Quadratic Residue Research Articles

Semi-primitive Roots and Irreducible Quadratic Forms

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any gcd calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [3].

ON CONSECUTIVE RESIDUES AND NONRESIDUES UNDER A LINEAR MAP IN A FINITE FIELD

Abstract For fixed m and a, we give an explicit description of those subsets of ${\mathbb F}_{q}$ , q odd, for which both x and $mx+a$ are quadratic residues (and other combinations). These results extend and refine results that date back to Gauss.

Nonvanishing of L-function of some Hecke characters on cyclotomic fields

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.

On the Paley graph of a quadratic character

Abstract Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound.

Infinite series of 3‐designs in the extended quadratic residue code

AbstractIn this paper, we show an infinite series of 3‐designs in the extended quadratic residue codes over for a prime .

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

Eight Error Correction for (57,29,17) Quadratic Residue Code Over Binary Field

Eight Error Correction for (57,29,17) Quadratic Residue Code Over Binary Field

About quadratic residues in a class of rings

About quadratic residues in a class of rings

A Blockchain Empowered Smart Home Access Scheme Based on Zero-trust Architecture

The rapid popularity of smart home devices worldwide has brought a lot of convenience to people’s lives and spawned numerous new applications. However, it has also elevated data and privacy security risks in household scenarios to an unprecedented level. In this paper, we propose a device access solution based on a zero-trust network architecture to address the cybersecurity risks in smart home scenarios. The proposed solution utilizes a zero-knowledge identity verification algorithm based on quadratic residues to achieve mutual authentication and authorization between devices and central control without exposing device privacy. To enhance the efficiency of verification and authorization, we introduce an incentive model based on information asymmetric algorithms, distributing a portion of the verification tasks to devices with redundant computing power. By comparing with traditional methods, this solution demonstrates higher security, improved verification efficiency, and optimized allocation of computing resources, all while protecting device privacy.

Insert loss experiment and acoustic impedance theoretical study on the noise barrier with MPP-QRD top structure

The effective edge processing is the key to the performance of noise barrier. To prevent the sound diffraction within the low-medium frequency range, the quadratic residue diffuser (QRD) was combined with microperforated panel (MPP) in this study. The impedance of MPP with different parameters and QRD structure with N = 7 was calculated respectively and the matching degree of the composite structure was obtained. The insertion loss of noise barriers with different MPP-QRD top structures was field measured and compared to the equi-height vertical noise barrier. The results show that the degree of impedance matching between microperforated panels and quadratic residue diffusers is significantly correlated with their performance as the top structure of noise barrier. The acoustic impedance matched MPP-QRD top structure can significantly improve the insertion loss of the vertical noise barrierwith the same height. When the microperforated panel perforation is 3 %, the thickness is 0.8 mm, and the aperture is 0.6 mm, performance of the MPP-QRD structure is the best, the relative insertion loss can reach 4.07 dB.

Understanding diffusion versus scattering devices and how they work

Nearly 50 years ago, Dr. Manfred Schroeder defined the word “Diffuser” as both a process and a product. Since then, many product manufacturers, acousticians and academics have come to use the term for many different products and designs. A careful reading of Dr. Schroeder's 1975 and 1978 papers reveals a lot of significant information as to why certain products are diffusers and why other products do not meet his criteria for diffusion. This paper will show, with the aid of Schroeder's interpretations and modern particle simulations, exactly what he had in mind when he, and others he was working with, created what ultimately became the Quadratic Residue Diffuser or QRD. Additionally, it will show his journey from primitive-root theory to his final quadratic designs. A discussion of the term “Scattering” will also be included to define how Schroeder uses the word and why we need to be careful as to how it is used as well. Test data showing phase models will be used to show the importance of phase manipulation in describing diffusion.

Exploring a Graph Complement in Quadratic Congruence

In this work, we investigate essential definitions, defining G as a simple graph with vertices in Zn and subgraphs Γu and Γq as unit residue and quadratic residue graphs modulo n, respectively. The investigation extends to the degree of G, Γu, and Γq, illuminating the properties of these subgraphs in the context of quadratic congruences.

Partition of quadratic residues and non-residues in $$\mathbb {Z}_p^*$$ for an odd prime p

Partition of quadratic residues and non-residues in $$\mathbb {Z}_p^*$$ for an odd prime p

A novel chaotic map partially blind signature scheme based on quadratic residue problems

A novel chaotic map partially blind signature scheme based on quadratic residue problems

Predição da área de olho de lombo da carcaça por imagens ultrassonográficas em ovinos utilizando visão computacional

ABSTRACT: The present research created a tool to measure ultrasound images of the rib eye area in sheep. One hundred twenty-one ultrasound images of sheep were captured, with regions of interest segmented using the U-Net algorithm. The metrics adopted to evaluate automatic segmentations were Dicescore and intersection over union. Finally, a regression analysis was performed using the AdaBoost Regressor and Random Forest Regressor algorithms and the fit of the models was evaluated using the Mean Square Residuals, mean absolute error and coefficient of determination. The values obtained for the Dice metric were 0.94, and for Intersection over Union it was 0.89, demonstrating a high similarity between the actual and predicted values, ranging from 0 to 1. The values of Mean Quadratic Residuals, mean absolute error and coefficient The determination of the regressor models indicated the best fit for the Random Forest Regressor. The U-Net algorithm efficiently segmented ultrasound images of the Longissimus Dorsi muscle, with greater precision than the measurements performed by the specialist. This efficient segmentation allowed the standardization of rib eye area measurements and, consequently, the phenotyping of beef sheep on a large scale.

On the solution set of additive and multiplicative congruences modulo primes

Let p be an odd prime. In this paper, analogs of Wilson?s and Wolstenholme?s theorems on the solution sets S+ = {n ? Z* p | n ? a + b ? ab (mod p)} and S? = {n ? Z* p | n ? a ? b ? ab (mod p)} are given, where Z* p denote a reduced residue system modulo p. We also establish congruences about sum and product of the quadratic residues in S+ or in S? modulo p. Finally, we raise a problem on how to solve Hadamard?s conjecture in the last section.

Block designs from self-dual codes obtained from Paley designs and Paley graphs

In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct \(t\)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.

Secure Groups for Threshold Cryptography and Number-Theoretic Multiparty Computation

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3max(a,b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation.

Families of multi-level Legendre-like arrays

AbstractFamilies of new, multi-level integer 2D arrays are introduced here as an extension of the well-known binary Legendre sequences that are derived from quadratic residues. We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size $$p\times p$$ p × p for many prime values p. Previously delta functions were used as the discrete projections which, when back-projected, build 2D perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is $$p+1$$ p + 1 . All members of these multi-level array families have perfect autocorrelation and constant, minimal cross-correlation. Proofs are given for four useful and general properties of these new arrays. 1) They are comprised of odd integers, with values between at most $$-p$$ - p and $$+p$$ + p , with a zero value at just one location. 2) They have the property of ‘conjugate’ spatial symmetry, where the value at location (i, j) is always the negative of the value at location $$(p-i, p-j)$$ ( p - i , p - j ) . 3) Any change in the value assigned to the array’s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of $$p+1$$ p + 1 , $$p\times p$$ p × p arrays can be compressed to size $$(p+1)^2$$ ( p + 1 ) 2 and each family member can be exactly and rapidly unpacked in a single $$p\times p$$ p × p decompression pass.

Retracted: On Symmetry of Complete Graphs over Quadratic and Cubic Residues

Retracted: On Symmetry of Complete Graphs over Quadratic and Cubic Residues