Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

A method is presented for generating random numbers with uniform distribution using linear recurrence sequences with very large period lengths. This method requires an irreducible polynomial modulo 2 to define the sequence. A suitable method for generating an infinite number of such polynomials is presented. The polynomials generated in this way can have an arbitrarily large degree, and a large enough order to make them suitable for practical applications.

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

The best known linear recurrence sequence is the Fibonacci Sequence 1,1,2,3,5,8,..., where each term (after the first two terms) is a sum of the two preceding terms. We may extend it to become a “doubly infinite” sequence $$ ..., - 8,5, - 3,2, - 1,1,0,1,1,2,3,5,8,... . $$ .

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.

Multivariate cryptography is one of the main candidates to guarantee the security of communication in a post quantum era. While multivariate signature schemes are very fast and require only modest computational resources, the key sizes of such schemes are quite large. In [17] Petzoldt et al. proposed a way to use Linear Recurring Sequences (LRS’s) for the key generation of the Unbalanced Oil and Vinegar (UOV) signature scheme by which they were able to reduce the public key size of this scheme by a factor of 7. In this paper we describe a modification of their scheme, which enables us not only to reduce the public key size, but also to speed up the verification process of the UOV scheme by a factor of 5.KeywordsMultivariate CryptographyUOV Signature SchemeKey Size ReductionFast Verification

On the Minimal Polynomial of the Product of Linear Recurring Sequences

We investigate the decidability of the monadic second-order (MSO) theory of the structure (N; <, P1,…,Pk), for various unary predicates P1,…,Pk ⊆ N. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers Powk = {kn : n ∈ N}, k-th powers Nk = {nk : n ∈ N}, and the set of terms of the Fibonacci sequence Fib = {0, 1, 2, 3, 5, 8, 13,…} (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following:

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

For a 1-dimensional semi-boundary link, Cochran constructed a sequence of Sato-Levine invariants of successively derived links.This is a linear recurrence sequence and conversely any linear recurrence sequence can be constructed in this way.An upper bound for the growth of this sequence is obtained.

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

We consider the Diophantine equation $U_n=p^xq^y$, where $U=(U_n)_{n\geq 0}$ is a linear recurrence sequence, $p$ and $q$ are distinct prime numbers and $x,y$ are non-negative integers not both zero. We show that under some technical assumptions the Dioph