Quantum Chaos and the Spectrum of Factoring. Issue 3 (22nd January 2021) - Rosales, Jose Luis; Briongos, Samira; Martín, Vicente
Abstract: The factorization ensemble is a set to which integer factorable numbers N ′ = x ′ y ′, having the same trivial factorization complexity, belong. Hence, the Rivest‐Shamir‐Adleman (RSA) cryptographic moduli pertain to this set. A function E [ x ′, y ′ ] can be defined therein which will be associated to the energy of a system of ions in a Penning trap. This is the quantum factoring simulator hypothesis connecting quantum mechanics with number theory. Here, a possible setup of the simulator from the magnetron energies of a Coulomb crystal in a cylindrical trap is described. Then, quantum mechanically, these energies may have only discrete values. To test the validity of the simulator hypothesis, evidence of this kind of discreteness from the statistics of the E [ x ′, y ′ ] s of a large random sample of RSA moduli is reported; indeed, their unfolded distance probability distribution fits to a Gaussian Unitary Ensemble, exactly as required if they actually correspond to the quantum energy levels spacing of a magnetically confined system that exhibits chaos. The confirmation of these predictions is consistent with the quantum simulator hypothesis and, thereby, it points to the existence of a liaison between quantum mechanics and number theory. Abstract : There exists a Hamiltonian formulation of the factorization problem of a number product of two primes. It corresponds to a classical system that exhibits chaos. The canonical quantization of this system connects quantum mechanics with number theory. The setup of this factoring simulator, from the collective radial motion energies of a Coulomb crystal in a cylindrical Penning trap, is prescribed.