On μ-symmetric polynomials

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root function [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

Integer dynamics

Let [Formula: see text] be an integer, and write the base [Formula: see text] expansion of any non-negative integer [Formula: see text] as [Formula: see text], with [Formula: see text] and [Formula: see text] for [Formula: see text]. Let [Formula: see text] denote an integer polynomial such that [Formula: see text] for all [Formula: see text]. Consider the map [Formula: see text], with [Formula: see text]. It is known that the orbit set [Formula: see text] is finite for all [Formula: see text]. Each orbit contains a finite cycle, and for a given [Formula: see text], the union of such cycles over all orbit sets is finite. Fix now an integer [Formula: see text] and let [Formula: see text]. We show that the set of bases [Formula: see text] which have at least one cycle of length [Formula: see text] always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.

Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients

Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

Analytical cryptanalysis upon N = p2q utilizing Jochemsz-May strategy

This paper presents a cryptanalytic approach on the variants of the RSA which utilizes the modulus N = p2q where p and q are balanced large primes. Suppose e∈Z+ satisfying gcd(e, ϕ(N)) = 1 where ϕ(N) = p(p − 1)(q − 1) and d < Nδ be its multiplicative inverse. From ed − kϕ(N) = 1, by utilizing the extended strategy of Jochemsz and May, our attack works when the primes share a known amount of Least Significant Bits(LSBs). This is achievable since we obtain the small roots of our specially constructed integer polynomial which leads to the factorization of N. More specifically we show that N can be factored when the bound δ<119−294+18γ. Our attack enhances the bound of some former attacks upon N = p2q.

New Jochemsz–May Cryptanalytic Bound for RSA System Utilizing Common Modulus N = p2q

This paper describes an attack on the Rivest, Shamir and Adleman (RSA) cryptosystem utilizing the modulus N=p2q where p and q are two large balanced primes. Let e1,e2<Nγ be the integers such that d1,d2<Nδ be their multiplicative inverses. Based on the two key equations e1d1−k1ϕ(N)=1 and e2d2−k2ϕ(N)=1 where ϕ(N)=p(p−1)(q−1), our attack works when the primes share a known amount of least significant bits (LSBs) and the private exponents share an amount of most significant bits (MSBs). We apply the extended strategy of Jochemsz–May to find the small roots of an integer polynomial and show that N can be factored if δ<1110+94α−12β−12γ−130180γ+990α−180β+64. Our attack improves the bounds of some previously proposed attacks that makes the RSA variant vulnerable.

Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences

Consider the system x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X2 + Y7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively.