# A Hybrid Key Generator Model Based on Multiscale Prime Sieve and Quantum-Inspired Approaches
## 3 Theoretical Background Useful for the Quantum-Inspired Subkey  
Quantum computing relies on fundamental principles of quantum mechanics, such as superposition and entanglement, to perform calculations. [PERSON]'s uncertainty principle is one of the most significant principles in quantum cryptography, according to which certain properties of a particle cannot be measured simultaneously with infinite precision. It is possible to generate a quantum encryption key using the superposition state of a quantum system. If the system must be quantised in terms of quantum state, then it must be represented as in (1), i.e., a superposition of basis states (see [20]):  
\[\Psi=\sum_{i}c_{i}|i> \tag{1}\]  
where \(c_{i}\) are complex coefficients. When observed, the wavefunction collapses and yields a random result, which can be employed to form a cryptographic key. Indeed, the collapse of the wavefunction is a fundamental principle of quantum mechanics stating that, upon measurement, a quantum state in superposition (e.g., \(|\,\psi\rangle=\alpha|\,0\rangle+\beta|\,1\rangle\)) collapses into one of its basis states (\(|\,0\rangle\) or \(|\,1\rangle\)) with a probability determined by the squared amplitudes of \(\alpha\) and \(\beta\)[20]. In computational and cryptographic contexts, this probabilistic behaviour can be emulated and used to generate non-deterministic bits or cryptographic key structures, as we will show in the next section in our EQRNG (Emulated Quantum Random Number Generator). It is important to clarify that in our approach, the reference to wavefunction collapse is strictly metaphorical and inspired by computation. We do not use any real quantum devices or entangled systems. Instead, we simulate the probabilistic behaviour of quantum systems using classical pseudo-random generators tuned to mimic the expected distributions. Therefore, the term quantum entropy source is redefined here as a simulation of pseudo-quantum entropy, intended to emulate, rather than replicate, quantum uncertainty within classical frameworks. We can also employ quantum key distribution (QKD) in its computational form (see [21]), as a quantum mechanics-inspired computational version of the famous BB84 protocol [22], in which photons are polarised and observed in different bases to arrive securely at a shared secret key. In [21], photons are replaced by \"computational photons\", or \"information pockets\", which mimic the behaviour of photons in BB84. In this implementation, we generate a stream of bits from the probability distribution of a quantum particle in a potential given by the one-dimensional Schrodinger Equation (2)  
\[\hat{H}\Psi=E\Psi \tag{2}\]  
where \(\hat{H}\) is the system's Hamiltonian given by the (3)  
\[\hat{H}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x) \tag{3}\]  
Let us consider a finite potential well, solve the equation numerically, and use the probability distribution to generate the encryption key. To generate a random encryption key, the method is as follows:  
* Solve the Schrodinger equation numerically for a particle in a finite potential well. * Sample the values of the wavefunction at discrete points. * Generate random bits based on the probability of measurement at each point. * Convert the bit sequence into a cryptographic key. Therefore, the Schrodinger equation of quantum mechanics can be used to create secure cryptographic keys based on its probabilistic nature and the quantum behaviour of physicalsystems. Below is a step-by-step repetition of how it can be applied to cryptographic key generation:  
1. Use quantum energy levels. The Schrodinger equation specifies the quantum states of a particle in a potential. Each solution of the equation produces energy eigenvalues that are typically distributed in a non-deterministic manner. How to use it: You can numerically simulate the Schrodinger equation for a complex physical system (e.g., an atom in an arbitrary potential) and use the calculated eigenvalues to obtain random bits for a cryptographic key. 2. Quantum random number generation (QRNG).