It is not entirely clear what you want, but suppose you need a trapdoor permutation - the function that is easy to invert only if you know a secret parameter - and which is not based on number-theoretic assumptions.
There are two well known families of such schemes: Multivariate Cryptography (MQ) and Code-based cryptosystems (for instance, McEliece cryptosystem).
An MQ scheme typically consists of a low-degree invertible polynomial transformation $S$ (public or secret) and two secret affine transformations $A_1,A_2$. The public key is exposed a polynomial(s) $A_2(S(A_1(x)))$. There are many insecure candidates for the nonlinear layer (see survey in this paper), but it is generally assumed that the HFE cryptosystem with sufficiently large parameters is secure.
The code-based cryptosystems hide the generator of some linear code in a similar manner by multiplying it by properly chosen matrices and expose the public key as a matrix.
Both families produce terribly large public keys up to several megabytes, and are of limited use because of that and the lack of systems that withstood a lot of cryptanalysis (except for the original McEliece proposal).
UPDATE:
You may also want to look at some bijective polynomials suggested in the recent algebraic papers. I do not know if there exist efficient inversion algorithms.
For example:
$\left(X^{2^k}+X+a\right)^{-l}+X$ over $\mathbb{F}_{2^n}$, where $n/\text{GCD}(n,k)$ is odd, $l(2^k+1)\equiv 2^{n/2-1}\pmod{2^n-1}$, Tr$(\delta)=1$ in this 2010 paper
For odd $p$: $\left(X^{p^k}-X+a\right)^{\frac{p^n+1}{2}}+X^{p^k}+X$ over $\mathbb{F}_{p^n}$ in this 2012 paper.
