We consider a finite field $\mathbb{F}_p$, where $p$ is a large prime e.g. 256-bit.
We have $b$ a fixed element of the field. We encode it as $b'=b||h(b)$, where $h(.)$ is a cryptographic hash function. Assume the output of $h(.)$ is of size 160-bit. We encode that way to distinguish the element $b'$ from a random element $r$ of the field.
So to check if the element has the above structure we do:
(1) Parse the value: $r=r'||r''$, where size of $r''$ is 160-bit.
(2) check: $h(r')\stackrel{?}=r''$
Question1: How can we show/prove that the random element only with a negligible probability can have such structure?
Question2: Can we reduce the $h(.)$ output size to 80-bit (which is not the standard hash function output size) and set $p$ as smaller prime number (e.g. 128-bit )and prove a random value can have the above structure with only negligible probability?
Citing a paper that use/prove such statement would suffice.