negligible

A negligible function is a function $\mu: \mathbb{N}\to\mathbb{R}$ such that, for any positive integer $c$ there exists an integer $N_c$ such that for all $n > N_c$,

i.e., that $\mu(n)$ eventually goes to zero faster than any polynomial, even polynomials $x^c$ of arbitrarily high degree.

In cryptography, $n$ is usually a security parameter (key size, or number of rounds of a zero knowledge proof protocol) and $\mu(n)$ represents the probability that the cryptographic scheme fails, which we would like to be negligible. The definition shows that by increasing the key size $n$, we can drive the failure probability to zero at a faster-than-polynomial rate. This means that our scheme remains secure against an attacker capable of repeating their attack a polynomial number of times.