Is there a mental poker algorithm that does not rely on commutative encryption? The algorithm for mental poker that I saw on Wikipedia and everywhere else relies on an encryption algorithm such that $E_k(E_j(P)) = E_j(E_k(P))$, but I can't find a modern and secure algorithm for which this is true. The only algorithm I found with this property is SRA (I believe that's what it's called) but it relies on modular exponentiation, which is slow and resource intensive compared to a more common algorithm such as AES, not to mention less secure.

Is there a mental poker algorithm that doesn't rely on this property?
Yes, there are some other algorithms that do not rely on commutative encryption.

The Wikipedia page for Mental poker lists some other examples. It describes a non-shuffling poker protocol that uses homomorphic encryption. It has this caveat, though: 


  However, the method needs all cards that have already been dealt to be known to all, which in most poker-style games would beat its very purpose.


So it's not practical.

There is also a very interesting paper titled How to Use Bitcoin to Play Decentralized Poker. The authors demonstrate how you could use secure multiparty computation with identifiable abort (ID-MPC) to play a decentralized game of poker.

(Use this Internet Archive link if the Mental poker page has been updated.)