Random thuds on cracking public key encryption

Lemme see now...

Let T be a plaintext which Alice must convey to Bob.

Let ¤ represent exclusive-or, or some similar operation which is reversible when twice applied (or at least symmetric, like TEA, where encode and decode work jes' fine.)

Let P be Alice's public key, and p be her private key.

Let P ← M¤p, where M is mysterious^†.

Let Q be Bob's public key, and q his private key.

Let Q ← M¤q, where M is still mysterious.

Alice cleverly enciphers C ← T¤Q¤p.

Alice pops C into an envelope and sends it to Bob, awaiting anxiously.

Bob deduces Alice's private key, p ← (M¤p)¤(M¤q)¤q ← P¤Q¤q.

Bob deciphers T ← C¤Q¤p.

Aaarrrgh!

Ok. This probably ain't how public key encryption works, but if it reduces to this case, kiddies, the jig is up!

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^†M is not my favorite pseudorandom number generator, Mersenne Twister. M is mysterious! Like this.