Calculating phi(k) for 1<k<N

Question

Given a large N, I need to iterate through all phi(k) such that 1 < k < N :

• time-complexity must be O(N logN)
• memory-complexity mus

Answer 1

No one has found a faster way to calculate phi(k) (aka, Euler's totient function) than by first finding the prime factors of k. The world's best mathematicians have thrown many CPU cycles at the problem since 1977, since finding a faster way to solve this problem would create a weakness in the RSA public-key algorithm. (Both the public and the private key in RSA are calculated based on phi(n), where n is the product of two large primes.)