How to write integers in non-integer base

2005 ◽  
pp. 154-164 ◽  
Author(s):  
Christiane Frougny
Keyword(s):  
Integer Base

scholarly journals Comparison and intelligent analysis of NTRU and ETRU signature algorithms for public key digital signature

2021 ◽  
Vol 2083 (4) ◽  
pp. 042009
Author(s):  
Zifeng Zhu ◽  
Fei Tian
Keyword(s):  
Digital Signature ◽  
Public Key ◽  
High Dimensional ◽  
Public Key Cryptosystem ◽  
Fast Speed ◽  
Intelligent Analysis ◽  
Small Footprint ◽  
Integer Base ◽  
Definition Of

Abstract Three American mathematicians made the NTRU public-key cryptosystem in 1996, it has a fast speed, small footprint, and also it is easy to produce key advantages. The NTRU signature algorithm is based on an integer base, the performance of the signature algorithm will change when the integer base becomes other bases. Based on the definition of “high-dimensional density” of lattice signatures, this paper chooses the ETRU signature algorithm formed by replacing the integer base with the Eisenstein integer base as a representative, and analyzes and compares the performance, security of NTRU and ETRU signature algorithms, SVP and CVP and other difficult issues, the speed of signature and verification, and the consumption of resources occupied by the algorithm.

scholarly journals Dimension of the intersection of certain Cantor sets in the plane

2021 ◽  
Vol 41 (2) ◽  
pp. 227-244
Author(s):  
Steen Pedersen ◽  
Vincent T. Shaw
Keyword(s):  
Complex Plane ◽  
Cantor Set ◽  
Cantor Sets ◽  
Gaussian Integer ◽  
Integer Base ◽  
Digit Expansions

In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.

On the representation of Fibonacci and Lucas numbers in an integer base

2013 ◽  
Vol 37 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Yann Bugeaud ◽  
Mihai Cipu ◽  
Maurice Mignotte
Keyword(s):  
Lucas Numbers ◽  
Integer Base ◽  
Fibonacci And Lucas Numbers

Hausdorff Dimension and Nonlinear Relations Between Frequencies of Digits

2012 ◽  
Vol 19 (03) ◽  
pp. 1250018
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Asymptotic Behavior ◽  
Hausdorff Dimension ◽  
Nonlinear Perturbations ◽  
Taylor Coefficients ◽  
Integer Base

For a class of sets defined in terms of frequencies of digits in some integer base m, we study their Hausdorff dimension. Our main aim is to consider nonlinear perturbations of the case when all frequencies are equal (and thus when the Hausdorff dimension is maximal). As a first step we consider the case when only one frequency is related to another, by the function x ↦ 1/m + εx + δx2, and for which the computations are already quite substantial. We show that the Hausdorff dimension is analytic in the parameters ε and δ, and we estimate the asymptotic behavior of the Taylor coefficients of the dimension in terms of m.

scholarly journals Irrationality measures for some automatic real numbers

2009 ◽  
Vol 147 (3) ◽  
pp. 659-678 ◽  
Author(s):  
BORIS ADAMCZEWSKI ◽  
TANGUY RIVOAL
Keyword(s):  
Rational Approximation ◽  
Generating Functions ◽  
Finite Automaton ◽  
Upper Bounds ◽  
Explicit Description ◽  
Real Numbers ◽  
Irrationality Exponent ◽  
Integer Base

AbstractThis paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.

The zeta function of the beta transformation

1994 ◽  
Vol 14 (2) ◽  
pp. 237-266 ◽  
Author(s):  
Leopold Flatto ◽  
Jeffrey C. Lagarias ◽  
Bjorn Poonen
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Symbolic Dynamics ◽  
Counting Function ◽  
Minimum Modulus ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Integer Base ◽  
Image Position ◽  
Totally Disconnected

AbstractThe β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta functionwhere Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.

scholarly journals Bifurcation sets arising from non-integer base expansions

2019 ◽  
Vol 6 (4) ◽  
pp. 301-341 ◽  
Author(s):  
Pieter Allaart ◽  
Simon Baker ◽  
Derong Kong
Keyword(s):  
Integer Base

scholarly journals Computation of L_⊕ for several cubic Pisot numbers

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Julien Bernat
Keyword(s):  
Central Symmetry ◽  
Maximal Length ◽  
Pisot Numbers ◽  
International Audience ◽  
Integer Base

International audience In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that dβ(1) = 0.k1d-1 kd with d ∈ ℕ, d ≥ 2 and k1 ≥ kd ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L_⊕. In particular, we prove that L_⊕ = 5 in the Tribonacci case.

scholarly journals Polynomial analogues of restricted multicolor b-ary partition functions

2020 ◽  
pp. 1-21
Author(s):  
Karl Dilcher ◽  
Larry Ericksen
Keyword(s):  
Recurrence Relation ◽  
Finite Sequence ◽  
Factorization Theorem ◽  
Partition Functions ◽  
Explicit Formulas ◽  
Number Formula ◽  
Integer Base ◽  
Positive Integers

Given an integer base [Formula: see text], a number [Formula: see text] of colors, and a finite sequence [Formula: see text] of positive integers, we introduce the concept of a [Formula: see text]-restricted [Formula: see text]-colored [Formula: see text]-ary partition of an integer [Formula: see text]. We also define a sequence of polynomials in [Formula: see text] variables, and prove that the [Formula: see text]th polynomial characterizes all [Formula: see text]-restricted [Formula: see text]-colored [Formula: see text]-ary partitions of [Formula: see text]. In the process, we define a recurrence relation for the polynomials in question, obtain explicit formulas, and identify a factorization theorem.

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