New Generalized Cyclotomic Quaternary Sequences with Large Linear Complexity and a Product of Two Primes Period

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

Ideals in B1(X) and residue class rings of B1(X) modulo an ideal

<p>This paper explores the duality between ideals of the ring B₁(X) of all real valued Baire one functions on a topological space X and typical families of zero sets, called Z_B-filters, on X. As a natural outcome of this study, it is observed that B₁(X) is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of B₁(X), complete descriptions of the fixed maximal ideals of both B₁(X) and B₁^* (X) are obtained. Though free maximal ideals of B₁(X) and those of B₁^* (X) do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. It is proved here that for a perfectly normal T₁ space X, free maximal ideals of B₁(X) are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of B₁(X) modulo an ideal, with special emphasize on real and hyper real maximal ideals of B₁(X).</p>

Construction of Fair Dice Pairs

An interesting and challenging problem in mathematics is how to construct fair dice pairs. In this paper, by means of decomposing polynomials in a residue class ring and applying the Discrete Fourier Transformation, we present all the 2000 fair dice pairs and their 8 equivalence classes in a four-person game, identifying what we call the mandarin duck property of fair dice pairs.

On ideal lattices, Gröbner bases and generalized hash functions

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, [Formula: see text] (here [Formula: see text] is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in [Formula: see text], and use its computational hardness to establish the collision resistance of the hash functions.