The Gaussian Integers

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

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Gaussian Integers

An Introduction to Number Theory with Cryptography ◽

10.1201/b15718-18 ◽

2016 ◽

pp. 445-470

Keyword(s):

Gaussian Integers

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Digital Signature Schemes over the Ring of Gaussian Integers

Proceedings of the 2016 International Conference on Education, Management and Computer Science ◽

10.2991/icemc-16.2016.167 ◽

2016 ◽

Author(s):

Xuedong Dong

Keyword(s):

Digital Signature ◽

Signature Schemes ◽

Gaussian Integers

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Generator of PRN's on the norm group

Researches in Mathematics and Mechanics ◽

10.18524/2519-206x.2020.2(36).233737 ◽

2021 ◽

Vol 25 (2(36)) ◽

pp. 26-39

Author(s):

P. Fugelo ◽

S. Varbanets

Keyword(s):

Prime Number ◽

Quadratic Field ◽

Exponential Sum ◽

Random Numbers ◽

Gaussian Integers ◽

New Family ◽

Serial Test ◽

Polynomial Representations ◽

Modulo P ◽

Norm Group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

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ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000270 ◽

2015 ◽

Vol 100 (1) ◽

pp. 21-32

Author(s):

ELLIOT BENJAMIN ◽

C. SNYDER

Keyword(s):

Class Group ◽

Number Fields ◽

Ideal Class Group ◽

Quadratic Number ◽

Gaussian Integers ◽

Real Quadratic Fields ◽

Order Four ◽

Real Quadratic ◽

Narrow Class

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

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