Chengdong TaoSouth China University of Technology, ChinaHong XiangChongQing University, ChinaAlbrecht PetzoldtTechnische Universität Darmstadt, GermanyJintai DingChongQing University, China; University of Cincinnati, OH, USA
Group Theory and Lie Theorymathscidoc:2207.17001
Finite Fields and Their Applications, 35, 352-368, 2015.9
Multivariate cryptography is one of the main candidates to guarantee the security of communication in the presence of quantum computers. While there exist a large number of secure and efficient multivariate signature schemes, the number of practical multivariate encryption schemes is somewhat limited. In this paper we present our results on creating a new multivariate encryption scheme, which is an extension of the original SimpleMatrix encryption scheme of PQCrypto 2013. Our scheme allows fast en- and decryption and resists all known attacks against multivariate cryptosystems. Furthermore, we present a new idea to solve the decryption failure problem of the original SimpleMatrix encryption scheme.
Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory.
Jérémie BrieusselInstitut/Laboratoire Montpelliérain Alexander Grothendieck (IMAG) (UMR 5149), Université de Montpellier, 34090 Montpellier, FranceTianyi Zheng Department of Mathematics, Stanford University, Stanford (Palo Alto) CA 94305
Group Theory and Lie TheoryMetric GeometryProbabilitymathscidoc:2203.17002
We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and Lp-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].