Notes on the Euler Phi Function (or Euler’s totient function)
Along with the factoring large number into 2 primes, the RSA encryption system also
relies heavily on the Euler phi function. Before we can define this function, we must define
the greatest common divisor, and what it means to be coprime.
Definition 1. The greatest common divisor (or GCD) of two integers aand bis a number
that divides both aand bwithout remainder and is the largest of such numbers.
Example 1. The GCD of 10 and 15 is 5, because 5 divides both 10 and 15 and no larger
number divides both.
The GCD of 144 and 54 is 18, because 18 divides both 144 and 54 and no larger number
divides both.
The GCD of 31 and 33 is 1, because they don’t share any common divisors but 1.
Definition 2. Two numbers aand bare coprime if their GCD is 1.
Example 2. Any two prime numbers are coprime.
31 and 33 are coprime (see above example).
8 and 12 are not coprime because their GCD is 4.
Definition 3. Euler’s phi function is defined as follows:
ϕ(n) = the number of positive integers less than or equal to n that are coprime to n.
Example 3.
ϕ(7) = 6
because the GCD of 7 with any positive integer less or equal to than 7 is 1 (since 7 is prime)
and hence 7 is coprime to any positive integer less than or equal to 7.
Example 4.
ϕ(8) = 4
because 8 is coprime to 1, 3, 5, and 7 (i.e. the GCD of 8 and 1 is 1, the GCD of 8 and 3 is
1, etc.)
Example 5.
ϕ(15) = 8
because 15 is coprime to 1,2,4,7,8,11,13,14.
There are many different results regarding how to compute the Euler phi function, so I
would like to give you a couple that make life a little easier when trying to compute this.
Theorem 1. 1. If pis a prime number, then ϕ(p) = p−1.
2. If pand qare different prime numbers, then ϕ(p×q)=(p−1) ×(q−1).
Example 6. This result makes it easy to see that ϕ(7) = 6 and ϕ(15) = ϕ(3×5) = 2 ×4 = 8.
