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

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.

Antipalindromic numbers

Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance, kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multi-base palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multi-base antipalindromic numbers. We provide a user-friendly application for all studied questions.

Dimension of the intersection of certain Cantor sets in the plane

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.

Polynomial analogues of restricted multicolor b-ary partition functions

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.

Benford’s Law for coefficients of newforms

Let [Formula: see text] be a newform of even weight [Formula: see text] on [Formula: see text] without complex multiplication. Let [Formula: see text] denote the set of all primes. We prove that the sequence [Formula: see text] does not satisfy Benford’s Law in any integer base [Formula: see text]. However, given a base [Formula: see text] and a string of digits [Formula: see text] in base [Formula: see text], the set [Formula: see text] has logarithmic density equal to [Formula: see text]. Thus, [Formula: see text] follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato–Tate Conjecture.

Hausdorff Dimension and Nonlinear Relations Between Frequencies of Digits

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.