Could one build a symmetric encryption system that is competitive with the AES by randomly applying Toffoli-like gates? 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.
It seems plausible that one builds a symmetric encryption system competitive with AES by randomly applying Toffoli(-like) gates as restricted in the question. I do not know that it has been studied.

Paraphrasing the question, it constructs a block cipher with 128-bit block and 128-bit key similar in function to AES-128 as follows:


number inputs as $b_i$ with $0\le i&lt;128$ for block bits and $128\le i&lt;256$ for key bits
repeat $r$ times


pick random distinct indices $i,j,k$ with $(k\ge128)\implies(i\ge128)\wedge(j\ge128)$
apply a random Toffoli-like gate to $(b_i,b_j,b_k)$



In the end, the result is the 128-bit $x_i$ with $0\le i&lt;128$. The rule $(k\ge128)\implies(i\ge128)\wedge(j\ge128)$ insures that for any fixed key, the block cipher is reversible.

Seen from far away with blurry vision, there are similarities between this construction and AES-128: both are iterated permutation-substitution ciphers with reversible key transformation along the rounds, and 256 variables.  If we chose $i$, $j$, $k$ randomly until acceptable, a key bit is chosen as $z$ (the potentially changed bit) with odds about $1/5$, which grossly mimics AES spending significantly less computational effort on key scheduling than on data encryption.

I do not know the number of rounds $r$ making the block cipher secure; nor a technique to derive this. But I would not be astonished if it was markedly lower than $100000$.