and its applications
While the Weil pairing is geometric, the Tate pairing is arithmetic: its value depends on the base field considered. Nevertheless, the étale topology allows to interpret the Galois action in a geometric manner. In this paper, we discuss this point of view for the Tate pairing: its natural geometric interpretation is that it gives étale -torsors. While well known to experts, this interpretation is perhaps less known in the cryptographic community.
As an application, we explain how to use the Tate pairing to study the fibers of an isogeny, and we prove a conjecture by Castryck and Decru on multiradical isogenies
ICTERI 2020: ІКТ в освіті, дослідженнях та промислових застосуваннях. Інтеграція, гармонізація та передача знань 2020: Матеріали 16-ї Міжнародної конференції. Том II: Семінари. Харків, Україна, 06-10 жовтня 2020 р.
This volume represents the proceedings of the Workshops co-located with the 16th International Conference on ICT in Education, Research, and Industrial Applications, held in Kharkiv, Ukraine, in October 2020. It comprises 101 contributed papers that were carefully peer-reviewed and selected from 233 submissions for the five workshops: RMSEBT, TheRMIT, ITER, 3L-Person, CoSinE, MROL. The volume is structured in six parts, each presenting the contributions for a particular workshop. The topical scope of the volume is aligned with the thematic tracks of ICTERI 2020: (I) Advances in ICT Research; (II) Information Systems: Technology and Applications; (III) Academia/Industry ICT Cooperation; and (IV) ICT in Education.Цей збірник представляє матеріали семінарів, які були проведені в рамках 16-ї Міжнародної конференції з ІКТ в освіті, наукових дослідженнях та промислових застосуваннях, що відбулася в Харкові, Україна, у жовтні 2020 року. Він містить 101 доповідь, які були ретельно рецензовані та відібрані з 233 заявок на участь у п'яти воркшопах: RMSEBT, TheRMIT, ITER, 3L-Person, CoSinE, MROL. Збірник складається з шести частин, кожна з яких представляє матеріали для певного семінару. Тематична спрямованість збірника узгоджена з тематичними напрямками ICTERI 2020: (I) Досягнення в галузі досліджень ІКТ; (II) Інформаційні системи: Технології і застосування; (ІІІ) Співпраця в галузі ІКТ між академічними і промисловими колами; і (IV) ІКТ в освіті