# Is power of non-negligible function non-negligible?

If I have a probability which is > negl(n), i.e., non-negligible, will be this probability raised to the power of n also non-negligible?

## 1 Answer

No and yes, depending on what do you mean by $n$.

Take $f(x)=1/2$ for example. It is a non-negligible function, but $g(x)=(f(x))^x=(1/2)^{x}$ is negligible, i.e. for every positive integer $c$, there exists an integer $N_c$ such that for all $x > N_c$,

$$|g(x)|<{\frac {1}{x^{c}}}.$$

If you mean a constant $n$, then $g(x)=(f(x))^n=(1/2)^n$ is non-negligible.

• Thank you very much for the answer! I try to prove an assertion by contradiction. I assumed that a function is not one-way, i.e. Pr[Invert(n)] > negl(n). When counting the probability of a contradicting distinguisher I got Pr[Invert(n)]^n. So I try to understand if it's non-negligible? If it is then my proof is okay. According to your answer, as n is a constant, then it should be non-negligibe. But i am not sure. Jun 7, 2018 at 15:09
• Just judging from the information you provided, If $n$ is a constant, then yes it is non-negligible. However, double check your proof because in proofs $n$ is often a security parameter that is not a constant. Jun 7, 2018 at 15:29
• Okay. Thank you again! In my case {0, 1}^n is a domain of the one-way function. So I will just hope that n can be considered constant. Jun 7, 2018 at 15:33