Recall that the Toffoli gate is the mapping $T(x,y,z)=(x,y,(x\wedge y)\oplus z)$. We shall say that a function $T$ is Toffoli-like if $T(x,y,z)=(x,y,((a\oplus x)\wedge(b\oplus y))\oplus z\oplus c)$ for some $a,b,c\in\{0,1\}$.
Suppose that our key is $m$ bits and the message we want to encrypt $\mathbf{x}$ is $n$ bits. Let $C$ be a circuit on the $m+n$ bits consisting of Toffoli-like gates which are selected at random where either the target is always in the $n$ message input bits or both the control and the target are in the $m$ key bits. Then for each key $\mathbf{b}$ there is some $\mathbf{c}$ along with functions $D_{\mathbf{b}},E_{\mathbf{b}}$ (the encryption and decryption functions) such that $(\mathbf{c},E_{\mathbf{b}}(\mathbf{x}))=C(\mathbf{b},\mathbf{x})$ and $(\mathbf{b},D_{\mathbf{b}}(\mathbf{x}))=C^{-1}(\mathbf{c},\mathbf{x})$.
Is there any symmetric encryption system similar to the above cryptosystem which has been deeply studied or is already in use?
If so, then how does such a cryptosystem compare against the AES or another symmetric cryptosystem?
If there is no such cryptosystem available, then what weaknesses do such cryptosystems have and why are they not in use?
Will such a randomly generated symmetric encryption system be practical once super efficient reversible computers or partially reversible computers hit the market?
In this paper, it takes 109664 reversible gates to implement the 128 bit key AES. By comparison 100,000 Toffoli gates seems to be more than sufficient when one randomly applies Toffoli and Fredkin gate, so I am wondering why this randomly generated symmetric cryptosystem is not in use.