A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

Recently the solution of multidimensional, linear, and homogeneous recurrence relations, or partial difference equations (PDE), was obtained via a multidimensional extension of the combinatorics function technique, developed by Antippa and Phares. Combinatorics functions of the first and second kind are representations of ’’restricted’’ paths connecting two points in an n-dimensional space. These functions are shown to give the solution of the most general linear and inhomogeneous PDE. The consistency of the PDE with the initial value conditions is also discussed. Applications of the method are given elsewhere.

In the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems.

Abstract. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a − p(a)Dm−2(a)/Dm−1(a)}, where Dm(a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm(a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m ≥ 2 and an appropriate input a0, the fixed-point iterates ak+1 = Bm(ak) converge to a simple root having order m. The Basic Sequence corresponds to the pointwise evaluation of the Basic Family. We present several fractal images that confirm the theoretical convergence results: as m increases, the basins of attractions to the roots, as computed with respect to the iteration function Bm(z), rapidly converge to the Voronoi regions. Thus the regions with chaotic behavior rapidly shrink to the boundaries of the Voronoi regions. The interplay between the Basic Sequence and the Basic Family result in new algorithms for approximation of polynomial roots. Also as a byproduct of the analysis we give determinantal representation of solution of homogeneous linear recurrence relations, easily computable bounds on the modulus of terms of such recurrence relations, and fast computation of determinant of certain Toeplitz matrices. Finally, we prove a relationship between the Basic Sequence and the iterates of Bernoulli Method for approximation of extreme roots.

The aim of this paper is to investigate the behavior of quotients of sequences w n defined by the linear homogeneous recurrence relation $$w_{n + 2} = pw_{n + 1} - qw_n$$ (1) with arbitrary initial values w 0 = a, w 1 = b belonging to a normed field K, and p, q ∈ K. We shall write w n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts $$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$ (2.1) is defined in K. If the field K is normed and α, β are the roots of the polynomial x 2 – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q n ’s, where it is defined, converges, and $$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$ (3)

In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A052529 that has strong relationships with triangular numbers.

We describe a prototype of RRTS (Recurrence Relation Tutorial System), a tool for students to use to practice solving homogeneous and non-homogeneous recurrence relations, both linear and divide-and-conquer, that arise in algorithm analysis.

CHAPTER 4 - Recurrence Relations and Algebraic Equations

This dissertation is a study on Generating Functions and the Linear Recurrence Relations and is divided in three chapters in the following manner:On the first chapter we present Ordinary Generating Functions and Exponential Generating Functions along with their definitions, properties and a few theorems with their respective demonstrations.Still on this chapter we explore the concepts of Partitions and some Partition Generating Functions.On the second chapter we explore the concept of Recurrence Relations, specifically the Linear ones.We show a few methods for the resolution of Recurrence Relations, highlighting the resolution based on Generating Functions and also an interesting method using matrix diagonalization.The methods are very well detailed and are easy to comprehend.Chapter 3 is made of four applications on Recurrence Relations and Generating Functions.(n,k)-block fountain is a method of piling up coins on a plane obeying a few criteria.Steiner's pizza consists of dividing the pizza and obtaining the greatest number of slices with n cuts, triangulation of a convex n-agon is a problem of diving polygons in triangles and lastly one of the most famous sequences in Mathematics, the Fibonacci Sequence, where we show the sequence recurrence relation and its respective generating function.

In an earlier paper [1] we show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. This also holds for Kauffman polynomials. Here we prove the result using a different construction which in some cases yields a recurrence relation of much lower order, thereby drastically reducing the amount of computation required to obtain explicit polynomials for individual members or a general formula for the entire family.

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

We consider the problem of computing a linear recurrence relation (or equivalently a Linear Feedback Shift Register) of minimum order for a finite sequence over a field, with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is non-zero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we will call it bidirectional. If the field is finite, a sequence is periodic if and only if it admits a bidirectional linear recurrence relation. For solving the above problem we propose an algorithm similar to the Berlekamp-Massey algorithm and prove its correctness. We also describe the set of all solutions to this problem and prove some properties of the minimal polynomials of initial segments of the sequence.

This paper introduces some algorithms for solving linear relationships, homogeneous and non-homogeneous recurrence with constant and non-constant coefficients, using software as the main resource in solving processes. The Mathematica commercial application has provided the technical support necessary for the implementation of the methods used. It also presents other examples of recurrence relations, showing the effectiveness and limitations of the algorithms created by the author and programmed in Mathematica environment that provides.

A hierarchical threshold scheme is a scheme to solve the secret sharing with a hierarchical access structure where participants are partitioned into different levels. In the past hierarchical threshold scheme participants are partitioned based on a single attribute. But in practice, each participant always has several attributes, and the group of participants always should be partitioned based on different attributes to satisfy the security requirements. For example, Distinguished by position, the bank employees can be tellers or department managers. Distinguished by department, the bank employees can be from account department, administration department or others. The bank policy could require the presence of 5 employees in opening the vault, at least one of whom must be department manager, and at least 3 of whom are from account department. Even though hierarchical threshold scheme has been studied extensively in the past, none of the existing solutions solves the above problem. We propose a multi-attribute hierarchical threshold scheme based on Tassa's scheme, which is based on Birkhoff interpolation, and Mignotte's scheme, which is based on Chinese Remainder Theorem, to solve this problem.

Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.

Recursive families of graphs