Phi-Ns Cryptosystem

This document describes Phi-Ns, a new asymmetric cryptographic primitive based on the structured decomposition of the quadratic gap between two primes. Phi-Ns defines a public key q and a private key p such that the difference q^2 - p^2 is expressed as a composite value T. The value T is then factored into small controlled components noted a, b, and R, and the final secret structure abR is encoded through a randomized serialization process. The security of Phi-Ns does not rely on integer factorization, the discrete logarithm problem, or elliptic curves. Instead, it depends on the difficulty of recovering p from q when the attacker has no oracle, no structural anchor, and no method to determine whether a candidate p is correct. Because T can be recursively decomposed and reassembled into multiple unpredictable forms, the resulting search space grows combinatorially. Phi-Ns further supports recursive expansion of the internal abR structure, allowing the construction of high-entropy secret states even when p and q have relatively small bit lengths. This enables compact public keys, efficient computation, and strong post-quantum resistance based on a non-factorization hardness assumption. This document specifies the algorithmic model of Phi-Ns, its parameter choices, and the serialization rules used to encode the abR structure. It also outlines two public-key variants, PK-p and PK-q, which expose either p or q as the public value depending on deployment constraints.