Quadratic Residue Research Articles

Berkovich–Uncu Type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect $$t^{th}$$ power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case $$t=1$$ . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of $$\mathbb {N}^t$$ , and Jensen’s inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

Cartesian Product of Quadratic Residue Graphs

Cartesian Product of Quadratic Residue Graphs

Schematic 4-designs

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.

Consecutive quadratic residues in Beatty sequences

Let [Formula: see text] be an odd prime. We study the distribution of consecutive quadratic residues modulo [Formula: see text] in non-homogeneous Beatty sequences [Formula: see text], where [Formula: see text] and [Formula: see text] is irrational. Furthermore, we study the pseudorandom properties of sequences formed by the Legendre symbol and Beatty sequences. In addition, we give upper bounds for certain character sums over [Formula: see text].

Single parity check node adapted to polar codes with dynamic frozen bit equivalent to binary linear block codes

<span lang="EN-US">In the context of decoding binary linear block codes by polar code decoding techniques, we propose in this paper a new optimization of the serial nature of decoding the polar codes equivalent to binary linear block codes. In addition to the special nodes proposed by the simplified successive-cancellation list technique, we propose a new special node allowing to estimate in parallel the bits of its sub-code. The simulation is done in an additive white gaussian noise channel (AWGN) channel for several linear block codes, namely bose–chaudhuri–hocquenghem codes (BCH) codes, quadratic-residue (QR) codes, and linear block codes recently designed in the literature. The performance of the proposed technique offers the same performance in terms of frame error rate (FER) as the ordered statistics decoding (OSD) algorithm, which achieves that of maximum likelihood decoder (MLD), but with high memory requirements and computational complexity.</span>

The <i>m</i>Θ Quadratic Character in the <i>m</i>Θ Set <img width="40" height="20" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />

The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (<i>m</i>Θ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> contains the set <img width="10" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image007.png" /> and the elements <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image003.png" /> such that the support of <i>x</i> is not congruent to 0 modulo <i>n</i>. In this paper the purpose is to define on <img width="40" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image008.png" />, <i>p</i> prime, a notion of quadratic residues and quadratic character which respects its structure of <i>m</i>Θs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>). We characterize the <i>m</i>Θ set (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) and we then give some arithmetical and intrinsic <i>m</i>Θ parameters of <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> which lead us to the notion of factorial of <i>m</i> without <i>n</i> in <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, the <i>m</i>Θ quotient of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) modulo (<img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />) and a complete system of <i>m</i>Θ residues modulo <img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />, <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image005.png" />. After that, we define a <i>p</i>-valent modal quadratic residue, <i>p</i> prime. We characterize some properties of <i>p</i>-valent modal quadratic character and <i>p</i>-valent modal quadratic residue of <i>p^k</i> which establish the difference between the <i>m</i>Θ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of <img width="65" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image006.png" /> with respect to <i>p^k</i>. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.

Gaussian phenomena for small quadratic residues and non-residues

Assuming the Generalized Riemann Hypothesis, it is known that the smallest quadratic non-residue modulo a prime p p is less than or equal to ( log ⁡ p ) 2 (\log p)^2 . Our aim in this paper is to establish the distribution of quadratic non-residues in even smaller intervals of size ( log ⁡ p ) A (\log p)^A with A &gt; 1 A &gt;1 , for almost all primes p p .

Five errors correcting (73, 61, 12) quadratic residue code over ternary field

Five errors correcting (73, 61, 12) quadratic residue code over ternary field

Regular Patterns of Quadratic Residues

For an odd prime p we determine four kinds of patterns concerning the distribution of quadratic residues depending on three parameters, according to the class of p modulo 8.

Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b &amp;gt; 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.&lt;/p&gt;&lt;/abstract&gt;

The Stochasticity Parameter of Quadratic Residues

Abstract Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of ${\mathbb {Z}}/M{\mathbb {Z}}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper, we study the stochasticity parameter of the set $R_{M}$ of quadratic residues modulo $M$. We present a method that allows to find the asymptotics of $S(R_{M})$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,{\mathbb {Z}}/M{\mathbb {Z}})$ the average value of $S(U)$ over all subsets $U\subseteq {\mathbb {Z}}/M{\mathbb {Z}}$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. Let ${\mathfrak {S}}(R_{M})=S(R_{M})/s(|R_{M}|)$. We show that (a) $\varliminf _{M\to \infty }{\mathfrak {S}}(R_{M})&amp;lt;1&amp;lt;\varlimsup _{M\to \infty }{\mathfrak {S}}(R_{M})$; (b) the set $\{ M\in {\mathbb {N}}: {\mathfrak {S}}(R_{M})&amp;lt;1 \}$ has positive lower density.

Wave localization in number-theoretic landscapes

We investigate the localization of waves in aperiodic structures that manifest the characteristic multiscale complexity of certain arithmetic functions with a central role in number theory. In particular, we study the eigenspectra and wave localization properties of tight-binding Schr\"{o}dinger equation models with on-site potentials distributed according to the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and the Legendre sequence of quadratic residues modulo a prime (QRs). We employ Multifractal Detrended Fluctuation Analysis (MDFA) and establish the multifractal scaling properties of the energy spectra in these systems. Moreover, by systematically analyzing the spatial eigenmodes and their level spacing distributions, we show the absence of level repulsion with broadband localization across the entire energy spectra. Our study introduces deterministic aperiodic systems whose eigenmodes are all strongly localized in realistic finite one-dimensional systems and provides opportunities for novel quantum and classical devices of particular importance to cold-atom experiments in engineered speckle potentials and enhanced light-matter interactions.

Efficient Decoder for Turbo Product Codes Based on Quadratic Residue Codes

In this letter, we study turbo product codes with quadratic residue codes (called QR-TPCs) as the component codes. We propose an efficient decoder based on Chase-II algorithm with two convergence conditions for the iterative decoding of QR-TPCs. For each row and column, the Chase-II decoder will stop immediately when one of the conditions is met. The simulation results show that the proposed algorithm has a lower computational complexity compared with existing decoding methods. Moreover, a comparison with 5G low-density parity-check codes shows that the proposed turbo product codes have better performance for short code lengths.

Sound diffusion with spatiotemporally modulated acoustic metasurfaces

Traditional sound diffusers are quasi-random phase gratings attached to reflecting surfaces to introduce spatiotemporal incoherence into the backscattered acoustic field. Early designs consisted of periodically tiled diffuser grating unit cells to cover large surfaces. However, spatial periodicity leads to coherent constructive and destructive interference, which is undesirable for achieving acoustic diffusivity. This problem was partially addressed by using aperiodic tiling of unit cells based on pseudorandom sequences. While an aperiodic diffuser spacing can overcome the problems introduced by spatial periodicity, the improvements in performance come at the expense of increased thickness. In this work, we investigate spatiotemporal modulation of the surface acoustic admittance of a metasurface diffuser to improve sound diffusion. Using semi-analytical and finite element models, we demonstrate that the effects of the spatial periodicity can be mitigated without introducing an aperiodic spatial spacing, thus simultaneously minimizing diffuser thickness and improving diffusivity of the backscattered field. We develop a semi-analytical model that employs Fourier series expansion to determine the scattered sound field from a surface admittance consisting of a quadratic residue diffuser whose individual well admittances are modulated in a traveling wave fashion with modulation frequency, ωm, amplitude, Ym, and a wavenumber that matches the unit cell length, Λ. We observe significant improvement in diffusion due to the fact that the spatiotemporal modulation scatters sound into additional frequency-wavenumber pairs associated with harmonics of ωm and their diffraction orders. The semi-analytical model results are verified using a time-domain finite element model and compared with periodic and aperiodic diffuser designs.

Development of a New Chaotic Maps Cryptosystem with Quadratic Residue Problem

A new fast public key cryptosystem is proposed, which is based on two dissimilar number-theoretic hard problems, namely the simultaneous chaotic maps (CM) problem and quadratic residue (QR) problem. The adversary has to solve the two hard problems simultaneously to recover the plaintext according to their knowledge about the public keys and the cipher-text. Cryptographic quadratic residue and chaotic system are employed to enhance the security of our cryptosystem scheme. The encryption, and decryption are discussed in details. Several security attacks are proposed to illustrate the system shield through chaotic maps and quadratic residue problems. The performance analysis of the proposed scheme show a much improved performance over existing techniques.

Decoding Quadratic Residue Codes Using Deep Neural Networks

In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity.

On some symmetries of the base n expansion of 1∕m : the class number connection

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Furthermore, we obtain corollaries connecting the $ 3 $ rank of the class number of the real quadratic field $ \mathbb{Q}(\sqrt{m}) $ to the $ 3 $ divisibility of the number of certain quadratic residues modulo $ m $. Secondly, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain distribution results of $ m $ modulo any prime $ p $ that properly divides $ n+1 $. Our result implies that the primes $ m $ for which $ n $ is a primitive root belong to one of two equivalence classes modulo any prime $ p $ as above.

An RFID Zero-Knowledge Authentication Protocol Based on Quadratic Residues

An RFID system is a contactless automatic identification technology and a solution to many business and personal applications. However, the privacy threats to RFID systems cause significant concerns in real RFID applications. Among the privacy issues in RFID systems, we are particularly interested in designing an RFID authentication protocol that achieves strong indistinguishability-based privacy (ind-privacy) and possesses the zero-knowledge property for active tags. In this article, a zero-knowledge authentication protocol (ZKAP) using quadratic residues is proposed. Then, an indistinguishability-based experiment using random oracles is modeled to prove that the proposed scheme achieves strong ind-privacy against active eavesdroppers. The zero-knowledge property is defined with respect to the tags’ private information against dishonest readers, and that property of our ZKAP is proven by constructing a polynomial time simulator that generates a transcript with the same distribution as that of ZKAP. Finally, the proposed protocol is implemented in a Raspberry Pi and a PC, and the performance is evaluated in terms of authentication time. The experimental results demonstrate that our ZKAP achieves its design goals.

The distribution of quadratic and higher residues

The distribution of quadratic and higher residues

Sound-scattering properties of Sierpinski triangle fractal structures in the near field

This paper proposes a new design of an acoustic diffuser based on the construction rules of the Sierpinski triangle in order to broaden the effective diffusion frequency range. The diffuser is made of triangular blocks of different sizes attached to a plane surface. The effects of the number of fractal iterations, the height of triangular blocks, and arrangements of the blocks on the normal-incidence diffusion coefficients in the near field are examined through numerical simulations based on the boundary element method (BEM) in the frequency range of 100 Hz – 5 kHz. Furthermore, measurement results will be presented to validate the diffusion performance presented by the numerical simulations. The diffusion performance of a conventional quadratic residue diffuser (QRD) is compared to confirm the advantage of the designed diffuser for broadening the effective frequency range. It shows that the fractal patterns with various sizes of blocks improve diffusion performance compared to the conventional QRD of the same size, especially in the mid-low frequency range below 1 kHz.