Time complexity of Euler's totient function

Asked May 17 '19 at 05:33

Active May 23 '19 at 11:53

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I believe there are different time complexities for Euler's totient function depending on how you execute the algorithm. The two I know of are:

Are there any other possibilities?

edited May 23 '19 at 11:53

Maarten Bodewes

asked May 17 '19 at 05:33

dkssud10

Both formulas are wrong. For the first, there seems to be a mix between $k$ and $n$, and calculating $\gcd(k,n)$ when $0<k<n$ has cost sizably larger than $O(\log n)$. For the second, factoring $n$ has cost much lower than $O(n)$. – fgrieu May 17 '19 at 06:25
3

You may want to note that "computing the euler totient function of a number" is (roughly) as difficult as "factoring that number", so don't expect anything better than "factor, then compute". – SEJPM May 17 '19 at 08:36
Euler's function is not itself an algorithm, and as such does not have a time complexity. When you specify an algorithm, that algorithm will have a time complexity. It depends, of course, on what units of ‘time’ you're counting ($\lg n$-bit multiplications? bit operations? communication costs?), including how much parallelism you can throw at it and what type of machine you have. – Squeamish Ossifrage May 17 '19 at 14:21

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