Abstract

This research is dedicated to formulating mathematical model for Falcon-M, an innovative post-quantum digital signature algorithm that operates without relying on trapdoor mechanisms. The primary intent is to mitigate the architectural complexity commonly associated with lattice-based signature schemes, while simultaneously ensuring their inherent structural integrity and computational efficacy. The findings delineate the convergence characteristics of discrete Gaussian sampling within the quotient polynomial ring  establish an upper bound for the likelihood of hash collisions within the signature scheme, and deduce explicit analytical boundaries for the propagation of floating-point errors during both forward and inverse Number-Theoretic Transform computations. It is observed that Falcon-M streamlines the key generation process when juxtaposed with trapdoor-based designs, all while maintaining an optimal quasilinear computational complexity of  O(n log n). These results advance the theoretical underpinnings of lattice-based post-quantum signatures and support the creation of cryptographic systems that are both numerically stable and structurally less intricate.