Cryptography based on #P-complete problems

Are there any examples of a cryptographic scheme based on (an average-case form of) a #P-complete problem?

We don't know how to base cryptography even on $\mathbf{NP}$-completeness let alone $\#\mathbf{P}$-completeness. Moreover, there are known barriers to basing cryptography on $\mathbf{NP}$-completeness: [AGGM,BT], and also [Chapter 7, B].

That said, it one is willing to make additional assumptions then $\#\mathbf{P}$-hardness can be useful: e.g., in [CHK+], it is shown that in the random-oracle model, hardness of $\#SAT$ (which is complete for $\#\mathbf{P}$) can yield hard distributions of Nash, the problem of computing Nash equilibrium (of, say, two-player games).

[AGGM]: Akavia, Goldreich, Goldwasser and Moshkovitz, On Basing One-Way Functions on NP-Hardness, STOC 2006

[B]: Brzuszka, On the Foundations of Key Exchange, PhD Thesis, 2013

[BT]: Bogdanov and Trevisan, On Worst-Case to Average-Case Reductions for NP Problems, FOCS 2003

[CHK+]: Choudhuri et al, Finding a Nash Equilibrium Is No Easier Than Breaking Fiat-Shamir, STOC 2019

Update: The following answer does not cover the original question. It is the result of confusing p-complete with #p-complete.

The first ever public key cryptographic algorithm was based on a p-complete problem. It is today known as Merkle Puzzle. Roughly speaking, the complexity of key exchange for the receiving and sending sides is $\mathcal{O}(n)$ while a successful attack complexity is $\mathcal{O}(n^2)$.

Although in modern cryptography, it is not considered secure anymore, Merkle's idea changed everything in cryptography.