Quadratic Residue Research Articles

A Counting Proof for When 2 Is a Quadratic Residue

Using the group consisting of the eight Möbius transformations x, – x, , , , and , we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq.

Inequalities between sums over prime numbers in progressions

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series.

A Scheme for Collective Encoding and Iterative Soft-Decision Decoding of Cyclic Codes of Prime Lengths: Applications to Reed–Solomon, BCH, and Quadratic Residue Codes

A novel scheme is presented for encoding and iterative soft-decision decoding of cyclic codes of prime lengths. The encoding of a cyclic code of a prime length is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords from the cyclic code. The joint-decoding allows for information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding Reed-Solomon and BCH codes of prime lengths, the proposed decoding scheme not only requires much lower decoding complexity than other soft-decision decoding algorithms for these codes, but also yields superior performance. The proposed decoding scheme can also achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords. The decoding scheme is also applied to quadratic residue codes.

Bilinear forms in Weyl sums for modular square roots and applications

Let q be a prime, P⩾1 and let Nq(P) denote the number of rational primes p⩽P that split in the imaginary quadratic field Q(−q). The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for Nq(P) in the range q1/4+ε⩽P⩽q, for any fixed ε>0. This improves upon what is implied by work of Pollack and Benli–Pollack.The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salié sums). Our estimate has a power saving in the so-called Pólya–Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in Fq.This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted L-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdős–Odlyzko–Sárközy.

An ultrathin planar acoustic metasurface diffuser with narrowband uniform reflection

Diffuse reflection of sound is desirable in many practical scenarios, such as architectural acoustics, but most existing designs of acoustic diffusers have bulky size, corrugated profile, or limited spatial resolution. We design an ultrathin planar acoustic diffuser for producing narrowband diffuse reflection with high uniformity via precise modulation of reflected wavefront and propose a metasurface-based implementation comprising a monolayer of Helmholtz-like resonators much smaller than the working wavelength in all three dimensions. Our design is benchmarked with a commercialized Schroeder diffuser and is numerically proven to be capable of scattering the illuminating wave more uniformly than the conventional mechanisms based on the quadratic residue sequence over wide incident angles. We anticipate our design with simplicity and capability to find important applications in diverse scenarios.

Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories

Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor N of an RCFT and the quadratic residues modulo N play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data transforms the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators.

Binary sequences derived from differences of consecutive quadratic residues

<p style='text-indent:20px;'>For a prime <inline-formula><tex-math id="M1">\begin{document}$ p\ge 5 $\end{document}</tex-math></inline-formula> let <inline-formula><tex-math id="M2">\begin{document}$ q_0,q_1,\ldots,q_{(p-3)/2} $\end{document}</tex-math></inline-formula> be the quadratic residues modulo <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula> in increasing order. We study two <inline-formula><tex-math id="M4">\begin{document}$ (p-3)/2 $\end{document}</tex-math></inline-formula>-periodic binary sequences <inline-formula><tex-math id="M5">\begin{document}$ (d_n) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ (t_n) $\end{document}</tex-math></inline-formula> defined by <inline-formula><tex-math id="M7">\begin{document}$ d_n = q_n+q_{n+1}\bmod 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ t_n = 1 $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ q_{n+1} = q_n+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ t_n = 0 $\end{document}</tex-math></inline-formula> otherwise, <inline-formula><tex-math id="M11">\begin{document}$ n = 0,1,\ldots,(p-5)/2 $\end{document}</tex-math></inline-formula>. For both sequences we find some sufficient conditions for attaining the maximal linear complexity <inline-formula><tex-math id="M12">\begin{document}$ (p-3)/2 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>Studying the linear complexity of <inline-formula><tex-math id="M13">\begin{document}$ (d_n) $\end{document}</tex-math></inline-formula> was motivated by heuristics of Caragiu et al. However, <inline-formula><tex-math id="M14">\begin{document}$ (d_n) $\end{document}</tex-math></inline-formula> is not balanced and we show that a period of <inline-formula><tex-math id="M15">\begin{document}$ (d_n) $\end{document}</tex-math></inline-formula> contains about <inline-formula><tex-math id="M16">\begin{document}$ 1/3 $\end{document}</tex-math></inline-formula> zeros and <inline-formula><tex-math id="M17">\begin{document}$ 2/3 $\end{document}</tex-math></inline-formula> ones if <inline-formula><tex-math id="M18">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sufficiently large. In contrast, <inline-formula><tex-math id="M19">\begin{document}$ (t_n) $\end{document}</tex-math></inline-formula> is not only essentially balanced but also all longer patterns of length <inline-formula><tex-math id="M20">\begin{document}$ s $\end{document}</tex-math></inline-formula> appear essentially equally often in the vector sequence <inline-formula><tex-math id="M21">\begin{document}$ (t_n,t_{n+1},\ldots,t_{n+s-1}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M22">\begin{document}$ n = 0,1,\ldots,(p-5)/2 $\end{document}</tex-math></inline-formula>, for any fixed <inline-formula><tex-math id="M23">\begin{document}$ s $\end{document}</tex-math></inline-formula> and sufficiently large <inline-formula><tex-math id="M24">\begin{document}$ p $\end{document}</tex-math></inline-formula>.</p>

Pseudo Random Binary Sequence Based on Cyclic Difference Set

With the increasing reliance on technology, it has become crucial to secure every aspect of online information where pseudo random binary sequences (PRBS) can play an important role in today’s world of Internet. PRBS work in the fundamental mathematics behind the security of different protocols and cryptographic applications. This paper proposes a new PRBS namely MK (Mamun, Kumu) sequence for security applications. Proposed sequence is generated by primitive polynomial, cyclic difference set in elements of the field and binarized by quadratic residue (QR) and quadratic nonresidue (QNR). Introduction of cyclic difference set makes a special contribution to randomness of proposed sequence while QR/QNR-based binarization ensures uniformity of zeros and ones in sequence. Besides, proposed sequence has maximum cycle length and high linear complexity which are required properties for sequences to be used in security applications. Several experiments are conducted to verify randomness and results are presented in support of robustness of the proposed MK sequence. The randomness of proposed sequence is evaluated by popular statistical test suite, i.e., NIST STS 800-22 package. The test results confirmed that the proposed sequence is not affected by approximations of any kind and successfully passed all statistical tests defined in NIST STS 800-22 suite. Finally, the efficiency of proposed MK sequence is verified by comparing with some popular sequences in terms of uniformity in bit pattern distribution and linear complexity for sequences of different length. The experimental results validate that the proposed sequence has superior cryptographic properties than existing ones.

Quadratic residues and quartic residues modulo primes

In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let [Formula: see text] be an odd prime and let [Formula: see text] be any integer. We determine completely the product [Formula: see text] modulo [Formula: see text]; for example, if [Formula: see text] then [Formula: see text] where [Formula: see text] denotes the Legendre symbol. We also determine [Formula: see text] modulo [Formula: see text].

SEOTP: a new secure and efficient ownership transfer protocol based on quadric residue and homomorphic encryption

In systems equipped with radio frequency identification (RFID) technology, several security concerns may arise when the ownership of a tag should be transferred from one owner to another, e.g. the confidentiality of information related to the old owner or the new owner. Therefore, this transfer is usually done via a security protocol called the ownership transfer protocol. If the ownership of several things together transmitted from one owner to another during a single session, the protocol is referred to as the group ownership transfer protocol. Lee et al. recently proposed a new group ownership transfer protocol by using a cloud server, as a trusted third-party, and based on homomorphic encryption and quadratic residue. In this paper, at first, we explain some essential security attacks against this recently proposed RFID group ownership transfer protocol. The success probability of all attacks that are presented in this paper is one, and the complexities are just a run of the protocol. Zhu et al. also, to providesimultaneoustransfer of a group of tags in a multi-owner environment, proposed a lightweight anonymous group ownership transfer protocol. In this paper, we show that it suffers from a desynchronization attack. The success probability of this attack is one, and its complexity is only five runs of group ownership transfer protocol. Besides, to overcome the Lee et al.’s protocol security weaknesses, we present a new group ownership transfer protocol called SEOTP that is resistant against all known active and passive attacks, including the attacks presented in this paper. The provided security proof through informal methods and also formal methods such as Barrows–Abadi–Needham logic and the Scyther tool show the proposed protocol’s security correctness.

Double quadratic residue codes and self-dual double cyclic codes

In this paper, we introduce double Quadratic Residue Codes (QRC) of length $$n=p+q$$ for prime numbers p and q in the ambient space $${{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.$$ We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when $$p=q.$$ We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting.

Algebraic decoding of the (41, 21, 9) quadratic residue code without determining the unknown syndromes

Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper presents a fast algebraic scheme for decoding the binary (41, 21, 9) QR code for correcting up to 4 errors on an one-case-by-one-case basis, in which the calculation of unknown syndromes is eliminated and the conditions for checking for the various numbers of errors that exist in the received word are also simplified compared to those from Lin T. C. et al.’s algorithm, which is a traditional algebraic decoding algorithm (ADA). The computational complexity of the decoder performing the binary (41, 21, 9) QR code is analyzed thoroughly, which demonstrates that the proposed decoding scheme is faster, simpler, and more suitable for implementation than Lin T. C. et al.’s algorithm. Numerical emulation results demonstrate that the proposed decoding scheme achieves the same error-rate performance as Lin T. C. et al.’s algorithm, but it has a significantly decreased the complexity of this decoder in terms of the CPU time.

An Improved Anonymous Authentication Protocol for Wearable Health Monitoring Systems

Wearable health monitoring system (WHMS), which helps medical professionals to collect patients’ healthcare data and provides diagnosis via mobile devices, has become increasingly popular thanks to the significant advances in the wireless sensor network. Because health data are privacy-related, they should be protected from illegal access when transmitted over a public wireless channel. Recently, Jiang et al. presented a two-factor authentication protocol on quadratic residues with fuzzy verifier for WHMS. However, we observe that their scheme is vulnerable to known session special temporary information (KSSTI) attack, privileged insider attack, and denial-of-service (DoS) attack. To defeat these weaknesses, we propose an improved two-factor authentication and key agreement scheme for WHMS. Through rigorous formal proofs under the random oracle model and comprehensive informal security analysis, we demonstrate that the improved scheme overcomes the disadvantages of Jiang et al.’s protocol and withstands possible known attacks. In addition, comparisons with several relevant protocols show that the proposed scheme achieves more security features and has suitable efficiency. Thus, our scheme is a reasonable authentication solution for WHMS.

Refined estimates concerning sumsets contained in the roots of unity

We prove that the clique number of the Paley graph is at most $\sqrt{p/2} + 1$, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.

The Quadratic Residues and Some of Their New Distribution Properties

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues mod p . The main methods we used are the properties of the Legendre’s symbol mod p , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.

On the Stochasticity Parameter of Quadratic Residues

Following V.I. Arnold, we define the stochasticity parameter S(U) of a set $$U \subseteq {{\mathbb{Z}}_{M}}$$ to be the sum of squares of consecutive distances between the elements of U. The stochasticity parameter of the set RM of quadratic residues modulo M is studied. We compare $$S({{R}_{M}})$$ with the average value $$s(k) = s(k,M)$$ of S(U) over all subsets of $$U \subseteq {{\mathbb{Z}}_{M}}$$ of size k. It is proved that (a) for a set of moduli of positive lower density, we have $$S({{R}_{M}}) < s{\text{(|}}{{R}_{M}}{\text{|}})$$ ; and (b) for infinitely many moduli, $$S({{R}_{M}}) > s\left( {\left| {{{R}_{M}}} \right|} \right)$$ .

Secure attribute-based search in RFID-based inventory control systems

We develop a secure attribute-based search protocol for Radio Frequency Identification (RFID) systems. This protocol can be used to simultaneously identify groups of items that share a set of attribute values. To the best of our knowledge, this is the first such work with the potential to significantly enhance the security and intelligence of RFID-enabled applications in inventory control and supply chain management. The protocol is designed to be lightweight, suited for resource-constrained basic passive tags, and compliant with the Electronic Product Code (EPC) standards. This is achieved by exploiting the zero knowledge properties of quadratic residues. The security and privacy properties offered by the protocol are rigorously proven through formal verification.

Modified quadratic residue constructions and new extremal binary self-dual codes of lengths 64, 66 and 68

In this work we consider modified versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary field and the rings F2+uF2 and F4+uF4 for different prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary self-dual codes of different lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8].

On the distribution of quadratic residues and non-residues modulo composite integers and applications to cryptography

We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form a+X={(a+x)modn∣x∈X}, where n is a prime or the product of two primes and X is a subset of integers with given Jacobi symbols modulo prime factors of n. We then present applications of these formulas to Cocks’ identity-based encryption scheme and statistical indistinguishability.

Fractal behavior of BDS-2 satellite clock offsets and its application to real-time clock offsets prediction

The satellite clocks of the BeiDou Navigation Satellite System (BDS) have characteristic differences compared to those of the global positioning system and Galileo. Therefore, the satellite clock offsets prediction method of BDS is different from those of these two systems. The basis for establishing a more appropriate prediction model is to clarify the statistical characteristics of the BDS satellite clock offsets, which can be reflected by fitting residuals of precise clock errors. Fractal behavior is generally not considered in existing studies. In this study, the rescaled range analysis method is improved by using the moving-average method in order to verify the fractal behavior of the BDS clock offsets. The computation results of the Hurst exponent show that the BDS clock offsets are fractal series with long-term memory, and the memory spans are obtained by V-statistic calculation. The quadratic polynomial fitting residuals of BDS clock offsets are fitted by using the periodic model and fractal interpolation model, where the latter approach has a higher accuracy. In the predictive modeling process, the quadratic polynomial forecasting model is improved by using the memory span, so that the forecasting model not only inherits the overall clock offsets trend but also considers the local memory trend. The fractal interpolation prediction model of the clock offsets is established by using the extension method of the affine iteration function system. The experimental results show that the average prediction accuracy of the fractal model in 3, 6, 12, and 24 h can reach 1.4890, 2.0222, 3.1609, and 4.9278 ns, which is 57.74%, 50.20%, 52.66%, and 49.42% higher than the products from the China iGMAS ultra-rapid prediction, respectively.