In public key cryptography, Alice knows functions $f$ and its inverse $f^{-1}$. $f$ is public and $f^{-1}$ is secret. To sign a message $m$, she gives $(m,a=f^{-1}(m))$. To verify a signature, the verifier checks if $f(a)=f(f^{-1}(m)) ?= m$.
$f$ and $f^{-1}$ are related by some secret $S$ and it is computationally infeasible an adversary to compute $f^{-1}$ or the secret $S$.
Examples of secrets $S$ are integer factorization and discrete logarithms.
One possible approach is to take the secret $S$ to be obfuscation permutation and $f$,$f^{-1}$ to be some "objects" for which isomorphism (and the permutation $S$) is hard to compute, while still have "some properties" for signature.
The question from 2016 Finite objects for which isomorphism is NP-hard or harder? appears to give hard isomorphism.
Q Can we make cryptography signature algorithm based on hardness of isomorphism with security at least NP-hard?
There is related work based Isomorphism of Polynomials, but we believe its security is only Graph Isomorphism (could be wrong on this).
NP-hardness of Hamiltonian Cycle is used in Zero-knowledge proof
Added 2020-09-15
From answer the linked question two candidates for NP-hardness are CIRCUIT ISOMORPHISM and FORMULA ISOMORPHISM.
On p.1: The formula isomorphism problem is in $\Sigma_2^p$ , NP-hard, and unlikely to be $\Sigma_2^p$.-complete