Cryptanalysis of Aperiodic Cut-and-Project Window Maps as Candidate One-Way Functions

Post-quantum cryptography relies on precisely stated computational hardness assumptions rather than on physical analogy or structural complexity alone. Motivated by lattice-based cryptography and by aperiodic cut-and-project geometry, this manuscript defines a deliberately conservative toy construction called a cut-and-project one-way map (CP-OWM). The construction combines a public modular linear map with a nonlinear acceptance-window predicate derived from a hidden perpendicular projection. The purpose is not to introduce a deployable encryption scheme, key-encapsulation mechanism, or signature algorithm. The purpose is narrower: to test whether a nonlinear aperiodic window predicate can produce measurable resistance to standard lattice-reduction recovery attacks beyond matched SIS/LWE-style baselines. We formulate the toy primitive, state the adversarial knowledge model, identify correctness and leakage risks, and define failure criteria. Preliminary benchmark logs indicate that uncalibrated window scaling can make the acceptance region too loose, causing the CP-OWM instance to remain as vulnerable as a baseline under BKZ-20 attack. We therefore introduce acceptance-rate calibration: the window bound is tuned dimension by dimension so that the valid-sample probability remains close to a fixed target, here approximately one percent. This does not prove security. It only creates a more controlled benchmark in which attacker recovery probability, legitimate valid-key generation cost, entropy, collision rate, and leakage can be compared at fixed acceptance probability. The manuscript should be read as a feasibility and falsification protocol. A positive benchmark would motivate further cryptanalysis; a negative benchmark would show that the proposed aperiodic predicate collapses to known lattice-reduction behavior or leaks too much information. No claim of post-quantum security, standardization readiness, or blockchain protection is made. The calibrated acceptance-window constraint did not reduce attacker recovery probability relative to the SIS-style baseline; under the tested LLL/BKZ configurations, CP-OWM remained fully vulnerable up to N=60. The proposed toy construction therefore does not currently provide evidence for a post-quantum-secure primitive.