Python: finding all generators for a cyclic group

Question

Take a cyclic group Z_n with the order n. The elements are:

Z_n = {1,2,...,n-1}

For each of the elements, let us call them a, you test if a^x % n gives us all numbers in Z_n; x is here all numbers from 1 to n-1. If the element does generator our entire group, it is a generator.

I need a program that gets the order of the group and gives back all the generators. Here is what I tried:

import math
active = True

def test(a,b):
    a.sort()
    b.sort()
    
    return a == b

while active:
    order = input("Order of the cyclic group: ")
    print
    group = []
    for i in range(1, order):
        group.append(i)
    
    res = []
    for x in group:
        foo = []
        foo.append(x)
        for y in group:
            foo.append((x**y) % order)
        if(test(group,foo)):
            res.append(res,foo,axis=0)

    print res

Sadly, it gives back an empty list.

Asked By: Heinrich Jensen

||

Source

Answer 1

One change you can make is to use sets instead of lists to store the results, which makes comparing them easier.

def generators(n):
    s = set(range(1, n))
    results = []
    for a in s:
        g = set()
        for x in s:
            g.add((a**x) % n)
        if g == s:
            results.append(a)
    return results

for i in range(100):
    gens = generators(i)
    if gens:
        print(f"Z_{i} has generators {gens}")

Prints

Z_2 has generators [1]
Z_3 has generators [2]
Z_5 has generators [2, 3]
Z_7 has generators [3, 5]
Z_11 has generators [2, 6, 7, 8]
Z_13 has generators [2, 6, 7, 11]
Z_17 has generators [3, 5, 6, 7, 10, 11, 12, 14]
Z_19 has generators [2, 3, 10, 13, 14, 15]
Z_23 has generators [5, 7, 10, 11, 14, 15, 17, 19, 20, 21]
Z_29 has generators [2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27]
Z_31 has generators [3, 11, 12, 13, 17, 21, 22, 24]
Z_37 has generators [2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35]
Z_41 has generators [6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35]
Z_43 has generators [3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34]
Z_47 has generators [5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39, 40, 41, 43, 44, 45]
Z_53 has generators [2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51]
Z_59 has generators [2, 6, 8, 10, 11, 13, 14, 18, 23, 24, 30, 31, 32, 33, 34, 37, 38, 39, 40, 42, 43, 44, 47, 50, 52, 54, 55, 56]
Z_61 has generators [2, 6, 7, 10, 17, 18, 26, 30, 31, 35, 43, 44, 51, 54, 55, 59]
Z_67 has generators [2, 7, 11, 12, 13, 18, 20, 28, 31, 32, 34, 41, 44, 46, 48, 50, 51, 57, 61, 63]
Z_71 has generators [7, 11, 13, 21, 22, 28, 31, 33, 35, 42, 44, 47, 52, 53, 55, 56, 59, 61, 62, 63, 65, 67, 68, 69]
Z_73 has generators [5, 11, 13, 14, 15, 20, 26, 28, 29, 31, 33, 34, 39, 40, 42, 44, 45, 47, 53, 58, 59, 60, 62, 68]
Z_79 has generators [3, 6, 7, 28, 29, 30, 34, 35, 37, 39, 43, 47, 48, 53, 54, 59, 60, 63, 66, 68, 70, 74, 75, 77]
Z_83 has generators [2, 5, 6, 8, 13, 14, 15, 18, 19, 20, 22, 24, 32, 34, 35, 39, 42, 43, 45, 46, 47, 50, 52, 53, 54, 55, 56, 57, 58, 60, 62, 66, 67, 71, 72, 73, 74, 76, 79, 80]
Z_89 has generators [3, 6, 7, 13, 14, 15, 19, 23, 24, 26, 27, 28, 29, 30, 31, 33, 35, 38, 41, 43, 46, 48, 51, 54, 56, 58, 59, 60, 61, 62, 63, 65, 66, 70, 74, 75, 76, 82, 83, 86]
Z_97 has generators [5, 7, 10, 13, 14, 15, 17, 21, 23, 26, 29, 37, 38, 39, 40, 41, 56, 57, 58, 59, 60, 68, 71, 74, 76, 80, 82, 83, 84, 87, 90, 92]

Answered By: Patrick Haugh

Answer 2

def gen(a,b):
  s = set(range(0,a))
  g = set()
  for i in s:
    g.add((i*b)%a)
  return g
a = int(input()) #order of Z, e.g Z4, Z5, etc...
s = set(range(0,a))
for i in s:
  if(gen(a,i) == s):
    print(i)

You, may try this. It will work.

Answered By: Aziz Lokhandwala

Python: finding all generators for a cyclic group

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