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I had a brief look at "On Defining Proofs of Knowledge" by Bellare and Goldreich and I am a little confused by their definitions.

I was under the impression a negligible function $f$ was defined as something like $$\forall\ polynomials\ p\ \exists k\ s.t.\ \forall x > k: f(x) < \frac{1}{p(x)}$$

And that non-negligible meant simply that it was not negligible. The paper however states: "Put in other words negligible is not the negation of non-negligible!" (p. 5) And this seems to be based on the definition "a non-negligible function in $n$ is a function which is asymptotically bounded from below by a function of the form $n^{-c}$ for some constant $c$" (p. 4) which I am losely translating as $$\exists\ polynomial\ p\ and\ k\ s.t.\ \forall x > k: f(x) > \frac{1}{p(x)}$$ With the difference being functions which are somehow alternating.

This is a bit of a weird question because the mathematics seem to be clear but I am confused by what the common usage is. And is this something that is generally important? I've never seen it discussed elsewhere.

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    $\begingroup$ What they seem to be calling "non-negligible" (I can't check the ps file on mobile) is usually called "noticable" and the standard caution is that "non-negligible" (in the sense you used it) is not equivalent to "noticable". $\endgroup$
    – Maeher
    May 17 at 13:14
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    $\begingroup$ See e.g. these lecture notes $\endgroup$
    – Maeher
    May 17 at 13:24
  • $\begingroup$ Or page 9 of these $\endgroup$
    – Maeher
    May 17 at 13:30
  • $\begingroup$ Yes, that is the problem with the negation of complex sentences. $\endgroup$
    – kelalaka
    May 17 at 14:29

1 Answer 1

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  • Negligible Function: A function $\mu$ is negligible iff $\forall c \in N \;\; \exists n_0 \in N$ such that $\forall n \geq n_0, \mu(n) < n^{−c}.$

    As we generally now, a negligible function is smaller than any polynomial. We have also an equivalent limit definition;

    $f(n)$ is negligible than for every polynomial $q(n)$ we have;

    $$\lim_{n \rightarrow \infty} q(n) f(n) =0$$

    The easy examples are the $2^{-n},2^{-\sqrt{n}}, \text{ and } n^{- \log n}$.

  • Non-Negligible Function:* a function $\mu(n)$ is non-negligible iff $\exists c \in N$ such that $\forall n_0 \in N, \exists n \geq n_0$ such that $\mu(n) \geq n^{-c}.$

    To be non-negligible, only one candidate is enough to show that $n \geq n_0$ for which $\mu(n) \geq n^{-c}$.

  • Noticeable Function: A function $\mu$ is noticeable iff $\exists c \in N, n_0 \in N$ such that $\forall n \geq n_0, \mu(n) \geq n^{-c}.$

    As we can see, the difference from non-negligibility is; for all $n \geq n_0$

    An example is $n^{-3}$ which is only polynomially slow ( like any polynomial)

    Weak One-Way Functions are defined on noticeable functions.

Interleaving is the key to generating distinguishing examples. Take any noticeable and negligible function and interleave them;

$$\mu(n) = \cases{ 2^{-n} & : $x$ is even \\ n^{-3} & : $x$ is odd}$$

$\mu$ is a non-negligible and non-noticeable function!.


*Quantifiers negation: In the negation $\neg\forall = \exists$ and $\neg \exists = \forall$

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  • $\begingroup$ Based on the comments, I've given a shot that our site has this written around... $\endgroup$
    – kelalaka
    May 17 at 19:31
  • $\begingroup$ This answer does not address the fact that the cited paper defines non-negligible differently. $\endgroup$
    – Maeher
    May 17 at 19:53
  • $\begingroup$ @Maeher I'm aware of this. These definitions are the same as in Oded Goldreich's Foundations of Cryptography Volume I. So, I can say this is common usage as OP asked. If we consider the date of the article as '92 and the book as '01-03 we can say that this is common then ( Oded co-auther of the article and the sole author of the book) $\endgroup$
    – kelalaka
    May 17 at 20:03

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