Searching for Public-Key Cryptosystems

This article suggests the use of undecidable problems in constructing public-key cryptosystems. Any such system must still be in NP, but intuitive arguments suggest that this approach might be a reasonable alternative to the use of NP-complete problems. A general approach based on the undecidable word problem for groups is discussed, though without enough detail to evaluate a specific implementation. The resulting cryptosystem, if successful, would be a randomized public-key encryption procedure with infinitely many ciphertexts corresponding to each plaintext. Also , it is possible that a specific implementation could have infinitely many secret keys corresponding to one public key.