Cyclotomic Number Fields Research Articles

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.

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.

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.

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.

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.

Non-existence of Some Nearly Perfect Sequences, Near Butson–Hadamard Matrices, and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) m-ary sequences via their connection to (near) Butson–Hadamard (BH) matrices and (near) conference matrices. We refine the idea of Brock on the unsolvability of certain equations in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain and get many new non-existence results for near BH matrices and near conference matrices. We also apply previous results on vanishing sums of roots of unity and self conjugacy condition to derive non-existence results for near BH matrices and near conference matrices.

Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields whose inverse different CN/E is a square. Let AN/E denote the square root of CN/E. Then AN/E is a locally free Z[G]-module, which is in fact free provided N/E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N/E has odd degree, the classes of AN/E and ON (the ring of integers of N) in Cl(Z[G]) are equal (and in fact both trivial). We show that the above equality of classes still holds when N/E has even order, assuming that N/E is locally abelian. This result is obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse–Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]).

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ Z 2 n to $$\mathbb {Z}_{m}$$ Z m

In this paper, several nonexistence results on generalized bent functions \(f:\mathbb {Z}_{2}^{n} \rightarrow \mathbb {Z}_{m}\) are presented by using the knowledge on cyclotomic number fields and their imaginary quadratic subfields.

An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields

Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi(k)+1$ an integer and $\Phi_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|\Phi_k(m)|\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.

On multiplicatively independent bases in cyclotomic number fields

Recently the authors [12] showed that the algebraic integers of the form $${-m+\zeta_k}$$ are bases of a canonical number system of $${\mathbb {Z}[{\zeta_k]}}$$ provided $${m \geqq \phi(k)+1}$$ , where $${\zeta_k}$$ denotes a k-th primitive root of unity and $${\phi}$$ is Euler’s totient function. In this paper we are interested in the questions whether two bases $${-m+\zeta_k}$$ and $${-n+\zeta_k}$$ are multiplicatively independent. We show the multiplicative independence in case that 0 1.

Arithmetic of characteristic p special L-values

Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of $\F_q(T)$. We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular units, and give a negative answer to Anderson's Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second author's class number formula for the Carlitz module.

Upper bounds on relative class numbers of cyclotomic fields

Abstract We explain how one can use the explicit formulas for the mean square values of L-functions which we established elsewhere to obtain explcit upper bounds on relative class numbers of cyclotomic number fields. As an example, we show that the relative class numbers of the cyclotomic fields of conductor 4p, p ≥ 3 a prime, are less than or equal to 8√p(p/16)(p−1)/2.

ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

Field switching in BGV-style homomorphic encryption

The security of contemporary homomorphic encryption schemes over cyclotomic number field relies on fields of very large dimension. This large dimension is needed because of the large modulus-to-noise ratio in the key-switching matrices that are used

Field switching in BGV-style homomorphic encryption

The security of contemporary homomorphic encryption schemes over cyclotomic number field relies on fields of very large dimension. This large dimension is needed because of the large modulus-to-noi...

A Herbrand-Ribet theorem for function fields

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with $\mu_p$-coefficients over the splitting field of $\mu_p$, and in our analogue both occurrences of $\mu_p$ are replaced with the $\mathfrak{p}$-torsion scheme of the Carlitz module for a prime $\mathfrak{p}$ in $\F_q[t]$.