# About Levin's constructure of universal OWF, why $\log n$ bits TM can work?

In Pass's lecture notes 2.13 section, a universal OWF is constructed. But I have some confusion about:

Let $$M_g$$ be the smallest machine which computes function $$g$$. Since $$M_g$$ is a uniform algorithm it has some constant description size $$|M_g|$$. Thus, on a random n-bit input $$y = $$, the probability that machine $$M = M_g$$ (with appropriate padding of 0) is $$2^{-\log n} = 1/n$$.

$$g$$ in the context is a strong OWF. $$|M| = \log n$$ in the quote.

I mean, $$\log n$$ bits just generate $$n$$ Turing machines, why the specific $$g$$ must be there? So the $$n$$ Turing machines are all the TM?

I know that Turing machine is countable when alphabet set and other settings are fixed. But why in this case $$n$$ TMs can work?

It confused me for days, can anyone help?

Since $$M_g$$ is a uniform algorithm it has some constant description size $$|M_g|$$.
Uniform means that the same machine can compute the function $$g$$ for arbitrary input lengths. This machine then has a constant size description that is in particular independent of $$n$$.
So for a large enough $$n$$, $$\log n \geq |M_g|$$.