Cyclotomic Numbers Research Articles

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Ternary cyclotomic numbers and ternary Jacobi sums

<p>Cyclotomic numbers and Jacobi sums, introduced over two centuries ago by Gauss and Jacobi, respectively, are pivotal in number theory and find wide applications in combinatorial designs, coding theory, cryptography, and information theory. The cyclotomic problem, focused on determining all cyclotomic numbers, or equivalently evaluating all Jacobi sums of a given order, has been a subject of extensive research. This paper explores their trivariate counterparts, termed "ternary cyclotomic numbers" and "ternary Jacobi sums", highlighting the fundamental properties that mirror those of the classical cases. We show the ternary versions of Fourier series expansions, two symmetry properties, and a summation equation. We further demonstrate that ternary Jacobi sums, with at least one trivial variable, can be evaluated in terms of classical Jacobi sums of the same order. These properties are established through elementary methods that parallel those utilized in classical cases. Based on these properties, then we offer explicit calculations for all ternary Jacobi sums and ternary cyclotomic numbers of order $ e = 2 $, and near-complete results for order $ e = 3 $, with the exception of the elusive integer $ J_{3}(1, 1, 2) $ for us.</p>

An upper bound on the covering radius of the logarithmic lattice for cyclotomic number fields

Recently, in coding theory and cryptography, it has been important the diversified use of lattices. One use of them is to cover a space. Each lattice has a covering radius, a number corresponding to the radius of a ball whose translations by all the points of the lattice cover efficiently the space generated by a basis of it. A way to obtain lattices algebraically is from subgroups of the multiplicative group of units of a number field via the logarithm embedding. This includes the logarithm lattice. In this work, it is presented an upper bound on the covering radius of the logarithm lattice obtained from the units of general cyclotomic number fields via the logarithm embedding, which generalizes an upper bound present in a previous work for cyclotomic number fields of prime-power indices.

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

On the differential spectrum of a differentially 3-uniform power function

In this paper, we investigate the cardinality of the intersection of four particular cyclotomic classes of order two over the finite field with characteristic 5 by making most use of the cyclotomic numbers of orders two and four. As a consequence, a class of power functions over F5n is shown to be differentially 3-uniform and its differential spectrum is also completely determined.

Several classes of exponential sums and three-valued Walsh spectrums over finite fields

Let r≡1(mod4) be a prime, m a positive integer, ϕ(rm)2 the multiplicative order of 2 modulo rm, and q=2ϕ(rm)2, where ϕ(⋅) is the Euler's function. Let γ be a primitive element of the finite field Fq and α=γq−1rm. In this paper, we shall explicitly calculate the exponential sums S(a)=∑i=0rm−1χ(aαi), a∈Fq, where χ is the canonical additive character of Fq. As applications, using elliptic curves and cyclotomic numbers over Fr, the three-valued Walsh spectrums of four classes of monomial functions fa(x)=Trq/2(axq−1rm) are determined.

Lattice packings through division algebras

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh’s lattice packing result Venkatesh (Int Math Res Notices 2013(7): 1628–1642, 2013). In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. We employ a probabilistic argument to show the existence of lattices in certain dimensions with good packing densities. The approach improves the best known lower bounds on the lattice packing problem for certain dimensions.We work with a moduli space of lattices that are invariant under the action of a finite group, one that can be embedded inside a division algebra. To obtain our existence result, we prove a division algebra variant of the Siegel’s mean value theorem. In order to establish this, we describe a useful description of the Haar measure on our moduli space and a coarse fundamental domain to perform the integration.

On average coherence of cyclotomic lattices

We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of interest in sphere packing problems. As such, coherence and orthogonality defect are different measures of the extent to which a lattice fails to be orthogonal, and maximizing their quotient (normalized for the number of minimal vectors with respect to dimension) gives lattices with particularly good optimization properties. While orthogonality defect is a fairly classical and well-studied notion on various families of lattices, coherence is not. We investigate coherence properties of a nice family of algebraic lattices coming from rings of integers in cyclotomic number fields, proving a simple formula for their average coherence. We look at some examples of such lattices and compare their coherence properties to those of the standard root lattices.

On abelian $$\ell $$-towers of multigraphs III

Let \(\ell \) be a rational prime. Previously, abelian \(\ell \)-towers of multigraphs were introduced which are analogous to \(\mathbb {Z}_{\ell }\)-extensions of number fields. It was shown that for towers of bouquets, the growth of the \(\ell \)-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for \(\mathbb {Z}_{\ell }\)-extensions of number fields). In this paper, we extend this result to abelian \(\ell \)-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in \(\mathbb {Z}_\ell \) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at \(u=1\) of the Artin–Ihara L-function, when the base multigraph is not necessarily a bouquet.

Large absolute values of cyclotomic polynomials at roots of unity

The nth cyclotomic polynomial Φn(X) is the minimal polynomial of ζn:=e2πi/n. Given an integer m≥1 and a prescribed set S of arithmetic progressions modulo m, we define nx as the product of the primes p≤x lying in those progressions. Let d(n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists jx such that log⁡|Φnx(ζmjx)|/d(nx) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime.Our motivation comes from a paper of Vaughan (1975). He studied the case where S={±2(mod5)} and used it to show that the maximum coefficient in absolute value of Φn can be very large.

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings.

Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring instantiations enlarges the scope of possible applications and favors the diversification of security assumptions. In this work, we extend the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. By doing so, we allow the Ring-LWE problem to be used over maximal real subfields of cyclotomic number fields. We prove that Twisted Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the twist factor does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. As a result, sampling from a spherical Gaussian distribution can be done directly in the respective number field while maintaining its format and standard deviation when seen in via twisted embeddings.

A lower bound on HMOLS with equal sized holes

It is known that N(n), the maximum number of mutually orthogonal latin squares of order n, satisfies the lower bound N(n)≥n1/14.8 for large n. For h≥2, relatively little is known about the quantity N(hn), which denotes the maximum number of ‘HMOLS’ or mutually orthogonal latin squares having a common equipartition into n holes of a fixed size h. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound N(hn)≥(log⁡n)1/δ for any δ>2 and all n>n0(h,δ).

On the p ‐Adic Complexity of the Ding‐Helleseth‐Martinsen Binary Sequences

The purpose of this paper is to determine the p-adic complexity of the Ding-Helleseth-Martinsen (DHM) sequences with period N = 2q, where q = 5 (mod 8) is a prime number. We firstly use the p-adic exponential valuation, cyclotomic numbers of order four, “Gauss periods” and “quadratic Gauss sums” on finite field 𝔽q, and valued in ℤN, to determine the p-adic complexity of the DHM sequences.

Three classes of partitioned difference families and their optimal constant composition codes

<p style='text-indent:20px;'>Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> prime or composite, cyclotomy on a residue class ring <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{Z}}_{v} $\end{document}</tex-math></inline-formula> can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [<xref ref-type="bibr" rid="b40">40</xref>], we introduce a generalized cyclotomy of order <inline-formula><tex-math id="M4">\begin{document}$ e $\end{document}</tex-math></inline-formula> on the ring <inline-formula><tex-math id="M5">\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q_j $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ i\neq j $\end{document}</tex-math></inline-formula>) may not be co-prime, which includes classical cyclotomy as a special case. Here, <inline-formula><tex-math id="M9">\begin{document}$ q_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ q_k $\end{document}</tex-math></inline-formula> are powers of primes with an integer <inline-formula><tex-math id="M13">\begin{document}$ e|(q_i-1) $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M14">\begin{document}$ 1\leq i\leq k $\end{document}</tex-math></inline-formula>. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and <inline-formula><tex-math id="M15">\begin{document}$ d $\end{document}</tex-math></inline-formula>-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.</p>

On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Abstract Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn ) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn ) for n = 2 kpℓ , where k, ℓ ≥ 0 are integers and p is an odd prime number.

Two kinds of constructions of directed strongly regular graphs

In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.

On the 2-Adic Complexity of the Ding-Helleseth-Martinsen Binary Sequences

We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, "Gauss periods" and "quadratic Gauss sum" on finite field $\mathbb{F}_q$ and valued in $\mathbb{Z}_{2^N-1}$ where $q \equiv 5\pmod 8$ is a prime number and $N=2q$ is the period of the DHM sequences.

On the RLWE/PLWE equivalence for cyclotomic number fields

We study the equivalence between the ring learning with errors and polynomial learning with errors problems for cyclotomic number fields, namely: we prove that both problems are equivalent via a polynomial noise increase as long as the number of distinct primes dividing the conductor is kept constant. We refine our bound in the case where the conductor is divisible by at most three primes and we give an asymptotic subexponential formula for the condition number of the attached Vandermonde matrix valid for arbitrary degree.