Pairwise Coprime Research Articles

Symmetric Encryption Algorithms in a Polynomial Residue Number System

In this paper, we develop the theoretical provisions of symmetric cryptographic algorithms based on the polynomial residue number system for the first time. The main feature of the proposed approach is that when reconstructing the polynomial based on the method of undetermined coefficients, multiplication is performed not on the found base numbers but on arbitrarily selected polynomials. The latter, together with pairwise coprime residues of the residue class system, serve as the keys of the cryptographic algorithm. Schemes and examples of the implementation of the developed polynomial symmetric encryption algorithm are presented. The analytical expressions of the cryptographic strength estimation are constructed, and their graphical dependence on the number of modules and polynomial powers is presented. Our studies show that the cryptanalysis of the proposed algorithm requires combinatorial complexity, which leads to an NP‐complete problem.

Fermi isospectrality for discrete periodic Schrödinger operators

AbstractLet , where , , are pairwise coprime. Let be the discrete Schrödinger operator, where Δ is the discrete Laplacian on and the potential is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension : If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentials V and Y are the same (this feature is referred to as Fermi isospectrality of V and Y), and Y is a separable function, then V is separable; If two complex‐valued Γ‐periodic potentials V and Y are Fermi isospectral and both and are separable functions, then, up to a constant, lower dimensional decompositions and are Floquet isospectral, ; If a real‐valued Γ‐potential V and the zero potential are Fermi isospectral, then V is zero. In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.

Triple Coprime Vector Array for DOA and Polarization Estimation: A Perspective of Mutual Coupling Isolation

Traditional polarization-sensitive sensors involve a triplet of spatially collocated, orthogonally oriented, and diversely polarized electric dipoles. However, this kind of sensor has the drawback of severe mutual coupling among the three dipoles due to the characteristic of collocation, as well as low radiation efficiency because of the short length of the dipoles. Based on this problem, in this study we designed a new array structure called a ‘triple coprime array (TCA)’, equipped with long electric dipoles to obtain higher radiation efficiency. In this structure, the dipoles within different subarrays have orthogonal polarization modes, leading to mutual coupling isolation. The dipole interval of the subarrays is enlarged by means of a pairwise coprime relationship, which further weakens the mutual coupling effect and extends the array aperture. Simultaneously, a stable direction-of-arrival (DOA) and polarization estimation method is proposed. DOA information is accurately refined from the three subarrays without ambiguity problems, with the triple coprime characteristic improving the estimation results. Subsequently, polarization estimates can be obtained using the reconstructed model matrix and the least squares method. Numerous theoretical analyses were conducted and extensive simulation results verified the advantages of the TCA structure in mutual coupling, along with the superiority of the proposed joint DOA and polarization estimation algorithm in terms of estimation accuracy.

Smooth Points of the Space of Plane Foliations with a Center

Abstract We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree $d$.

On linear independence of Dirichlet L values

The study of linear independence of L(k,χ) for a fixed integer k>1 and varying χ depends critically on the parity of k vis-à-vis χ. This has been investigated by a number of authors for Dirichlet characters χ of a fixed modulus and having the same parity as k. The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a single number field lurking in the background). This entails a very careful and hands-on dealing with the arithmetic of compositum of number fields which we undertake in this work. Our results extend earlier works of the first author with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata.

Equivariant embeddings of manifolds into Euclidean spaces

Suppose a finite group G acts on a manifold M. By a theorem of Mostow, also Palais, there is a G-equivariant embedding of M into the m-dimensional Euclidean space Rm for some m. We are interested in some explicit bounds of such m.First we provide an upper bound: there exists a G-equivariant embedding of M into Rd|G|+1, where |G| is the order of G and M embeds into Rd. Next we provide a lower bound for finite cyclic group action G: If there are l points having pairwise co-prime lengths of G-orbits greater than 1 and there is a G-equivariant embedding of M into Rm, then m≥2l.Some applications to surfaces are given.

On an extension of a question of Baker

It is an open question of Baker whether the numbers [Formula: see text] for nontrivial Dirichlet characters [Formula: see text] with period [Formula: see text] are linearly independent over [Formula: see text]. The best known result is due to Baker, Birch and Wirsing which affirms this when [Formula: see text] is co-prime to [Formula: see text]. In this paper, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] values as [Formula: see text] varies over nontrivial Dirichlet characters with period [Formula: see text]. Then for any finite set of pairwise co-prime natural numbers [Formula: see text] with [Formula: see text] [Formula: see text], we show that the set [Formula: see text] is linearly independent over [Formula: see text]. In the process, we also extend a result of Okada about linear independence of the cotangent values over [Formula: see text] as well as a result of Murty–Murty about [Formula: see text] linear independence of such [Formula: see text] values. Finally, we prove [Formula: see text] linear independence of such [Formula: see text] values of Erdösian functions with distinct prime periods [Formula: see text] for [Formula: see text] with [Formula: see text] [Formula: see text].

Bent functions in the partial spread class generated by linear recurring sequences

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all mathcal{PS}mathcal{}^- and mathcal{PS}mathcal{}^+ bent functions of n=8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d=2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function.

Integral points on the congruent number curve

We study integral points on the quadratic twists E D : y 2 = x 3 − D 2 x \mathcal {E}_D:y^2=x^3-D^2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2 E D ( Q ) 2\mathcal {E}_D(\mathbb {Q}) in E D ( Q ) \mathcal {E}_D(\mathbb {Q}) and show that their total is ≪ ( 3.8 ) rank ⁡ E D ( Q ) \ll (3.8)^{\operatorname {rank} \mathcal {E}_D(\mathbb {Q})} . We further show that the average number of non-torsion integral points in this family is bounded above by 2 2 . As an application we also deduce from our upper bounds that the system of simultaneous Pell equations a X 2 − b Y 2 = d , b Y 2 − c Z 2 = d aX^2-bY^2=d,\ bY^2-cZ^2=d for pairwise coprime positive integers a , b , c , d a,b,c,d , has at most ≪ ( 3.6 ) ω ( a b c d ) \ll (3.6)^{\omega (abcd)} integer solutions.

A characterization of strong semilattices of periodic groups and rectangular bands by disjunctions of identities

Each completely regular semigroup is a semilattice of completely simple semigroups. The more specific concept of a strong semilattice provides the concrete product between two arbitrary elements. We characterize strong semilattices of rectangular groups by so-called disjunctions of identities. Disjunctions of identities generalize the classical concept of an identity and of a variety, respectively. The rectangular groups will be on the one hand left zero semigroups and right zero semigroups and on the other hand groups of exponent [Formula: see text], where [Formula: see text] is any set of pairwise coprime natural numbers.

A perfect secret sharing scheme for general access structures

The secret sharing (SS) scheme has been widely used as the de facto paradigm for group key management in cryptography and distributed computation. An SS scheme for a general access structure (GSS) has drawn more and more attention since it allows flexible access control. However, previous solutions to GSS need to either assign multiple shares to each participant or to solve a complex nonlinear integer programming. In this paper, we propose a novel strategy to address the two problems simultaneously based on the Chinese Remainder Theorem (CRT) for a polynomial ring over a finite field. We classify general access structures satisfying the monotone property into the two families of maximum forbidden subsets and minimum qualified subsets conforming to the security condition and the revealing condition. The moduli used in the scheme are pairwise coprime polynomials and can take the irreducible polynomials. To find the degrees of these polynomials, we only need to solve an integer linear programming (ILP) by minimizing the sum of the degrees of all the moduli. The proposed scheme is inherently a weighted SS scheme and may have no solution which commonly exists in all GSS schemes. To ensure a solution, we put forth a preprocessing algorithm which separates the original access structure into several subaccess structures based on graph theory. The proposed scheme only assigns a share for each participant and achieves perfect security. It has good generalization and provides a universal approach to program the scheme construction for SS schemes with general access structures, threshold access structures and weighted access structures.

A Bombieri–Vinogradov-type theorem with prime power moduli

In 2020, Roger Baker proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal {S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribut

On the classification of Smale–Barden manifolds with Sasakian structures

Smale–Barden manifolds [Formula: see text] are classified by their second homology [Formula: see text] and the Barden invariant [Formula: see text]. It is an important and difficult question to decide when [Formula: see text] admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all [Formula: see text] with [Formula: see text] and [Formula: see text] provided that [Formula: see text], [Formula: see text], [Formula: see text] are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

Expansions of abelian square-free groups

We investigate finitary functions from [Formula: see text] to [Formula: see text] for a square-free number [Formula: see text]. We show that the lattice of all clones on the square-free set [Formula: see text] which contain the addition of [Formula: see text] is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all [Formula: see text]-linearly closed clonoids, [Formula: see text], to the [Formula: see text] power, where [Formula: see text]. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most [Formula: see text].

The number of maximum primitive sets of integers

AbstractA set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

АЛГОРИТМ ВЫЧИСЛЕНИЯ ИДЕАЛА ШТИКЕЛЬБЕРГЕРА ДЛЯ МУЛЬТИКВАДРАТИЧНЫХ ПОЛЕЙ

We present an algorithm for computing the Stickelberger ideal for multiquadratic fields K = Q(√d1,√d2, . . . , √dn), where the integers di ≡ 1 mod 4 for i ∈ {1, . . . , n} or dj ≡ 2 mod 8 for one j ∈ {1, . . . , n}; all di’s are pairwise co-prime and squarefree. Our result is based on the paper of Kuˇcera [J. Number Theory, no. 56, 1996]. The algorithm we present works in time O(lg ∆K • 2n• poly(n)), where ∆K is the discriminant of K. As an interesting application, we show a connection between Stickelberger ideal and the class number of a multiquadratic field

Cubic forms, powers of primes and classification of elliptic curves

We prove that for almost all primes $p$ the equation $(x+y)(x^2+Bxy+y^2)=p^\alpha z^n$ with $B\in \{0,1,4,6\}$ has no solution in pairwise coprime nonzero integers $x$, $y$, $z\neq \pm 1$, integer $\alpha \gt 0$ and prime $n\geq p^{16p}$. Proving this we

$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$-additive cyclic codes are asymptotically good

We construct a class of $${\mathbb {Z}}_p{\mathbb {Z}}_p[v]$$ -additive cyclic codes, where p is a prime number and $$v^2=v$$ . We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number $$0<\delta <1$$ such that the p-ary entropy at $$\frac{k+l}{2}\delta $$ is less than $$\frac{1}{2}$$ , the relative minimum distance of the random code is convergent to $$\delta $$ and the rate of the random code is convergent to $$\frac{1}{k+l}$$ , where p, k, l are pairwise coprime positive integers.

ℤ4-Double Cyclic Codes Are Asymptotically Good

We construct a class of Z4-double cyclic codes generated by pairs of polynomials. Based on the probabilistic method, we prove the asymptotic properties of this class of codes: for any positive real number 0 <; δ <; 1 such that the 4-ary entropy at k+t/2 δ is less than 14, the rate of the random code is convergent to 1/k+t and the relative distance of the code is convergent to δ, where k and l are pairwise coprime positive odd integers. As a result, the Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -double cyclic codes are asymptotically good.

On a problem of Sárközy and Sós for multivariate linear forms

We prove that for pairwise co-prime numbers $k\_1,\dots,k\_d \geq 2$ there does not exist any infinite set of positive integers $\mathcal{A}$ such that the representation function $r\_{\mathcal{A}}(n) = # { (a\_1, \dots, a\_d) {\in} \mathcal{A}^d : k\_1 a\_1 + \cdots + k\_d a\_d = n }$ becomes constant for $n$ large enough. This result is a particular case of our main theorem, which poses a further step towards answering a question of Sárközy and Sós and widely extends a previous result of Cilleruelo and Rué for bivariate linear forms (Bull. of the London Math. Society, 2009).