Existence of finite commutative modular partition rings for cryptology

Determine whether, for a given natural number N, there exists a finite commutative ring derived from the ring of partitions (a modular ring of partitions modulo N) that can be used for cryptology; if such a ring exists, construct it explicitly.

Background

The authors propose exploring cryptographic applications based on the ring of partitions and raise the question of whether a finite commutative ‘modular’ version of this ring can be constructed for a given modulus N. They point out that in cryptographic contexts it is useful to have efficient exponentiation and greatest common divisor operations within such a structure.

They note potential relevance to protocols like Diffie–Hellman and RSA if an appropriate finite commutative ring of partitions can be defined, but emphasize that this direction remains unexplored.