Cryptography
random-number-generator probability
Updated Fri, 20 May 2022 20:18:16 GMT

# On The Next Bit Test

I would like to know what $$O(v(n))$$ really means in detailed and simple words please. I found it everywhere in the literature I am reviewing but I cannot find what the intuition of it (especially if it means the Big O Notation, then what it has to do with the Probability here; also what is $$v$$, it is never defined except in another paper as a constant -but maybe not related directly to v in this formula-).

Many Thanks.

References 1 , 2

## Solution

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.

• +0 – Thanks and +1. So a successful predictor should predict with a probability greater than or equal $1-O((n))$, How is the latter evaluated numerically. That's from where should I derive the $O((n))$ numerically? A simple example of a simple algorithm will be very much appreciated. — Sep 01, 2020 at 09:57