Seminar

Title: “Lattice-Based Zero-Knowledge Arguments for Integer Relations” by Benoit Libert, San Ling, Khoa Nguyen, and Huaxiong Wang

Abstract:

We provide lattice-based protocols allowing to prove relations among committed integers. While the most general zero-knowledge proof techniques can handle arithmetic circuits in the lattice setting, adapting them to prove statements over the integers is non-trivial, at least if we want to handle exponentially large integers while working with a polynomial size modulus q.

Title: Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k

Abstract:

Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb F$, the list contains only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)].

Title: Polynomial ideals, association schemes, and the Q-polynomial property

Abstract:

Let X ⊆ S^{m−1} be a spherical code in C^m. We study the ideal I ⊆ C[z_1, . . . , z_m] of polynomials that vanish on the points of X: I = { F(z) | (∀a ∈ X) (F(a) = 0) }. The primary example of interest is where the Gram matrix of X is proportional to the first idempotent in some Q-polynomial ordering of an association scheme (X, R) defined on X.

Title: Counting planar maps, 50 years after William Tutte

Abstract:

Every planar map can be properly coloured with four colours. But how many proper colourings has, on average, a planar map with $n$ edges? What if we allow a prescribed number of "monochromatic" edges, the endpoints of which share the same colour? What if we have $q$ colours rather than four?

Title: Lineup polytopes and exclusion principles

Abstract:

The set of all possible spectra of 1-reduced density operators for systems of N particles on a d-dimensional Hilbert space is a polytope called hypersimplex and this is related to Pauli's exclusion principle. If the spectrum of the original density operators is fixed, the set of spectra (ordered decreasingly) of 1-reduced density operators is also a polytope.

Title: Packing and covering balls in planar graphs

Abstract:

The set of all vertices at distance at most r from a vertex v in a graph G is called an r-ball. We prove that the minimum number of vertices hitting all r-balls in a planar graph G is at most a constant (independent of r) times the maximum number of vertex-disjoint r-balls in G.

Title: Homomorphism counts in robustly sparse graphs

Abstract:

For a fixed graph H and for arbitrarily large host graphs G, the number of homomorphisms from H to G and the number of subgraphs isomorphic to H contained in G have been extensively studied when the host graphs are allowed to be dense.

Title: On reverse mixing in quantum walks

Abstract:

A random walk on a connected undirected graph mixes to the principal eigenvector. We consider a continuous-time quantum walk to reverse this process: starting with the principal eigenvector, can we reach an arbitrary vertex if we were to place a loop on the target vertex?

Title: Rotational Differential-Linear Cryptanalysis of ARX Ciphers

Abstract:

The differential-linear attack, combining the power of the two most effective techniques for symmetric-key cryptanalysis, was proposed by Langford and Hellman at CRYPTO 1994. We further extend this framework by replacing the differential part of the attack by rotational differentials.

Title: Densest Subgraph: Supermodularity, Iterative Peeling, and Flow

Abstract:

The densest subgraph problem in a graph (DSG), in the simplest form, is the following. Given an undirected graph G = (V, E) find a subset S of vertices that maximizes the ratio |E(S)|/|S| where E(S) is the set of edges with both endpoints in S. DSG and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis.