(Z/nZ)*——Z/nZ的乘法群

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这篇文章介绍了模n乘法群(Z/nZ)*的概念，它由与n互质的整数组成。文章解释了如何计算这个群的元素，以及如何找到乘法逆元。通过Python代码示例展示了计算Z/nZ从1到100的乘法群，并给出了结果。

![](https://img.learnblockchain.cn/2025/12/25/1JwNaL-whP7vWq0sfYZ_Mbg.png)

## (Z/nZ)\* — Z/nZ 的乘法群

如果说我的职业生涯中有什么遗憾，那就是我希望自己能更早地发现密码学，并且花时间去理解群论的基础知识。

模 _n_ 的整数的乘法群表示为：

![](https://img.learnblockchain.cn/2025/12/25/1JwNaL-whP7vWq0sfYZ_Mbg.png)

表示与 _n_ 没有公因子的数字。如果 _n_ 是一个素数，那么这些值将从 1 到 _n_ −1。如果 _n_ =6，我们有两个因子 2 和 3，因此 ℤ∗6 将是 {1,5}，而其他值与 2 或 3 有共同的因子。

让我们尝试 _n_ =10，它有 2 和 5 的素因子：

```
Computing Z_n (showing first 50) and Z^*_n
# 计算 Z_n (显示前 50 个) 和 Z^*_n
p=10
Z_10= {1, 2, 3, 4, 5, 6, 7, 8, 9}
Z*_10= [1, 3, 7, 9]
phi(n)= 4
```

在这种情况下，阶 _(φ(_ n)) 是 4，因为 ℤ∗10 中有四个元素。由于 2、4、5、6 和 8 与 10 有共同的因子，因此它们不包括在 _Z_ ∗10 中。

![](https://img.learnblockchain.cn/2025/12/25/1uxHPmda-wyt27sqCPVk91w.png)

因此，我们可以为每个输入映射到集合中的每个值。对于乘法，我们有：

3 x 1= 3 (mod 10)

3 x 3 = 9 (mod 10)

3 x 7 = 21 = 1 (mod 10)

3 x 9 = 27 = 7 (mod 10)

乘数 a 的逆是 a^{-1} mod 10。则逆值为：

```
>>> pow(1,-1,10)
1
>>> pow(3,-1,10)
7
>>> pow(7,-1,10)
3
>>> pow(9,-1,10)
9
```

应用逆得到：

3 x 1 = 3 (mod 10)

9 x 7 = 63 = 3 (mod 10)

1 x 3 = 3 (mod 10)

7 x 9 = 63 = 3 (mod 10)

在这里，我们反转了乘法运算。

对于素数，ℤ∗ _n_ 将与 ℤ _n_ 相同。例如，对于 _n_ =13：

```
Computing Z_n (showing first 100) and Z^*_n
# 计算 Z_n (显示前 100 个) 和 Z^*_n
p=13
Z_13= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Z*_13= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
phi(n)= 12
```

在这种情况下，**阶是 φ(n)** = _n_ −1=12。

现在，让我们尝试 Z\_1 到 Z\_100：

```
import math
import sys

if (len(sys.argv)>1):
    N=int(sys.argv[1])

for i in range(100):
 a=set()
 for x in range(i):
      if (math.gcd(x,i)!=1): continue;
      if (x not in a ): a.add(x)

print(f"(Z/nZ)* is the multiplicative group of Z/nZ n={i}")
 print (sorted(a))
 print(f"Order is {len(a)}")
```

这给出了以下结果：

```
(Z/nZ)* is the multiplicative group of Z/nZ n=1
# (Z/nZ)* 是 Z/nZ n=1 的乘法群
[0]
Order is 1
# 阶是 1

(Z/nZ)* is the multiplicative group of Z/nZ n=2
# (Z/nZ)* 是 Z/nZ n=2 的乘法群
[1]
Order is 1
# 阶是 1

(Z/nZ)* is the multiplicative group of Z/nZ n=3
# (Z/nZ)* 是 Z/nZ n=3 的乘法群
[1, 2]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=4
# (Z/nZ)* 是 Z/nZ n=4 的乘法群
[1, 3]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=5
# (Z/nZ)* 是 Z/nZ n=5 的乘法群
[1, 2, 3, 4]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=6
# (Z/nZ)* 是 Z/nZ n=6 的乘法群
[1, 5]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=7
# (Z/nZ)* 是 Z/nZ n=7 的乘法群
[1, 2, 3, 4, 5, 6]
Order is 6
# 阶是 6

(Z/nZ)* is the multiplicative group of Z/nZ n=8
# (Z/nZ)* 是 Z/nZ n=8 的乘法群
[1, 3, 5, 7]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=9
# (Z/nZ)* 是 Z/nZ n=9 的乘法群
[1, 2, 4, 5, 7, 8]
Order is 6
# 阶是 6

(Z/nZ)* is the multiplicative group of Z/nZ n=10
# (Z/nZ)* 是 Z/nZ n=10 的乘法群
[1, 3, 7, 9]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=11
# (Z/nZ)* 是 Z/nZ n=11 的乘法群
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Order is 10
# 阶是 10

(Z/nZ)* is the multiplicative group of Z/nZ n=12
# (Z/nZ)* 是 Z/nZ n=12 的乘法群
[1, 5, 7, 11]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=13
# (Z/nZ)* 是 Z/nZ n=13 的乘法群
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
Order is 12
# 阶是 12

(Z/nZ)* is the multiplicative group of Z/nZ n=14
# (Z/nZ)* 是 Z/nZ n=14 的乘法群
[1, 3, 5, 9, 11, 13]
Order is 6
# 阶是 6
```(Z/nZ)* 是 Z/nZ 的乘法群 n=15
[1, 2, 4, 7, 8, 11, 13, 14]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=16
[1, 3, 5, 7, 9, 11, 13, 15]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=17
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=18
[1, 5, 7, 11, 13, 17]
阶数为 6
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=19
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=20
[1, 3, 7, 9, 11, 13, 17, 19]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=21
[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=22
[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
阶数为 10
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=23
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]
阶数为 22
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=24
[1, 5, 7, 11, 13, 17, 19, 23]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=25
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=26
[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=27
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=28
[1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=29
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]
阶数为 28
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=30
[1, 7, 11, 13, 17, 19, 23, 29]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=31
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]
阶数为 30
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=32
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=33
[1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=34
[1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=35
[1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=36
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=37
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=38
[1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 37]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=39
[1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=40
[1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=41
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=42
[1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=43
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]
阶数为 42
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=44
[1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=45
[1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=46
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]
阶数为 22
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=47
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]
阶数为 46
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=48
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=49
[1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48]
阶数为 42
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=50
[1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=51
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=52
[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=53
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]
阶数为 52
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=54
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=55
[1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=56
[1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=57
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=58
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57]
阶数为 28
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=59
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]
阶数为 58
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=60
[1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=61
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]
阶数为 60
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=62
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61]
阶数为 30
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=63
[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 50, 52, 53, 55, 58, 59, 61, 62]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=64
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=65
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64]
阶数为 48
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=66
[1, 5, 7, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 65]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=67
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]
阶数为 66
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=68
[1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 59, 61, 63, 65, 67]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=69
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68]
阶数为 44
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=70
[1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=71
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]
阶数为 70
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=72
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=73
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72]
阶数为 72
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=74
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=75
[1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=76
```
[1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 37, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]
顺序是 78
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=80
[1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79]
顺序是 32
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=81
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80]
顺序是 54
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=82
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81]
顺序是 40
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=83
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]
顺序是 82
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=84
[1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83]
顺序是 24
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=85
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84]
顺序是 64
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=86
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85]
顺序是 42
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=87
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86]
顺序是 56
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=88
[1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87]
顺序是 40
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=89
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]
顺序是 88
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=90
[1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89]
顺序是 24
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=91
[1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90]
顺序是 72
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=92
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91]
顺序是 44
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=93
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92]
顺序是 60
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=94
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93]
顺序是 46
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=95
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94]
顺序是 72
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=96
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95]
顺序是 32
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=97
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96]
顺序是 96
(Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=98
[1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 97]
顺序是 42
```$(Z/nZ)^*$ 是 $Z/nZ$ 的乘法群，$n=99$
[1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98]
阶数为 60
```

这是一个[演示](https://asecuritysite.com/principles_pub/z_group)。 有兴趣了解更多吗？ 试试，“群，环和离散对数”：

还有，“群运算”：

还有“群生成器”：

>- 原文链接： [medium.com/asecuritysite...](https://medium.com/asecuritysite-when-bob-met-alice/z-nz-the-multiplicative-group-of-z-nz-8c74a01f327e)
>- 登链社区 AI 助手，为大家转译优秀英文文章，如有翻译不通的地方，还请包涵～

(Z/nZ)* — Z/nZ 的乘法群

如果说我的职业生涯中有什么遗憾，那就是我希望自己能更早地发现密码学，并且花时间去理解群论的基础知识。

模 n 的整数的乘法群表示为：

表示与 n 没有公因子的数字。如果 n 是一个素数，那么这些值将从 1 到 n −1。如果 n =6，我们有两个因子 2 和 3，因此 ℤ∗6 将是 {1,5}，而其他值与 2 或 3 有共同的因子。

让我们尝试 n =10，它有 2 和 5 的素因子：

Computing Z_n (showing first 50) and Z^*_n
# 计算 Z_n (显示前 50 个) 和 Z^*_n
p=10
Z_10= {1, 2, 3, 4, 5, 6, 7, 8, 9}
Z*_10= [1, 3, 7, 9]
phi(n)= 4

在这种情况下，阶 (φ( n)) 是 4，因为 ℤ∗10 中有四个元素。由于 2、4、5、6 和 8 与 10 有共同的因子，因此它们不包括在 Z ∗10 中。

因此，我们可以为每个输入映射到集合中的每个值。对于乘法，我们有：

3 x 1= 3 (mod 10)

3 x 3 = 9 (mod 10)

3 x 7 = 21 = 1 (mod 10)

3 x 9 = 27 = 7 (mod 10)

乘数 a 的逆是 a^{-1} mod 10。则逆值为：

>>> pow(1,-1,10)
1
>>> pow(3,-1,10)
7
>>> pow(7,-1,10)
3
>>> pow(9,-1,10)
9

应用逆得到：

3 x 1 = 3 (mod 10)

9 x 7 = 63 = 3 (mod 10)

1 x 3 = 3 (mod 10)

7 x 9 = 63 = 3 (mod 10)

在这里，我们反转了乘法运算。

对于素数，ℤ∗ n 将与 ℤ n 相同。例如，对于 n =13：

Computing Z_n (showing first 100) and Z^*_n
# 计算 Z_n (显示前 100 个) 和 Z^*_n
p=13
Z_13= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Z*_13= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
phi(n)= 12

在这种情况下，阶是 φ(n) = n −1=12。

现在，让我们尝试 Z_1 到 Z_100：

import math
import sys

if (len(sys.argv)>1):
    N=int(sys.argv[1])

for i in range(100):
 a=set()
 for x in range(i):
      if (math.gcd(x,i)!=1): continue;
      if (x not in a ): a.add(x)

 print(f"(Z/nZ)* is the multiplicative group of Z/nZ n={i}")
 print (sorted(a))
 print(f"Order is {len(a)}")

这给出了以下结果：

(Z/nZ)* is the multiplicative group of Z/nZ n=1
# (Z/nZ)* 是 Z/nZ n=1 的乘法群
[0]
Order is 1
# 阶是 1

(Z/nZ)* is the multiplicative group of Z/nZ n=2
# (Z/nZ)* 是 Z/nZ n=2 的乘法群
[1]
Order is 1
# 阶是 1

(Z/nZ)* is the multiplicative group of Z/nZ n=3
# (Z/nZ)* 是 Z/nZ n=3 的乘法群
[1, 2]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=4
# (Z/nZ)* 是 Z/nZ n=4 的乘法群
[1, 3]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=5
# (Z/nZ)* 是 Z/nZ n=5 的乘法群
[1, 2, 3, 4]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=6
# (Z/nZ)* 是 Z/nZ n=6 的乘法群
[1, 5]
Order is 2
# 阶是 2

(Z/nZ)* is the multiplicative group of Z/nZ n=7
# (Z/nZ)* 是 Z/nZ n=7 的乘法群
[1, 2, 3, 4, 5, 6]
Order is 6
# 阶是 6

(Z/nZ)* is the multiplicative group of Z/nZ n=8
# (Z/nZ)* 是 Z/nZ n=8 的乘法群
[1, 3, 5, 7]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=9
# (Z/nZ)* 是 Z/nZ n=9 的乘法群
[1, 2, 4, 5, 7, 8]
Order is 6
# 阶是 6

(Z/nZ)* is the multiplicative group of Z/nZ n=10
# (Z/nZ)* 是 Z/nZ n=10 的乘法群
[1, 3, 7, 9]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=11
# (Z/nZ)* 是 Z/nZ n=11 的乘法群
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Order is 10
# 阶是 10

(Z/nZ)* is the multiplicative group of Z/nZ n=12
# (Z/nZ)* 是 Z/nZ n=12 的乘法群
[1, 5, 7, 11]
Order is 4
# 阶是 4

(Z/nZ)* is the multiplicative group of Z/nZ n=13
# (Z/nZ)* 是 Z/nZ n=13 的乘法群
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
Order is 12
# 阶是 12

(Z/nZ)* is the multiplicative group of Z/nZ n=14
# (Z/nZ)* 是 Z/nZ n=14 的乘法群
[1, 3, 5, 9, 11, 13]
Order is 6
# 阶是 6
```(Z/nZ)* 是 Z/nZ 的乘法群 n=15
[1, 2, 4, 7, 8, 11, 13, 14]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=16
[1, 3, 5, 7, 9, 11, 13, 15]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=17
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=18
[1, 5, 7, 11, 13, 17]
阶数为 6
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=19
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=20
[1, 3, 7, 9, 11, 13, 17, 19]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=21
[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=22
[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
阶数为 10
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=23
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]
阶数为 22
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=24
[1, 5, 7, 11, 13, 17, 19, 23]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=25
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=26
[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=27
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=28
[1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=29
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]
阶数为 28
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=30
[1, 7, 11, 13, 17, 19, 23, 29]
阶数为 8
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=31
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]
阶数为 30
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=32
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=33
[1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=34
[1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=35
[1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=36
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=37
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=38
[1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 37]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=39
[1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=40
[1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=41
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=42
[1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41]
阶数为 12
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=43
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]
阶数为 42
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=44
[1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=45
[1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=46
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]
阶数为 22
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=47
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]
阶数为 46
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=48
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=49
[1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48]
阶数为 42
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=50
[1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=51
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=52
[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=53
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]
阶数为 52
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=54
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53]
阶数为 18
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=55
[1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=56
[1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=57
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=58
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57]
阶数为 28
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=59
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]
阶数为 58
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=60
[1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59]
阶数为 16
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=61
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]
阶数为 60
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=62
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61]
阶数为 30
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=63
[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 50, 52, 53, 55, 58, 59, 61, 62]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=64
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=65
[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64]
阶数为 48
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=66
[1, 5, 7, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 65]
阶数为 20
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=67
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]
阶数为 66
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=68
[1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 59, 61, 63, 65, 67]
阶数为 32
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=69
[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68]
阶数为 44
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=70
[1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=71
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]
阶数为 70
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=72
[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71]
阶数为 24
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=73
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72]
阶数为 72
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=74
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73]
阶数为 36
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=75
[1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74]
阶数为 40
(Z/$n$Z)* 是 Z/$n$Z 的乘法群 $n$=76

[1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 37, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78] 顺序是 78 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=80 [1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79] 顺序是 32 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=81 [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80] 顺序是 54 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=82 [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81] 顺序是 40 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=83 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82] 顺序是 82 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=84 [1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83] 顺序是 24 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=85 [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84] 顺序是 64 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=86 [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85] 顺序是 42 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=87 [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86] 顺序是 56 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=88 [1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87] 顺序是 40 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=89 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88] 顺序是 88 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=90 [1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89] 顺序是 24 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=91 [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90] 顺序是 72 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=92 [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91] 顺序是 44 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=93 [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92] 顺序是 60 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=94 [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93] 顺序是 46 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=95 [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94] 顺序是 72 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=96 [1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95] 顺序是 32 (Z/$n$Z) is the multiplicative group of Z/$n$Z $n$=97 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96] 顺序是 96 (Z/$n$Z)* is the multiplicative group of Z/$n$Z $n$=98 [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 97] 顺序是 42

[1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98]
阶数为 60

这是一个演示。有兴趣了解更多吗？试试，“群，环和离散对数”：

还有，“群运算”：

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原文链接： medium.com/asecuritysite...

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(Z/nZ)*——Z/nZ的乘法群

(Z/nZ)* — Z/nZ 的乘法群

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