解题方法
1 . 离散对数在密码学中有重要的应用.设
是素数,集合
,若
,记
为
除以
的余数,
为
除以
的余数;设
,
两两不同,若
,则称
是以
为底
的离散对数,记为
.
(1)若
,求
;
(2)对
,记
为
除以
的余数(当
能被
整除时,
).证明:
,其中
;
(3)已知
.对
,令
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f05bea470ae14b90937f6f71dc9a6242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b2b0dcbc27df9950b26028e46f6c17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e5865fd0fb7c35e8a4a1d311163290b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbe6ebc6c1d1a214f5ca478ae666cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67a1f88ae28ecdb67c7f9c4ae61481b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae890dd5b6300cf23b4905e86410317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff99d1615f90ff71b56ca1dfebd626d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/420a12638f77a27c696f63ff946e8684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b0087ea124b6fd98fbbcb9bc4c2e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac54185ed8bb89c774ceb685408156c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca7b54c31c5ab3831f260012758ffa12.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099d389a1c0e5877350e62c52c4a724c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28ab2ad5d8b72e3f26bef4be0697ec70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34fc26e532b65641a53eaa7e127aa683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4d45dbe0a914249371aed3641515123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53ace23b21d7b119ad7ac5cf877c19f0.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2793be26b839ae9f8f83cf2b5a597cd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1a6740a4f2378965bc019bc6aacd44a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f278b4fd6ed264265e3ccfac4ab7ef02.png)
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2024-01-19更新
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