Complex Geometric Cryptanalysis of RSA

This research introduces a novel cryptanalytic framework for RSA by embedding its operations into the complex plane. RSA encryption is expressed as a mapping of the form: \Psi_a(b) = (a + bi)^e \bmod N with fixed real part a, where residues are studied in the Gaussian integer domain. The resulting distributions show structured lattice patterns, clustering behavior, and Fourier spectral features that correlate with the prime factors of N = p * q. We present both theoretical derivations and simulation results. In particular, Gaussian prime parameters p = 3+2i and q = 5+i (yielding N = 13+13i) are tested, alongside classical small moduli N = 21, 55, and 77. The experiments include: Residue distributions in the complex plane, showing clustering and periodicity Fourier spectrum analysis, where peaks correspond to factor-related frequencies Noise sensitivity tests, demonstrating fragility of complex RSA under perturbation This framework provides geometric intuition for RSA vulnerabilities, connects classical leakage analysis to quantum Fourier methods (such as Shor’s algorithm), and opens pathways for hybrid classical–quantum cryptanalysis. While not a practical attack on large RSA keys, the results highlight theoretical leakage and encourage further exploration of physical-layer or complex-domain security models.