What operations provide perfect secrecy other than modulo addition As far as I know OTP is the only algorithm proven to provide perfect secrecy. It can work with XOR which is addition modulo 2 and obviously it can work with additions modulo N. What other operations enable perfect secrecy? Can we for example have randomized tables to define the mixing of data and key and still prove the perfect secrecy property of OTP?
Any group operation works for a one-time-pad if you can shoehorn a message uniquely into a group element—addition modulo $n$, multiplication modulo $n$ if you can uniquely encode a message block as an integer coprime with $n$, addition or multiplication in finite extension fields, invertible matrix multiplication over finite fields, whatever group you want—and can be extended by a direct product to arbitrary-length messages: if you have a group operation $m \oplus k$, then you can extend it to an $\ell$-block message $(m_1, m_2, \dotsc, m_\ell)$ with an $\ell$-block key $(k_1, k_2, \dotsc, k_\ell)$ by $(m_1 \oplus k_1, \dotsc, m_\ell \oplus k_\ell)$.  Conventionally we talk about addition modulo 26 in English, or use addition modulo 2 for computers, of course.

Going beyond the one-time pad approach, to encrypt a $t$-bit message, you could choose a permutation $\pi$ uniformly at random from all $2^t!$ possibilities; then the one-time cipher $m \mapsto \pi(m)$ has ‘perfect secrecy’ too.  Of course, it's tremendously inefficient even compared to a one-time pad!  (Bonus: You could get built-in authentication with forgery probability bounded by $2^{-f}$ by using $m \mapsto \pi(m \mathbin\| 0^f)$.  Of course, still much cheaper to compose a one-time pad and one-time authenticator…)