302 research outputs found
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
Construction of irreducible polynomials through rational transformations
Let be the finite field with elements, where is a power
of a prime. We discuss recursive methods for constructing irreducible
polynomials over of high degree using rational transformations.
In particular, given a divisor of and an irreducible polynomial
of degree such that is even or , we show how to obtain from a sequence of
irreducible polynomials over with .Comment: 21 pages; comments are welcome
A Non-commutative Cryptosystem Based on Quaternion Algebras
We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion
algebras. This cryptosystem uses bivariate polynomials as the underling ring.
The multiplication operation in our cryptosystem can be performed with high
speed using quaternions algebras over finite rings. As a consequence, the key
generation and encryption process of our cryptosystem is faster than NTRU in
comparable parameters. Typically using Strassen's method, the key generation
and encryption process is approximately times faster than NTRU for an
equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure
that makes inefficient standard lattice attacks on the private key. This
entails a higher computational complexity for attackers providing the
opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is
more resistant than NTRU against known attacks at an equivalent parameter set.
Moreover, message protection is feasible through larger polynomials and this
allows us to obtain the same security level as other NTRU-like cryptosystems
but using lower dimensions.Comment: Submitted for possible publicatio
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects
Let and be the random variable representing the size of the
smallest component of a combinatorial object generated uniformly and randomly
over elements. A combinatorial object could be a permutation, a monic
polynomial over a finite field, a surjective map, a graph, and so on. It is
understood that a component of a permutation is a cycle, an irreducible factor
for a monic polynomial, a connected component for a graph, etc. Combinatorial
objects are categorized into parametric classes. In this article, we focus on
the exp-log class with parameter (permutations, derangements, polynomials
over finite field, etc.) and (surjective maps, -regular graphs,
etc.) The generalized Buchshtab function plays an important role
in evaluating probabilistic and statistical quantities. For , Theorem
from \cite{PanRic_2001_small_explog} stipulates that
for some and
sufficiently large . We revisit the evaluation of using
different methods: analytic estimation using tools from complex analysis,
numerical integration using Taylor expansions, and computation of the exact
distributions for using the recursive nature of the counting
problem. In general for any , Theorem from \cite{BenMasPanRic_2003}
connects the quantity for with the asymptotic
proportion of -objects with large smallest components. We show how the
coefficients of the Taylor expansion of for depends on those for . We use this family of coefficients to evaluate
.Comment: 16 pages, 2 tables, 15 reference
Infinitude of palindromic almost-prime numbers
It is proven that, in any given base, there are infinitely many palindromic
numbers having at most six prime divisors, each relatively large. The work
involves equidistribution estimates for the palindromes in residue classes to
large moduli, offering upper bounds for moments and averages of certain
products closely related to exponential sums over palindrome
The number of irreducible polynomials of degree n over Fq with given trace and constant terms
AbstractWe study the number Nγ(n,c,q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ(n,c,q) varying c and γ. We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any Nγ(n,c,q) when n=a(q−1) for a nonzero constant term c and a nonzero trace γ. As a byproduct, we give a simple and explicit formula for the number N(n,c,q) of irreducible polynomials over Fq of degree n=q−1 with a prescribed primitive constant term c
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