Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

ℤ[] is Euclidean

Abstract: We provide the first unconditional proof that the ring is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of . It is proved that if is a real quadratic field (modulo the existence of two special primes of ) or if is a cyclotomic extension of then: the ring of integers of is a Euclidean domain if and only if it is a principal ideal domain. The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

Threshold decoding of cyclic codes

Abstract: It is shown that every cyclic code over GF(p) can be decoded up to its minimum distance by a threshold decoder employing general parity checks and a single threshold element. This result is obtained through the application of a general decomposition theorem for complex-valued functions defined on the space of all n -tuples with elements from the ring of integers modulo p .

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Abstract: We consider the secure computation problem in a minimal model, where Alice and Bob each holds an input and wish to securely compute a function of their inputs at Carol without revealing any additional information about the inputs. For this minimal secure computation problem, we propose a novel coding scheme built from two steps. First, the function to be computed is expanded such that it can be recovered while additional information might be leaked. Second, a randomization step is applied to the expanded function such that the leaked information is protected. We implement this expand-and-randomize coding scheme with two algebraic structures - the finite field and the modulo ring of integers, where the expansion step is realized with the addition operation and the randomization step is realized with the multiplication operation over the respective algebraic structures.

Fourier series on the ring of integers in a $p$-series field

Abstract: 1. In this note we will describe a natural setting for harmonic analysis on the dyadic group, 2, (also known as the Walsh-Paley group) and give a few illustrative results. Details and proofs will appear elsewhere. The dyadic group is viewed classically as the set of all sequences of zeroes and ones with addition (mod 2) defined pointwise, and is supplied with the usual product topology. From our point of view, 2 will be the additive subgroup of the ring of formal power series in one variable over G F (2). The subject of this note is harmonic analysis on the ring of integers, O, in the field, K (called a ^-series field), of formal Laurent series (with finite principal part) in one variable over GF(p), where p is a prime. Such a field K is a particular instance of a local field ; that is, a locally compact, totally disconnected, nondiscrete, complete field. The £-adic fields are other examples of local fields. The results in this note have extensions to Fourier series on the ring of integers in any local field and also to multiple Fourier series. These extensions will not be given here. The idea that 2 might be an instance of a ring of integers in a local field developed in a conversation with E. M. Stein.

Perfect cuboids and irreducible polynomials

Abstract: The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$, and $u$ would mean the non-existence of perfect cuboids. This irreducibility is conjectured and then verified numerically for approximately 10000 instances of $a$, $b$, and $u$.