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# System prepended metadata

title: Ring(環)定義
tags: [crypto]

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---
title: Ring(環)定義
tags: crypto
lang: zh_tw
---

* [筆記總覽](https://hackmd.io/@LJP/rkerFdnqS)

[TOC]

# Ring(環) 定義
A ring $(R,+,\times)$ is a set R with two binary operations $+$ and $\times$ such that
- $(R,+)$ is an **abelian group**
- Closed under $\times$: 
    - $a\times b\in R$ for all $a,\ b\in R$
- Associative under $\times$
    - $a\ \times(b\ \times\ c)\ =\ (a\ \times\ b)\ \times\ c$ for all $a,\ b,\ c\in G$
- Distributive laws
    - $a\times(b+c)=a\times b+a\times c$
    - $(a+b)\times c=a\times c+b\times c$
    - for all $a,\ b,\ c\in R$

$\star$ **這裡的加法乘法定義跟一般的加法乘法不同**
根據 [wiki](https://en.wikipedia.org/wiki/Ring_(mathematics)#Example:_Integers_modulo_4)
- The sum ${\displaystyle {\overline {x}}+{\overline {y}}}$ is the remainder when the integer $x+y$ is divided by rank
- The product ${\displaystyle {\overline {x}}\times{\overline {y}}}$ is the remainder when the integer $x\times y$ is divided by rank

簡單說就是先加\乘後取餘數