Real Subfield Research Articles

Construction of lattices over the real sub-field of ℚ(ς8) for block fading (wiretap) coding

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field [Formula: see text] over [Formula: see text]. We employ binary codes to generate a lattice over [Formula: see text], which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

An upper bound on the number of classes of perfect unary forms in totally real number fields

Let K be a totally real number field of degree n over [Formula: see text], with discriminant and regulator [Formula: see text], respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by [Formula: see text], where [Formula: see text] is the discriminant of the field K, [Formula: see text] is the additive Hermite–Humbert constant over positive-definite unary forms for K and [Formula: see text] is the covering radius of the log-unit lattice. In particular, when K is Galois over [Formula: see text] and n is a prime number, the number of homothety classes of unary forms is upper bounded by [Formula: see text], where [Formula: see text] is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by [Formula: see text].

Dihedral Artin representations and CM fields

For a fixed CM field K K with maximal totally real subfield F F , we consider isomorphism classes of dihedral Artin representations of F F which are induced from K K , distinguishing between those which are “canonically” induced from K K and those which are “noncanonically” induced from K K . The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.

Characterization of Totally Real Subfields of 2-Power Cyclotomic Fields and Applications to Signal Set Design

A classification of all totally real subfields K of cyclotomic field Q(zeta_{2^r}), for any r ≥ 4, and the fully-diverse related versions of the Z^n-lattice are presented along with closed-form expressions for their minimum product distance. Any totally real subfield K of Q(zeta_{2^r}) must be of the form K=Q(zeta_{2^2} + zeta_{2^2}^{-1}), where s = r − j for some 0 ≤ j ≤ r − 3. Signal constellations for transmitting information over both Gaussian and Rayleigh fading channels (which can be useful for mobile communications) can be carved out of those lattices.

Asymptotic Fermat for signatures (r, r, p) using the modular approach

Let K be a totally real field, and rge 5 a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions (x,y,z) in mathcal {O}_K^3 of the signature (r, r, p) equation x^r+y^r=z^p (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of K(zeta _r). When K=mathbb {Q} we get that for most of the primes r<150 with r not equiv 1 mod 8 there are no non-trivial, primitive integer solutions (x, y, z) with 2|z for signatures (r, r, p) when p is sufficiently large. Similar results hold for quadratic fields, for example when K=mathbb {Q}(sqrt{2}) there are no non-trivial, primitive solutions (x,y,z)in mathcal {O}_K^3 with sqrt{2}|z for signatures (5, 5, p), (11, 11, p), (13, 13, p) and sufficiently large p.

Form Class Groups and Class Fields of CM-Fields

Let F be a totally real number field of class number one, and let K be a CM-field with F as its maximal real subfield. For each positive integer N, we construct a class group of certain binary quadratic forms over F which is isomorphic to the ray class group of K modulo N. Assuming further that the narrow class number of F is one, we construct a class field of the reflex field of K in terms of the singular values of Hilbert modular functions.

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.

Level-raising for automorphic forms on 𝐺𝐿_{𝑛}

Let E E be a CM number field and F F its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of G L n ( A E ) GL_{n}(\mathbb {A}_E) . This generalizes previously known results of Thorne [Forum Math. Sigma 2 (2014)] by removing certain hypotheses occurring in that work. In particular, the level-raising prime p p is allowed to be unramified as opposed to inert in F F , the field E / F E/F is not assumed to be everywhere unramified, and the field E E is allowed to be a general CM field.

Unit groups of some multiquadratic number fields and 2-class groups

Let pequiv -q equiv 5pmod 8 be two prime integers. In this paper, we investigate the unit groups of the fields L_1 =mathbb {Q}(sqrt{2}, sqrt{p}, sqrt{q}, sqrt{-1} ) and L_1^+=mathbb {Q}(sqrt{2}, sqrt{p}, sqrt{q} ). Furthermore , we give the second 2-class groups of the subextensions of L_1 as well as the 2-class groups of the fields L_n =mathbb {Q}( sqrt{p}, sqrt{q}, zeta _{2^{n+2}} ) and their maximal real subfields.

On p-rationality of [formula omitted] for Sophie Germain primes l

The p-rationality of a totally real abelian number field can be checked from the values L(2−p,χ) of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of Q(ζ2l+1)+, the maximal real subfield of Q(ζ2l+1), for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that Q(ζ2l+1)+ is p-rational for such pairs if p<4l. We also prove that the Siegel's heuristics on the equidistribution of the residues of Bernoulli numbers modulo p imply that Q(ζ2l+1)+ is p-rational for all but finitely many p that are primitive roots modulo l.

Binary self-dual codes and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Binary self-dual codes and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Lafforgue pseudocharacters and parities of limits of Galois representations

Let $F$ be a CM field with totally real subfield $F^+$ and let $\pi$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $\pi$ a Galois representation $R_\pi:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_\pi(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (essentially) conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_n(\mathbf A_F)$ are odd in the sense of Bella\"iche--Chenevier.

Self-dual codes over $${\mathbb {F}}_2[u]/\langle u^4 \rangle $$ and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Let $$K={\mathbb {Q}}(\zeta _8)$$ be the cyclotomic field over $${\mathbb {Q}}$$ of the extension degree 4. We give an integral lattice construction on $${\mathbb {Q}}(\zeta _8)$$ induced from codes over the ring $${\mathcal {R}}= {\mathbb {F}}_2[u]/\langle u^4 \rangle $$ . We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a code over $${\mathcal {R}}$$ . If C is a Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield $$k={\mathbb {Q}}(\zeta _8+\zeta _8^{-1})$$ of K. Also, we see that Hilbert–Siegel modular form of weight n and genus g can be seen in terms of the complete joint weight enumerator of codes $$C_j$$ , for $$1\le j\le g$$ over $${\mathcal {R}}$$ .

Infinite families of cyclotomic function fields with any prescribed class group rank

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F q ( T ) whose class groups can have arbitrarily large ℓ n -rank, where F q is the finite field of prime power order q . We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k ( Λ ) + of cyclotomic function fields k ( Λ ) whose ideal class groups have arbitrary ℓ n -rank for n = 1, 2, and 3, where ℓ is a prime divisor of q − 1 . We also obtain a tower of cyclotomic function fields K i whose maximal real subfields have ideal class groups of ℓ n -ranks getting increased as the number of the finite places of k which are ramified in K i get increased for i ≥ 1 . Our main idea is to use the Kummer extensions over k which are subfields of k ( Λ ) + , where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k ( Λ ) + of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k . We demonstrate our results by presenting some examples calculated by MAGMA at the end.

An intriguing hyperelliptic Shimura curve quotient of genus 16

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E/F$ of class number $17$ contained in $\mathbf{Q}(\zeta_{64})$, the cyclotomic field of $64$th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N/\mathbf{Q}$ unramified outside $2$ and $\infty$, with Galois group $\mathrm{Gal}(N/\mathbf{Q})\simeq F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. In fact, the Jacobian $\mathrm{Jac}(X_0^D(1))$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$, is that it is also hyperelliptic over $\mathbf{Q}$.

Rotated $ A_n $-lattice codes of full diversity

&lt;p style='text-indent:20px;'&gt;Some important properties of lattices are packing density and full diversity, which may be good for signal transmission over both Gaussian and Rayleigh fading channel, respectively. The algebraic lattices are constructed through twisted homomorphism of some modules in the ring of integers of a number field &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{K} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In this paper, we present a construction of some families of rotated &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ A_n- $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;lattices, for &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ n = 2^{r-2}-1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ r \geq 4 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, via totally real subfield of cyclotomic fields. Furthermore, closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties.&lt;/p&gt;

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Trace forms of certain subfields of cyclotomic fields and applications

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices $ \Lambda_9, \Lambda_{10}$ and $\Lambda_{11}$ and they are images of $\mathbb{Z}$-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to $7$ over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.

Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

A method for constructing rotated Z^n-lattices, with n a power of 2, based on totally real subfields of the cyclotomic field Q(\zeta_{2^r}), where r\geq 4 is an integer, is presented. Lattices exhibiting full diversity in some dimensions n not previously addressed are obtained.