41,167 research outputs found
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
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 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
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 , 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 , where
is an alphabet with 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
computations in order to invert them, only 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
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