Physical meaning of Negligible and Non-Negligible Functions

A typical nerd

10/15/23, 5:14 PM

I've been itching my head over this for a while despite going through the queries related to the topic. Can someone explain me negligible and non-negligible function in a concise way?

As of my naive understanding (correct me if I've the wrong take);

A non-negligible function is one which approaches zero relatively slow (eg: reciprocal of polynomial function as compared to exponential function) and given enough computationally possible time and space, such function is breakable after, poly(n) time.

A negligible function is one which approaches zero quickly and is computationally infeasible to break as the function grows faster with time. e.g. $\frac{1}{k^n}$

I don't understand how $\epsilon(\lambda) \geq \frac{1}{\lambda^n}$ is non-negligible and $\lambda \geq \lambda_d: \epsilon(\lambda) \leq \frac{1}{\lambda^n}$ is negligible. Also, what's $\lambda_d$ suppose to mean?

0 + 3

stream-cipher

negligible

probability

kelalaka

10/15/23, 5:28 PM

Do you want something like these? **1**) [What exactly is a negligible (and non-negligible) function?](https://crypto.stackexchange.com/q/5832/18298) 2) [How small is negligible?](https://crypto.stackexchange.com/q/61739/18298) 3) [What are not non-negligible functions?](https://crypto.stackexchange.com/q/100177/18298) 4) [Identifying negligible functions](https://crypto.stackexchange.com/q/63284/18298)

Mikero

10/15/23, 7:10 PM

I don't think $\lambda_d$ is a typo. It is saying, for all $d$, there exists a number $\lambda_d$, such that for all $\lambda \ge \lambda_d$, the function is smaller than $1/\lambda^d$. In other words, for all $d$, the function is *eventually* smaller than $1/\lambda^d$, and "eventually" means "for all but finitely many $\lambda$", i.e., "for all $\lambda$ larger than some finite limit $\lambda_d$"

kelalaka

10/15/23, 7:26 PM

@Mikero you are right about it. Instead of $\lambda_d$ another symbol much easier to grasp.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Physical meaning of Negligible and Non-Negligible Functions

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.