Prime numbers between squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a conditional proof of the conjecture assuming a hypothesis about the behavior of Selberg's integral in short interval

On ELSV-type formulae, Hurwitz numbers and topological recursion

We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, "Chiodo formulas for the r-th roots and topological recursion", Lett. Math. Phys. (2016).Comment: 18 pages, comments are welcom

Stream cipher based on quasigroup string transformations in Zp∗Z_p^*

In this paper we design a stream cipher that uses the algebraic structure of the multiplicative group \bbbz_p^* (where p is a big prime number used in ElGamal algorithm), by defining a quasigroup of order $p-1$ and by doing quasigroup string transformations. The cryptographical strength of the proposed stream cipher is based on the fact that breaking it would be at least as hard as solving systems of multivariate polynomial equations modulo big prime number $p$ which is NP-hard problem and there are no known fast randomized or deterministic algorithms for solving it. Unlikely the speed of known ciphers that work in \bbbz_p^* for big prime numbers $p$, the speed of this stream cipher both in encryption and decryption phase is comparable with the fastest symmetric-key stream ciphers.Comment: Small revisions and added reference

The Crossed Product by a Partial Endomorphism and the Covariance Algebra

Given a local homeomorphism \sigma:U -> X where U is a clopen subset of an compact and Hausdorff topological space X, we obtain the possible transfer operators L_\rho which may occur for \al:C(X) -> C(U) given by \al(f)=f\sigma. We obtain examples of partial dynamical systems (X_A,\sigma_A) such that the construction of the covariance algebra C^*(X_A,\sigma_A) and the crossed product by partial endomorphism O(X_A,\al,L) associated to this system are not equivalent, in the sense that there does not exists invertible function \rho in C(U) such that O(X_A,\al,L_\rho)=C^*(X_A,\sigma).Comment: 13 pages, no figure

Candidate One-Way Functions and One-Way Permutations Based on Quasigroup String Transformations

In this paper we propose a definition and construction of a new family of one-way candidate functions ${\cal R}_N:Q^N \to Q^N$, where $Q=\{0,1,...,s-1\}$ is an alphabet with $s$ elements. Special instances of these functions can have the additional property to be permutations (i.e. one-way permutations). These one-way functions have the property that for achieving the security level of $2^n$ computations in order to invert them, only $n$ bits of input are needed. The construction is based on quasigroup string transformations. Since quasigroups in general do not have algebraic properties such as associativity, commutativity, neutral elements, inverting these functions seems to require exponentially many readings from the lookup table that defines them (a Latin Square) in order to check the satisfiability for the initial conditions, thus making them natural candidates for one-way functions.Comment: Submitetd to conferenc