> For the complete documentation index, see [llms.txt](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/yuan-gen-yu-zhi-biao/zhi-biao.md).

# 指标

## **定义**

设$$m>1$$为整数，$$a$$是与$$m$$互素的正整数，$$g$$为模$$m$$的一个原根，则存在唯一的$$1\leq r\leq \varphi{\left(m\right)}$$，使得$$g^r\equiv a\left(\mod m\right)$$，记作$$r=\mathrm{ind}\_{g}{a}$$

## **定理**

* 整数$$r$$满足$$g^r\equiv a\left(\mod m\right)\Longrightarrow r\equiv \mathrm{ind}\_{g}{a}\left(\mod \varphi{\left(m\right)}\right)$$
* 以$$g$$为底的对模$$m$$有相同指标$$r$$的所有整数全体是模$$m$$的一个简化剩余类
* $$\mathrm{ind}*{g}{a\_1\cdots a\_n}\equiv \mathrm{ind}*{g}{a\_1}+\cdots +\mathrm{ind}\_{g}{a\_n}\left(\mod \varphi{\left(m\right)}\right)$$
* $$\mathrm{ind}*{g}{a^n}\equiv n\cdot \mathrm{ind}*{g}{a}\left(\mod \varphi{\left(m\right)}\right)$$