Due to Shor's algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.
We study applications of a quantum procedure called Simon's algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon's algorithm: finding a collision requires Ω(2n/2) queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.
We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB) are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure when using a quantum-secure PRF.
Second, we show that slide attacks can also be sped up using Simon's algorithm. This is the first exponential speed up of a classical symmetric cryptanalysis technique in the quantum model.

Quasimorphisms (QM) of groups to the reals are well studied and are linked to stable commutator length (scl) via Bavard Duality- Theorem. The notion of QM can be generalized to yield maps  between groups such that each QM from one group pulls back to a QM in the other.

We will give both a short overview of features of scl and investigate these generalized QMs with large scale properties of the commutator group. 

This is a survey (with some proofs) of chapter 2 of the notes http://renyi.mta.hu/~szamuely/beilintronew.pdf of T. Szamuely and G. Zabradi on Beilinson's approach to the p-adic Hodge decomposition theorem.

Given an integer $d$ such that $2 \leq d \leq 50$, we want to
answer the question: When is the sum of
$d$ consecutive cubes a perfect power? In other words, we want to find all
integer solutions to the equation
$(x+1)^3 + (x+2)^3 + \cdots + (x+d)^3 = y^p$. In this talk, we present some
of the techniques used to tackle such diophantine problems.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.

Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will first introduce a notion
of cuspidality for these pairs. The cuspidal pairs are expected to correspond to the supercuspidal irreducible representations of G via the local Langlands correspondence.
We will next describe a construction of  a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig.
Finally, we will use this map to decompose  the set of enhanced Langlands parameters into Bernstein series.
This is joint work with Ahmed Moussaoui and Maarten Solleveld.