Convolutions over residue classes of quadratic integers

For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fréchet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.

We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF(pm) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method for multiplication with the best known asymptotic complexity, i.e. O(n log n log log n). Unfortunately, the original FFT method bears significant overhead due to the conversions between the time and the frequency domains, which makes it impractical to perform multiplication of relatively short (160 - 1024 bits) integer operands as used in many applications. In this work, we introduce an efficient way of performing polynomial multiplication in finite fields using the FFT. We show that, with careful selection of parameters, all the multiplications required for the FFT computations can be avoided and polynomial multiplication in finite fields can be achieved with only O(m) multiplications in addition to O(m log m) simple shift, addition and subtraction operations. We show that, especially in constrained devices where multiplication is expensive, polynomial multiplication in the suggested finite fields using the FFT outperforms both the schoolbook and Karatsuba methods for practically small finite fields, e.g., relevant to elliptic curve cryptography.

Let k=Q(d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(d)={x+yd|x,y∈Q}. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either wd=d=[a0;a1,a2,⋯,al(d)−1,2a0¯] in the case of d ≡ 2,3(mod 4) or wd=1+d2=[a0;a1,a2,⋯,al(d)−1,2a0−1¯] in the case of d ≡ 1(mod 4) where l(d) is the period length of continued fraction expansion.The purpose of this paper is to obtain classification of some types of real quadratic fields Q(d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is l=l(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain...

We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank $ \geq 1$, of imaginary or unusual discriminant $D$, for a fixed $|D| = q^{\deg(D)}$. We present numerical data for quadratic function fields over $\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$ with $\deg(D) \leq 11$. Our algorithm produces quadratic function fields of minimal genus for any given 3-rank. Our numerical data mostly agrees with the Friedman-Washington heuristics for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv -1 \pmod{3}$. The data for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv 1 \pmod{3}$ does not agree closely with Friedman-Washington, but does agree more closely with some recent conjectures of Malle.

Let K be a quadratic extension field of the rational numbers Q. Let C K be the 2-class group of K in the narrow sense. It is a classical result that rank C K= t l , where t is the number of primes that ramify in K/Q. Now let C~ = {ai: a e CK}, and let R K denote the 4-class rank of K in the narrow sense; i.e., RK= rank ~Kc z = dimv2 ttc'Z/t~41~K/~KJ" Here F 2 is the finite field with two elements, and 2 4 CK/C K is an elementary abelian 2-group which we are viewing as a vector space o v e r F 2. Given a quadratic field K, one can compute R K by computing the rank (over F2) of a certain matrix of Legendre symbols (cf. [11]). Now assume K is imaginary quadratic. So K=Q(1/~mm), where m is a square-free positive integer. For each positive integer t, each nonnegative integer e, and each positive real number x, we define

Recent research presents a technique to enumerate all valid assignments of “twiddle factors” for power-of-two fast Fourier transform (FFT) flow graphs. Brute-force search employing state-of-the-art Boolean satisfiability (SAT) solvers can then be used to find FFT algorithms within this large solution space which have desirable characteristics. Surprisingly, this approach has discovered FFT algorithms requiring fewer operations than the split-radix algorithm even when all twiddle factors are nth roots of unity. This paper reviews and then extends this prior research to examine fast discrete convolution algorithms when implemented via FFT and inverse FFT (IFFT) algorithms. In particular, we find that the combination of FFT and IFFT algorithms in fast convolution permits greater freedom when selecting valid twiddle factor assignments. We exploit this freedom and use SAT solvers to find new fast convolution algorithms with the lowest operation counts known.

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

In this paper we derive some elements of the rings of integers in the cubic\nfields of the form $\\mathbf{Q}(\\sqrt[3]{d})$, where $d$ is even, which cannot be\nwritten as a difference of two squares in the considered ring. We show that\ncorresponding Diophantine quadruples do not exist for such elements, what\nsupports the hypothesis mainly proved for the ring of integers and for certain\nquadratic fields.

The radio (SWR) concept has become a topic of widespread interest for reconfigurable mobile architecture design. It is seen as the next evolutionary step in the mobile communications. In this context of SWR, a way to decrease the runtime of the software reconfiguration and to optimize the sharing between the software and the hardware of the execution platform called was introduced. This technique is based on two approaches, the first one is called the common function approach, the second one is called the common operator approach. Being interested on the second parametrization technique, we propose in this paper a reconfigurable FFT (fast Fourier transform) operator. This operator can be reconfigured to switch from an operator dedicated to compute the FFT in the complex field (i.e for OFDM modulation or frequential equalization) to an operator which computes the FFT in the Galois field in order to perform Reed-Solomon (RS) encoding and two steps of the decoding process.

Splitting quaternion algebras over quadratic number fields

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

Let d be a squarefree integer. Does there exist four squares in arithmetic pro- gression over Q( p d )? We shall give a partial answer to this question, depending on the value of d. In the a±rmative case, we construct explicit arithmetic progressions consisting of four squares over Q( p d ).

Let L be a number field and OL the ring of algebraic integers in L. For apolynomial f with coefficients in OL, the content of f in L is the ideal of OLgenerated by coefficients of f. The polynomial f is primitive in L if the contentof f in L is OL. In 2005, Arturo Magidin and David McKinnon proved the Gauss’ lemma fornumber fields, the product of two primitive polynomials is also primitive, andsome applications following from Gauss’ lemma for number fields. A function field K over a finite field k is a finite separable field extension overk(x) where x is a transcendantal element. In this research, we study Magidin and McKinnon’s work on the function fields.

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

A coprime antenna array consists of two or more sparse subarrays featuring enhanced degrees of freedom (DOF) and reduced mutual coupling. This paper introduces a new class of planar coprime arrays, based on the theory of ideal lattices. In quadratic number fields, a splitting prime p can be decomposed into the product of two distinct prime ideals, which give rise to the two sparse subarrays. Their virtual difference coarray enjoys a quadratic gain in DOF, thanks to the generalized Chinese remainder theorem (CRT). To enlarge the contiguous aperture of the coarray, we present hole-free symmetric CRT arrays with simple closed-form expressions. The ring of Gaussian integers and the ring of Eisenstein integers are considered as examples to demonstrate the procedure of designing coprime arrays. With Eisenstein integers, our design yields a difference coarray that is a subset of the hexagonal lattice, offering a significant gain in DOF over the rectangular lattice, given the same physical areas. Maximization of CRT arrays in the aspect of essentialness and the superior performance in the context of angle estimation will be presented in the companion paper (Part II).