scholarly journals Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Valentin Ovsienko
Keyword(s):  
Chebyshev Polynomials ◽  
Existence And Uniqueness ◽  
Modular Group ◽  
Picard Group ◽  
Complex Numbers ◽  
Gaussian Integers ◽  
Group I

This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials. Comment: 21 pages

Complex Numbers with Three Radix Expansions

1982 ◽  
Vol 34 (6) ◽  
pp. 1335-1348 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Complex Numbers ◽  
Natural Numbers ◽  
Gaussian Integers

1. Introduction. This paper deals with the lack of uniqueness of the representations of the complex numbers in positional notation using Gaussian integers as bases.Kátai and Szabó [3] proved that all the complex numbers can be written in radix form using the base –n + i with the natural numbers 0, 1, 2, …, n2 as digits. They remarked that they did not assert the uniqueness of these representations but gave no further indications of any multiple expansions. The geometry of these complex bases [2] indicates that some numbers have two expansions in a given base, while a few numbers even have three different expansions.

scholarly journals A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2050
Author(s):  
Malek Safieh ◽  
Johann-Philipp Thiers ◽  
Jürgen Freudenberger
Keyword(s):  
Elliptic Curve ◽  
Modular Arithmetic ◽  
Complex Numbers ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Point Multiplication ◽  
Hardware Implementations ◽  
Secure Hardware ◽  
Prime Fields ◽  
High Flexibility

This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.

scholarly journals The Division Algorithm in Complex Bases

1996 ◽  
Vol 39 (1) ◽  
pp. 47-54 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Iterated Function System ◽  
Complex Numbers ◽  
Integer Part ◽  
Division Algorithm ◽  
Fractal Boundary ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Linear Maps ◽  
Iterated Function ◽  
Long Division

AbstractComplex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

Quadratic Integers and Coxeter Groups

1999 ◽  
Vol 51 (6) ◽  
pp. 1307-1336 ◽  
Author(s):  
Norman W. Johnson ◽  
Asia Ivić Weiss
Keyword(s):  
Hyperbolic Plane ◽  
Modular Group ◽  
Coxeter Groups ◽  
Picard Group ◽  
Symmetry Groups ◽  
Discrete Groups ◽  
Algebraic Integers ◽  
Linear Fractional Transformations ◽  
Rings Of Algebraic Integers ◽  
Groups Of Transformations

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]).

scholarly journals Montgomery Reduction for Gaussian Integers

Cryptography ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 6
Author(s):  
Malek Safieh ◽  
Jürgen Freudenberger
Keyword(s):  
Elliptic Curve ◽  
Initial Point ◽  
Modular Arithmetic ◽  
Complex Numbers ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Point Multiplication ◽  
Prime Fields ◽  
Integer Rings ◽  
Montgomery Reduction

Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

Fuchsian Subgroups of the Picard Group

1976 ◽  
Vol 28 (3) ◽  
pp. 481-485 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Free Product ◽  
Semidirect Product ◽  
Finite Index ◽  
Function Theory ◽  
Picard Group ◽  
Automorphic Function ◽  
Linear Transformations ◽  
Gaussian Integers ◽  
Abstract Group ◽  
Generalized Free Product

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformationswith a, b, c, d Gaussian integers.Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

On the algebra of modular forms on a congruence subgroup

1979 ◽  
Vol 86 (3) ◽  
pp. 461-466 ◽  
Author(s):  
A. J. Scholl
Keyword(s):  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Expansion ◽  
Congruence Subgroup ◽  
Complex Numbers ◽  
Graded Algebra

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.

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