Legendre Symbol Research Articles

A Novel VLSI Architecture for Multi-Constellation and Multi-Frequency GNSS Acquisition Engine

This paper proposes a novel VLSI architecture of GNSS acquisition engine based on short-time correlation combined with an FFT scheme. The architecture supports multi-constellation systems and multi-frequency satellite signals flexibly by the manner of time-division multiplexing. The supported signals include GPS, BDS, GLONASS, GALILEO, QZSS, IRNSS, and SBAS. Compared with other direct acquisition structures, the search efficiency of the acquisition engine is improved by using the IF playback structure. Based on the characteristics of L1C and B1C signal spread spectrum codes, an efficient generation method and the circuit structure of Legendre sequence which is compatible with L1C and B1C spread spectrum codes are proposed. It can effectively reduce the die area of the spread spectrum code generator and the generation time of the L1C/B1C spread spectrum code. Combining with the acquisition scheme adopted in this paper, the structure of the short-time correlators’ array is optimized, and the maximum clock frequency of the acquisition engine is significantly improved. The acquisition engine proposed in this paper is implemented in a 55-nm CMOS technology. The system occupied a silicon area of 2.1 mm2 and consumes only 72.02-mW power while realizing the maximum clock frequency at about 333.33 MHz.

Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences

Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.

R-th order nonlinearity, correlation measure and least significant bit of the discrete logarithm

Each finite binary sequence (sh) is associated with a Boolean function B. The correlation measure of order k and the r-th order nonlinearity are figures of merit for the unpredictability of (sh) and B, respectively. We estimate the r-th order nonlinearity of B in terms of the correlation measure of order 2r of (sh). We apply our result to Boolean functions associated with the Legendre sequence, that is, the binary sequence describing the least significant bit of the discrete logarithms in the finite field $\mathbb {F}_{p}$ of p elements, where p > 2 is a prime.

Large values of short character sums

TextIn this paper, we prove that for any A>0 there exist infinitely many primes p for which sums of the Legendre symbols modulo p over an interval of length (ln⁡p)A can take large values. VideoFor a video summary of this paper, please visit https://youtu.be/wNvv8wujF3o.

Linear complexity of Legendre‐polynomial quotients

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict $\left(\frac{0}{p}\right)=1$. In an earlier work, the linear complexity of $(f_u)$ was determined for $w=p-1$ under the assumption of $2^{p-1}\not\equiv 1 \pmod {p^2}$. In this work, we give possible values on the linear complexity of $(f_u)$ for all $1\le w<p-1$ under the same conditions. We also state that the case of larger $w(\geq p)$ can be reduced to that of $0\leq w\leq p-1$.

Distribution questions for trace functions with values in cyclotomic integers and their reductions

We consider ℓ \ell -adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.

On the 2-Adic Complexity of the Two-Prime Generator

Hu introduced a simple method to compute the 2-adic complexity of any periodic binary sequence with ideal two-level autocorrelation. We extend this approach to some other sequences. First, we provide a substantially shorter proof of the maximality of the 2-adic complexity of the Legendre sequence of period $N\equiv 1\bmod 4$ first proved by Xiong et al . Then, we show that the 2-adic complexity of the two-prime generator of period $pq$ with two odd primes $p\ne q$ attains the maximum if $(q+1)/4\le p\le 4q-1$ . This result was only known for twin primes $q=p+2$ before. For arbitrary odd primes $p\ne q$ , we can still prove that the 2-adic complexity of the two-prime generator is close to its period.

Infinite family of equi-isoclinic planes in Euclidean odd dimensional spaces and of complex symmetric conference matrices of odd orders

A n-set of equi-isoclinic planes in Rr is a set of n planes spanning Rr each pair of which has the same non-zero angle arccos⁡λ. We prove that for any odd integer k≥3 such that 2k=pα+1, p an odd prime, α a positive integer the maximum number of equi-isoclinic planes with angle arccos⁡12k−2 in R2k−1 is equal to 2k−1. It is shown that the solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2k−1, and that all these constructions are performed by use of the Legendre symbol of the Galois field GF(pα).

Elliptic Curve Integral Points on y2 = x3 + 19x + 46

By using elementary number theory methods, such as congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 19x + 46 has no integer point.

Trace representation of Legendre sequences over non-binary fields

For distinct odd primes N and p, we view the N-periodic binary Legendre sequence as a p-ary sequence and present its trace representation via trace functions over \({\mathbb {F}}_p\). We use a skill to calculate the Mattson–Solomon polynomials of Legendre sequences and then describe the Mattson–Solomon polynomials by means of trace functions over \({\mathbb {F}}_p\).

Hadamard matrices and the spectrum of quadratic symmetric polynomials over finite fields

In this article, we present a beautiful connection between Hadamard matrices and exponential sums of quadratic symmetric polynomials over Galois fields. This connection appears when the recursive nature of these sequences is analyzed. We calculate the spectrum for the Hadamard matrices that dominate these recurrences. The eigenvalues depend on the Legendre symbol and the quadratic Gauss sum over finite field extensions. In particular, these formulas allow us to calculate closed formulas for the exponential sums over Galois field of quadratic symmetric polynomials. Finally, in the particular case of finite extensions of the binary field, we show that the corresponding Hadamard matrix is a permutation away from a classical construction of these matrices.

Construction of Period &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$qp$ &lt;/tex-math&gt; &lt;/inline-formula&gt; PGISs With Degrees Equal to or Larger Than Four

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of $N = qp$ , where $q$ and $p$ are distinct primes. The study begins with the partitioning of a ring $\mathbb {Z}_{N}$ into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of $\mathbb {Z}_{N}$ should be carried out; here, two cases, the even period $N = 2p$ and the odd period $N = qp$ , are treated separately. We can adopt the Legendre sequences of the prime period $p$ to construct PGISs of period $2p$ with degrees larger than four. For the case of period $qp$ , we introduce the Jacobi symbols to partition $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.

On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation

Let \begin{document}$q$\end{document} be a prime greater than 4. In this paper, we determine the coefficients of the discrete Fourier transform over the finite field \begin{document}$\mathbb {F}_q$\end{document} of two classes of quaternary sequences of even length with optimal autocorrelation. They are quaternary sequence with period \begin{document}$2p$\end{document} derived from binary Legendre sequences and quaternary sequence with period \begin{document}$2p(p+2)$\end{document} derived from twin-prime sequences pair. As applications, the linear complexities over the finite field \begin{document}$\mathbb {F}_q$\end{document} of both of the quaternary sequences are determined.

Improved Lower Bound for the Mahler Measure of the Fekete Polynomials

We show that there is an absolute constant $$c > 1/2$$ such that the Mahler measure of the Fekete polynomials $$f_p$$ of the form $$\begin{aligned} f_p(z) := \sum _{k=1}^{p-1}{\left( \frac{k}{p} \right) z^k} \end{aligned}$$ (where the coefficients are the usual Legendre symbols) is at least $$c\sqrt{p}$$ for all sufficiently large primes p. This improves the lower bound $$\left( \frac{1}{2} - \varepsilon \right) \sqrt{p}$$ known before for the Mahler measure of the Fekete polynomials $$f_p$$ for all sufficiently large primes $$p \ge c_{\varepsilon }$$ . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

Low Correlation Sequences From Linear Combinations of Characters

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with a crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this paper, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.

Lq Norms of Fekete and Related Polynomials

AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

Power series approximations to Fekete polynomials

We study how well Fekete polynomials Fp(X)=∑n=0p−1npXn∈Z[X]with the coefficients given by Legendre symbols modulo a prime p can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.

The least number with prescribed Legendre symbols and representation by binary quadratic forms of small discriminant

In this article we estimate the number of integers up to X which can be properly represented by a positive-definite, binary, integral quadratic form of small discriminant. This estimate follows from understanding the vector of signs that arises from computing the Legendre symbol of small integers n at multiple primes.

Trigonometric series and special values of L-functions

Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L-functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L-functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions.

Quadratic nonresidues below the Burgess bound

For any odd prime number [Formula: see text], let [Formula: see text] be the Legendre symbol, and let [Formula: see text] be the sequence of positive nonresidues modulo [Formula: see text], i.e. [Formula: see text] for each [Formula: see text]. In 1957, Burgess showed that the upper bound [Formula: see text] holds for any fixed [Formula: see text]. In this paper, we prove that the stronger bound [Formula: see text] holds for all odd primes [Formula: see text] provided that [Formula: see text] where the implied constants are absolute. For fixed [Formula: see text], we also show that there is a number [Formula: see text] such that for all odd primes [Formula: see text], there are [Formula: see text] natural numbers [Formula: see text] with [Formula: see text] provided that [Formula: see text]