Number Theory | Generators of finite cyclic group under addition

Last Updated : 20 Feb, 2023

Given a number n, find all generators of cyclic additive group under modulo n. Generator of a set {0, 1, ... n-1} is an element x such that x is smaller than n, and using x (and addition operation), we can generate all elements of the set.
Examples:

Input : 10  Output : 1 3 7 9  The set to be generated is {0, 1, .. 9}  By adding 1, single or more times, we   can create all elements from 0 to 9.  Similarly using 3, we can generate all  elements.  30 % 10 = 0, 21 % 10 = 1, 12 % 10 = 2, ...  Same is true for 7 and 9.    Input  : 24  Output : 1 5 7 11 13 17 19 23

A simple solution is to run a loop from 1 to n-1 and for every element check if it is generator. To check generator, we keep adding element and we check if we can generate all numbers until remainder starts repeating.
An Efficient solution is based on the fact that a number x is generator if x is relatively prime to n, i.e., gcd(n, x) =1.
Below is the implementation of above approach:

C++

// A simple C++ program to find all generators #include <bits/stdc++.h> using namespace std;  // Function to return gcd of a and b int gcd(int a, int b) {     if (a == 0)         return b;     return gcd(b%a, a); }  // Print generators of n int printGenerators(unsigned int n) {     // 1 is always a generator     cout << "1 ";      for (int i=2; i < n; i++)          // A number x is generator of GCD is 1         if (gcd(i, n) == 1)             cout << i << " "; }  // Driver program to test above function int main() {     int n = 10;     printGenerators(n);     return 0; }

Java

// A simple Java program to find all generators  class GFG {       // Function to return gcd of a and b static int gcd(int a, int b) {     if (a == 0)         return b;     return gcd(b%a, a); }  // Print generators of n static void printGenerators(int n) {     // 1 is always a generator     System.out.println("1 ");      for (int i=2; i < n; i++)          // A number x is generator of GCD is 1         if (gcd(i, n) == 1)             System.out.println(i +" "); }  // Driver program to test above function public static void main(String args[]) {     int n = 10;     printGenerators(n); } }

Python3

# Python3 program to find all generators  # Function to return gcd of a and b def gcd(a, b):     if (a == 0):         return b;     return gcd(b % a, a);  # Print generators of n def printGenerators(n):          # 1 is always a generator     print("1", end = " ");      for i in range(2, n):          # A number x is generator          # of GCD is 1         if (gcd(i, n) == 1):             print(i, end = " ");  # Driver Code n = 10; printGenerators(n);      # This code is contributed by mits

// A simple C# program to find all generators using System;  class GFG  {      // Function to return gcd of a and b static int gcd(int a, int b) {     if (a == 0)         return b;     return gcd(b % a, a); }  // Print generators of n static void printGenerators(int n) {     // 1 is always a generator     Console.Write("1 ");      for (int i = 2; i < n; i++)          // A number x is generator of GCD is 1         if (gcd(i, n) == 1)             Console.Write(i +" "); }  // Driver code public static void Main(String []args) {     int n = 10;     printGenerators(n); } }  // This code contributed by Rajput-Ji

PHP

<?php // PHP program to find all generators  // Function to return gcd of a and b  function gcd($a, $b) {     if ($a == 0)         return $b;     return gcd($b % $a, $a); }  // Print generators of n function printGenerators($n) {          // 1 is always a generator     echo "1 ";      for ($i = 2; $i < $n; $i++)          // A number x is generator          // of GCD is 1         if (gcd($i, $n) == 1)             echo $i, " "; }  // Driver program to test // above function     $n = 10;     printGenerators($n);      // This code is contributed by Ajit ?>

JavaScript

<script>  // A simple Javascript program to  // find all generators  // Function to return gcd of a and b function gcd(a, b) {     if (a == 0)         return b;              return gcd(b % a, a); }  // Print generators of n function printGenerators(n) {          // 1 is always a generator     document.write("1 ");      for(var i = 2; i < n; i++)          // A number x is generator of          // GCD is 1         if (gcd(i, n) == 1)             document.write(i + " "); }  // Driver Code var n = 10;  printGenerators(n);  // This code is contributed by Kirti  </script>

Output :

1 3 7 9

Time Complexity: O(nlogn)
Auxiliary space: O(1)

How does this work?
If we consider all remainders of n consecutive multiples of x, then some remainders would repeat if GCD of x and n is not 1. If some remainders repeat, then x cannot be a generator. Note that after n consecutive multiples, remainders would anyway repeat.
Interesting Observation :
Number of generators of a number n is equal to ?(n) where ? is Euler Totient Function.

Number Theory | Generators of finite cyclic group under addition

Ujjwal Goyal

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Number Theory | Generators of finite cyclic group under addition

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