Imaginary Quadratic Integers Research Articles

A Simpler Proof on the Existence of Good Nested Lattice Codes Over Imaginary Quadratic Integers for AWGN Channel

A Simpler Proof on the Existence of Good Nested Lattice Codes Over Imaginary Quadratic Integers for AWGN Channel

Lattice Reduction Over Imaginary Quadratic Fields

Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice if the chosen ring is Euclidean. Secondly, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, we numerically show the time-advantage of using algebraic LLL by considering lattice bases generated from wireless communications and cryptography.

Lattices Over Algebraic Integers With an Application to Compute-and-Forward

In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

On the equivariant $K$-homology of PSL$_2$ of the imaginary quadratic integers

We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.

SELF SIMILAR TILES ARISING FROM THE UNITARY NUMBER SYSTEMS IN EUCLIDEAN RINGS OF IMAGINARY QUADRATIC INTEGERS

Abstract. In [7], it was shown that Euclidean rings R of imaginary qua-dratic integers admit unitary number systems. In this paper, as an appli-cation of the result, we obtain all self similar tiles arising from the unitarynumber systems of R. 1. IntroductionLet Rbe a ring of imaginary quadratic integers, which consists of all algebraicintegers in the eld Qpdfor some square-free negative integer d, and let bbe an element of Rwhose norm N(b) is greater than one. Then a completerepresentatives Dof R=(b) is called a digit set for a base b, where (b) denotesan ideal generated by b. Then a unitary number system (b;D) in Ris de nedso that all non-zero elements of digits in Dconsist of units of R.From [7] we see that among the nine principal ideal domains of imaginaryquadratic integers, only the ve ringsZp 1; Zp 2; Z1 +p 32; Z1 +p 72; Z1 + 112;equipped with Euclidean functions de ned by norms admit unitary numbersystems. As proposed in [7], the concept of a unitary number system can be re-placement of a ‘universal side divisor’ by Motzkin. Furthermore unitary numbersystems are already available in various applications of complex based numbersystems. In particular, Khmelnik [3] and Penny [6] independently showed thata number system with a base b= 1 +p 1 and a digit set D= f0;1ginZp 1yields so called the twin dragon (see Figure 2 in Appendix). Knuth[5] proposed another binary number system with a base b=p 2 and a digit

The mod 2 cohomology rings of [formula omitted] of the imaginary quadratic integers

We establish general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL2 groups over imaginary quadratic integers on their associated symmetric space. By way of doing this, we extend the torsion subcomplex reduction technique to cases where the kernel of the group action is nontrivial. Using the equivariant and Lyndon–Hochschild–Serre spectral sequences, we investigate the second page differentials and show how to obtain the mod 2 cohomology rings of our groups from this information.

NUMBER SYSTEMS PERTAINING TO EUCLIDEAN RINGS OF IMAGINARY QUADRATIC INTEGERS

Abstract. For a ring R of imaginary quadratic integers, using a conceptof a unitary number system in place of the Motzkin’s universal side divisor,we show that the following statements are equivalent:(1) R is Euclidean.(2) R has a unitary number system.(3) R is norm-Euclidean.Through an application of the above theorem we see that R admits binaryor ternary number systems if and only if R is Euclidean. 1. IntroductionIt is well known that among rings of imaginary quadratic integers, only nineringsZp 1; Zp 2; Z1 +p 32; Z1 + 72; Z1 +p 112;Z1 +p 192; Z1 +p 432; Z1 +p 672; Z1 +p 1632are principal ideal domains, which was conjectured by Gauss and settled com-pletely by Stark [15]. Furthermore, only the rst ve examples of those are Eu-clidean domains, whose Euclidean functions are induced by the norms; whereas,the other four have no Euclidean functions whatsoever. A brilliant proof for thelatter claim was presented by Motzkin [12] around 1949, who came up with acriterion for an integral domain to be Euclidean. But the proof seems too tersefor laymen. And lling details of Motzkin’s proof especially for non-existence ofEuclidean algorithm of ring Zh

Optimality of the Width-w Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given.

The homological torsion of PSL$_2$ of the imaginary quadratic integers

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.

Quadratic Integers and Coxeter Groups

AbstractMatrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive n-space or hyperbolic (n+1)-space Hn+1. For small n, thesemay be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of Hn+1. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group PSL2(), the Gaussian modular (Picard) group PSL2([i]), and the Eisenstein modular group PSL2([ω]).

Quaternionic modular groups

Matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n-space or hyperbolic ( n+1)-space H n+1 . In particular, groups operating in the hyperbolic plane or hyperbolic 3-space may be represented by 2×2 matrices whose entries are rational integers or real or imaginary quadratic integers. The theory is extended here to groups operating in H 4 or H 5 and matrices over one of the three basic systems of quaternionic integers. Quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of H 4 and H 5. For four-dimensional groups the division ring of quaternions is treated as a Clifford algebra. Results in hyperbolic 5-space derive from the homeomorphism of inversive 4-space and the quaternionic projective line.

Automorphisms of SL 2 of Imaginary Quadratic Integers

Automorphisms of SL 2 of Imaginary Quadratic Integers

Automorphisms of 𝑆𝐿₂ of imaginary quadratic integers

We determine the outer automorphism groups of the two-dimensional special linear and projective special linear groups of the ring of integers in an imaginary quadratic field.

On the minimal algorithm in rings of imaginary quadratic integers

A good approximation for the minimal algorithm of the euclidian ring of integers of Q (√− d) is given for the cases d = 2, 7 and 11. These approximations are derived from results of H. W. Lenstra, Jr. by computing methods.

A nonvanishing theorem for the cuspidal cohomology of SL2 over imaginary quadratic integers

A nonvanishing theorem for the cuspidal cohomology of SL2 over imaginary quadratic integers