Euler’s Totient Function

2.8

Euler’s Totient function Φ(n) for an input n is count of numbers in {1, 2, 3, …, n} that are relatively prime to n, i.e., the numbers whose GCD (Greatest Common Divisor) with n is 1.

Examples:

Φ(1) = 1  
gcd(1, 1) is 1

Φ(2) = 1
gcd(1, 2) is 1, but gcd(2, 2) is 2.

Φ(3) = 2
gcd(1, 3) is 1 and gcd(2, 3) is 1

Φ(4) = 2
gcd(1, 4) is 1 and gcd(3, 4) is 1

Φ(5) = 4
gcd(1, 5) is 1, gcd(2, 5) is 1, 
gcd(3, 5) is 1 and gcd(4, 5) is 1

Φ(6) = 2
gcd(1, 6) is 1 and gcd(5, 6) is 1, 

How to compute Φ(n) for an input n?

A simple solution is to iterate through all numbers from 1 to n-1 and count numbers with gcd with n as 1. Below is C implementation of the simple method to compute Euler’s Totient function for an input integer n.

// A simple C program to calculate Euler's Totient Function
#include <stdio.h>

// Function to return gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b%a, a);
}

// A simple method to evaluate Euler Totient Function
int phi(unsigned int n)
{
    unsigned int result = 1;
    for (int i=2; i < n; i++)
        if (gcd(i, n) == 1)
            result++;
    return result;
}

// Driver program to test above function
int main()
{
    int n;
    for (n=1; n<=10; n++)
      printf("phi(%d) = %d\n", n, phi(n));
    return 0;
}

Output:

phi(1) = 1
phi(2) = 1
phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4 
phi(9) = 6
phi(10) = 4

The above code calls gcd function O(n) times. Time complexity of the gcd function is O(h) where h is number of digits in smaller number of given two numbers. Therefore, an upper bound on time complexity of above solution is O(nLogn) [How? there can be at most Log10n digits in all numbers from 1 to n]

Below is a Better Solution. The idea is based on Euler’s product formula which states that value of totient functions is below product over all prime factors p of n.
eulersproduct
The formula basically says that the value of Φ(n) is equal to n multiplied by product of (1 – 1/p) for all prime factors p of n. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2.

We can find all prime factors using the idea used in this post.

1) Initialize : result = n
2) Run a loop from 'p' = 2 to sqrt(n), do following for every 'p'.
     a) If p divides n, then 
           Set: result = result  * (1.0 - (1.0 / (float) p));
           Divide all occurrences of p in n.
3) Return result  

Below is C implementation of Euler’s product formula.

// C program to calculate Euler's Totient Function
// using Euler's product formula
#include <stdio.h>

int phi(int n)
{
    float result = n;   // Initialize result as n

    // Consider all prime factors of n and for every prime
    // factor p, multiply result with (1 - 1/p)
    for (int p=2; p*p<=n; ++p)
    {
        // Check if p is a prime factor.
        if (n % p == 0)
        {
            // If yes, then update n and result
            while (n % p == 0)
                n /= p;
            result *= (1.0 - (1.0 / (float) p));
        }
    }

    // If n has a prime factor greater than sqrt(n)
    // (There can be at-most one such prime factor)
    if (n > 1)
        result *= (1.0 - (1.0 / (float) n));

    return (int)result;
}

// Driver program to test above function
int main()
{
    int n;
    for (n=1; n<=10; n++)
      printf("phi(%d) = %d\n", n, phi(n));
    return 0;
}

Output:

phi(1) = 1
phi(2) = 1
phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4
phi(9) = 6
phi(10) = 4

We can avoid floating point calculations in above method. The idea is to count all prime factors and their multiples and subtract this count from n to get the totient function value (Prime factors and multiples of prime factors won’t have gcd as 1)

1) Initialize result as n
2) Consider every number 'p' (where 'p' varies from 2 to √n). 
   If p divides n, then do following
   a) Subtract all multiples of p from 1 to n [all multiples of p
      will have gcd more than 1 (at least p) with n]
   b) Update n by repeatedly dividing it by p.
3) If the reduced n is more than 1, then remove all multiples
   of n from result.

Below is C implementation of above algorithm.

// C program to calculate Euler's Totient Function
#include <stdio.h>

int phi(int n)
{    
    int result = n;   // Initialize result as n

    // Consider all prime factors of n and subtract their
    // multiples from result
    for (int p=2; p*p<=n; ++p)
    {
        // Check if p is a prime factor.
        if (n % p == 0)
        {
            // If yes, then update n and result 
            while (n % p == 0)
                n /= p;
            result -= result / p;
        }
    }

    // If n has a prime factor greater than sqrt(n)
    // (There can be at-most one such prime factor)
    if (n > 1)
        result -= result / n;
    return result;
}

// Driver program to test above function
int main()
{
    int n;
    for (n=1; n<=10; n++)
      printf("phi(%d) = %d\n", n, phi(n));
    return 0;
}

Output:

phi(1) = 1
phi(2) = 1
phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4
phi(9) = 6
phi(10) = 4

Let us take an example to understand the above algorithm.

n = 10. 
Initialize: result = 10

2 is a prime factor, so n = n/i = 5, result = 5
3 is not a prime factor.

The for loop stops after 3 as 4*4 is not less than or equal
to 10.

After for loop, result = 5, n = 5
Since n > 1, result = result - result/n = 4

Some Interesting Properties of Euler’s Totient Function
1) For a prime number p, Φ(p) is p-1. For example Φ(5) is 4, Φ(7) is 6 and Φ(13) is 12. This is obvious, gcd of all numbers from 1 to p-1 will be 1 because p is a prime.

2) For two numbers a and b, if gcd(a, b) is 1, then Φ(ab) = Φ(a) * Φ(b). For example Φ(5) is 4 and Φ(6) is 2, so Φ(30) must be 8 as 5 and 6 are relatively prime.

3) For any two prime numbers p and q, Φ(pq) = (p-1)*(q-1). This property is used in RSA algorithm.

4) If p is a prime number, then Φ(pk) = pk – pk-1. This can be proved using Euler’s product formula.

5) Sum of values of totient functions of all divisors of n is equal to n.
gausss
For example, n = 6, the divisors of n are 1, 2, 3 and 6. According to Gauss, sum of Φ(1) + Φ(2) + Φ(3) + Φ(6) should be 6. We can verify the same by putting values, we get (1 + 1 + 2 + 2) = 6.

6) The most famous and important feature is expressed in Euler’s theorem :

The theorem states that if n and a are coprime
(or relatively prime) positive integers, then

aΦ(n) ≡ 1 (mod n) 

The RSA cryptosystem is based on this theorem:

In the particular case when m is prime say p, Euler’s theorem turns into the so-called Fermat’s little theorem :

ap-1 ≡ 1 (mod p) 

7) Number of generators of a finite cyclic group under modulo n addition is Φ(n).

Related Article:
Euler’s Totient function for all numbers smaller than or equal to n
Optimized Euler Totient Function for Multiple Evaluations

References:
http://e-maxx.ru/algo/euler_function

http://en.wikipedia.org/wiki/Euler%27s_totient_function

This article is contributed by Ankur. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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