On The Next Bit Test

But this notation is defined (informally) in the first paper.

The notation $O(\nu(n))$ is used for any function, $f(n)$, that vanishes faster than the inverse of any polynomial, that is for every polynomial, $\mathrm{poly} (n)$, and $n$ large enough, $f(n) \leq 1/\mathrm{poly}(n)$

Therefore, what it means is no probabilistic polynomial time (PPT) algorithm $A$ can guess the next bit at inverse polynomially decreasing error rate.

Given any PPT algorithm $A$ this error probability decays super-polynomially, for example at a rate $\exp(-\log^2 n)$ which goes to zero faster than any polynomial.

Claim: $\exp(-\log^2 n)$ is less than $n^{-c}$ for any constant $c$ for $n$ large enough.

Proof: Look at the reciprocal. $$\exp(\log^2 n) > n^c$$ if and only if $$\log^2 n > c \log n$$ which will clearly happen as soon as $$\frac{\log^2 n}{\log n}>c$$ i.e., as soon as $\log n>c$.

Note that $\exp(\log^{1+\epsilon} n)$ is superpolynomial for all $\epsilon>0.$

Edit: The superpolynomial convergence is essentially what is referred to as negligibility.

Solved by user12016 (16,085 )
Sep 01, 2020 at 00:32