Legendre Symbol Research Articles

Legendre-signed partition numbers

Let f:N→{0,±1}, for n∈N let Π[n] be the set of partitions of n, and for all partitions π=(a1,a2,…,ak)∈Π[n] letf(π):=f(a1)f(a2)⋯f(ak). With this we define the f-signed partition numbersp(n,f)=∑π∈Π[n]f(π).In this paper, for odd primes p we derive asymptotic formulae for p(n,χp) as n→∞, where χp(n) is the Legendre symbol (np) associated to p. A similar asymptotic formula for p(n,χ2) is also established, where χ2(n) is the Kronecker symbol (n2). Special attention is paid to the sequence (p(n,χ5))N, and a formula for p(n,χ5) supporting the recent discovery that p(10j+2,χ5)=0 for all j≥0 is discussed. As a corollary, our main results imply that the periodic vanishing displayed by (p(n,χ5))N does not occur in any sequence (p(n,χp))N for p≠5 such that p≢1(mod8). In addition, work of Montgomery and Vaughan on exponential sums with multiplicative coefficients is applied to establish a uniform upper bound on certain doubly infinite series involving multiplicative functions f with |f|≤1.

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.

On the Baillie PSW-conjecture

The Baillie PSW hypothesis was formulated in 1980 and was named after the authors R. Baillie, C. Pomerance, J. Selfridge and S. Wagstaff. The hypothesis is related to the problem of the existence of odd numbers n \equiv \pm 2 (mod 5), which are both Fermat and Lucas-pseudoprimes (in short, FL-pseudoprimes). A Fermat pseudoprime to base a is a composite number n satisfying the condition an - 1 \equiv 1 (mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying Fn - e(n) \equiv 0 (mod n), where n(e) is the Legendre symbol e(n) = \bigl( n 5 \bigr) , Fm the mth term of the Fibonacci series. According to Baillie’s PSW conjecture, there are no FL-pseudoprimes. If the hypothesis is true, the combined primality test checking Fermat and Lucas conditions for odd numbers not divisible by 5 gives the correct answer for all numbers of the form n \equiv \pm 2 (mod 5), which generates a new deterministic polynomial primality test detecting the primality of 60 percent of all odd numbers in just two checks. In this work, we continue the study of FL-pseudoprimes, started in our article "On a combined primality test" published in the journal "Izvestia VUZov.Matematika" No. 12 in 2022. We have established new restrictions on probable FL-pseudoprimes and described new algorithms for checking FL-primality, and with the help of them we proved the absence of such numbers up to the boundary B = 1021, which is more than 30 times larger than the previously known boundary 264 found by J. Gilchrist in 2013. An inaccuracy in the formulation of theorem 4 in the mentioned article has also been corrected.

Design of Universal Code Generator for Multi-Constellation Multi-Frequency GNSS Receiver

A multi-constellation, multi-frequency Global Navigation Satellite System (GNSS) receiver is capable of simultaneously receiving signals from multiple satellite constellations across various frequency bands. This allows for increased observations, thereby enhancing navigation accuracy, continuity, effectiveness, and reliability. The spread spectrum code structures used in satellite navigation signals differ among systems. Compatible code generators are employed in multi-constellation, multi-frequency GNSS receivers to support tasks such as signal acquisition and tracking. There are three main types of spread spectrum code structures: Linear Feedback Shift Register (LFSR), Legendre sequences, and Memory codes. The Indian Regional Navigation Satellite System (IRNSS) released the L1-SPS (Standard Positioning Service) signal format in August 2023, which utilizes the Interleaved Z4-linear ranging code (IZ4 code) as its spread spectrum code. Currently, there is no universal code generator design compatible with the IZ4 code. In this paper, a proposed universal code generator is based on the hardware structure of the IRNSS IZ4 code generator. It achieves compatibility with all LFSR-based spread spectrum codes and enables parallel generation of multiple sets of GNSS signal spread spectrum codes, thereby improving hardware utilization efficiency. The proposed structure is implemented and validated using FPGA design, and resource consumption is provided as part of the validation results.

On the symmetric 2-adic complexity of periodic binary sequences

Symmetric 2-adic complexity is a better measure for assessing the strength of a periodic binary sequence against the rational approximation attack. We study the symmetric 2-adic complexity of the sequences with known 2-adic complexity. To this end, we propose an equivalent definition of the symmetric 2-adic complexity and two sufficient conditions for a periodic sequence to achieve the maximum symmetric 2-adic complexity. As a first application, several types of sequences with maximum 2-adic complexity are easily shown to have the maximum symmetric 2-adic complexity. In addition, with the new definition, by analysing the algebraic structure of the sequences and applying the method for the calculation of their 2-adic complexity, we determine the symmetric 2-adic complexity of the Ding-Helleseth-Martinsen sequence and the sequence constructed by interleaving a pair of Legendre sequences.

The Integral Points on the Elliptic Curve y 2 = 7 qx(x 2 + 128)

The positive integer points of elliptic curves are very important in the theory of numbers and arithmetic algebra; it has a wide range of applications in cryptography and other fields.The main purpose of this paper is to apply elementary methods, the properties of congruence and Legendre symbols, to study the elliptic curve y 2 = 7 qx(x 2 + 128)and proved that the elliptic curve has at most three integer points when q = 5(mod8) is an odd prime number.

A Study on Positive Integer Points of Elliptic Curve y 2 = 7mx(x 2 + 512)

By using elementary methods such as the properties of congruence and Legendre symbols, it is proved that the elliptic curve y 2 = 7mx(x 2 + 512) has at most 1 positive integer point when q ≡ 5(mod8) is an odd prime number.

Pseudorandom Binary Sequences: Quality Measures and Number-Theoretic Constructions

In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order k, expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

On the q-binomial identities involving the Legendre symbol modulo 3

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

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.

On Graham partitions twisted by the Legendre symbol

Abstract We investigate when there is a partition of a positive integer n n , n = f ( λ 1 ) + f ( λ 2 ) + ⋯ + f ( λ ℓ ) , n=f\left({\lambda }_{1})+f\left({\lambda }_{2})+\cdots +f\left({\lambda }_{\ell }), satisfying that 1 = χ p ( λ 1 ) λ 1 + χ p ( λ 2 ) λ 2 + ⋯ + χ p ( λ ℓ ) λ ℓ , 1=\frac{{\chi }_{p}\left({\lambda }_{1})}{{\lambda }_{1}}+\frac{{\chi }_{p}\left({\lambda }_{2})}{{\lambda }_{2}}+\cdots +\frac{{\chi }_{p}\left({\lambda }_{\ell })}{{\lambda }_{\ell }}, where χ p {\chi }_{p} is the Legendre symbol modulo prime p p and f ( k ) = k f\left(k)=k or the k k th m m -gonal number with m = 3 m=3 , 4, or 5.

Evaluations of three symmetric Toeplitz determinants

In this paper, we evaluate the following three symmetric Toeplitz determinants: det [ | j − k | + δ jk ] 1 ≤ j , k ≤ n , det [ F | j − k | + δ jk ] 1 ≤ j , k ≤ n and det [ ( | j − k | 3 ) − δ jk ] 1 ≤ j , k ≤ n , where δ jk is the Kronecker delta, ( F i ) i ≥ 0 is the Fibonacci sequence, and ( ⋅ 3 ) is the Legendre symbol.

Algebraic Attacks against Grendel: An Arithmetization-Oriented Primitive with the Legendre Symbol

The rise of modern cryptographic protocols such as Zero-Knowledge proofs and secure Multi-party Computation has led to an increased demand for a new class of symmetric primitives. Unlike traditional platforms such as servers, microcontrollers, and desktop computers, these primitives are designed to be implemented in arithmetical circuits. In terms of security evaluation, arithmetization-oriented primitives are more complex compared to traditional symmetric cryptographic primitives. The arithmetization-oriented permutation Grendel employs the Legendre Symbol to increase the growth of algebraic degrees in its nonlinear layer. To analyze the security of Grendel thoroughly, it is crucial to investigate its resilience against algebraic attacks. This paper presents a preimage attack on the sponge hash function instantiated with the complete rounds of the Grendel permutation, employing algebraic methods. A technique is introduced that enables the elimination of two complete rounds of substitution permutation networks (SPN) in the sponge hash function without significant additional cost. This method can be combined with univariate root-finding techniques and Gröbner basis attacks to break the number of rounds claimed by the designers. By employing this strategy, our attack achieves a gain of two additional rounds compared to the previous state-of-the-art attack. With no compromise to its security margin, this approach deepens our understanding of the design and analysis of such cryptographic primitives.

Artin-Schreier curves given by [formula omitted]-linearized polynomials

Let Fq be a finite field with q elements, where q is a power of an odd prime p. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve yq−y=x⋅F(x)−λ, where F(x) is a Fq-linearized polynomial and λ∈Fq. Our results provide a characterization of the number of affine rational points of this curve in the extension Fqr of Fq, for gcd⁡(q,r)=1. In the case F(x)=xqi−x we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.

Random Polynomials in Legendre Symbol Sequences

Abstract It is important in cryptographic applications that the “key” used should be generated from a random seed. Thus, if the Legendre symbol sequence generated by a polynomial (as proposed by Hoffstein and Lieman) is used, that is { ( f ( 1 ) p ) , ( f ( 2 ) p ) , ( f ( 3 ) p ) , ⋯ , ( f ( p ) p ) } , \left\{ {\left( {{{f\left( 1 \right)} \over p}} \right),\left( {{{f\left( 2 \right)} \over p}} \right),\left( {{{f\left( 3 \right)} \over p}} \right), \cdots ,\left( {{{f\left( p \right)} \over p}} \right)} \right\}, then it is important to choose the polynomial f “almost” at random. Goubin, Mauduit, and Sárközy presented some not very restrictive conditions on the polynomial f, but these conditions may not be satisfied if we choose a “truly” random polynomial. However, how can it be guaranteed that the pseudorandom measures of the sequence should be small for almost "random" polynomials? These semirandom polynomials will be constructed with as few modifications as necessary from a truly random polynomial.

A note on complex conference matrices

Let Fq be the Galois field of order q=pm, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of Fq⁎ and for any b∈Fq⁎ we have∑a∈Fq⁎ϰ(a)ϰ(a+b)‾=−1. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that C⁎C=(n−1)I. Paley used finite fields with odd orders q=pm, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders q+1 whenever q≡1(mod4) and real skew-symmetric conference matrices of orders q+1 whenever q≡−1(mod4). In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order q+1 whenever q≥4 is any prime power as well as a complex skew-symmetric conference matrix of order q+1 whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ϰpk for any integer k=1,...m−1 are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.

On congruences involving Apéry numbers

In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime p > 3 p>3 and positive odd integer m m , we prove that there is a p p -adic integer c m c_m only depending on m m such that ∑ k = 0 p − 1 ( 2 k + 1 ) m ( − 1 ) k A k ≡ c m p ( p 3 ) ( mod p 3 ) , \begin{equation*} \sum _{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left (\frac {p}{3}\right )\pmod {p^3}, \end{equation*} where A k = ∑ j = 0 k ( k j ) 2 ( k + j j ) 2 A_k=\sum _{j=0}^{k}\binom {k}{j}^2\binom {k+j}{j}^2 is the Apéry number and ( . p ) (\frac {.}{p}) is the Legendre symbol.

The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications

AbstractSequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

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].

On certain congruences involving central binomial coefficients

Let p be an odd prime. In this paper, using some properties of Fibonacci numbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, we establish some congruences involving central binomial coefficient modulo p and p^2. We also give some new identities for hyperbolic functions.