Prime-Composite Periodic Function : From the Prime Periodicity Model to Unified Quantum Verification of RSA/ECC

Starting from the generative philosophy of the ancient Chinese classic Tao Te Ching, this paper discovers the periodicity and phase of prime numbers in the modulo-30 orbits and establishes the “prime periodicity model”. Inspired by the harmonious thought of the Doctrine of the Mean, the model is named the “Prime- Composite Periodic Function” . Two key quantum simulation verifications are com- pleted: Quantum simulation of integer factorization (RSA) based on the modulo-30 sieve: successfully factorized the composite number 1891 on a 20-qubit circuit in 18 seconds, consistent with classical results. Solution of discrete logarithm (ECC) based on quantum phase estimation: on the PyQPanda simulator, the phase corresponding to the private key x is extracted from h = gx mod p, recovering x = 9, with multiple tests confirming universality. The two experiments together demonstrate the unified explanatory power of the Prime-Composite Periodic Function for the underlying mathematical structures of RSA and ECC. This paper also discusses potential applications in post-quantum cryptography, low-probability-of-intercept radar waveforms, and room-temperature quantum error correction.