# Multiscale Sieve for Smart Prime Generation and Application in Info-Security, IoT and Blockchain
## 6 Applications: GMP for Big Numbers, IoT and Blockchain  
A version of the algorithm has been implemented via the GMP library, demonstrating efficiency very similar to that analyzed above, which allows us to work with high precision on numbers that cannot be represented in the range \([0,2^{64}-1]\). This implementation exploits the advanced optimizations and capabilities of GMP to rigorously handle numbers with thousands or even millions of digits, which is essential for applications requiring extremely high precision, such as cryptography, i.e., RSA, XTR and elliptic curve cryptosystems (ECC); numerical analysis; and scientific computing. For instance, RSA is a widely used public key encryption system that underpins much of the secure communication on the internet today. The security of RSA depends on the mathematical challenge of factoring large composite numbers, specifically those that are the product of two large prime numbers, randomly and independently of each other, denoted as \(N=p\cdot q\). The difficulty in breaking RSA lies in the fact that determining \(p\) and \(q\) from N is computationally demanding and time-consuming with the current algorithms and hardware. The multiscale sieve offers a more efficient way to approach the factorization problem; it could significantly reduce the time needed to decompose \(N\) into its prime factors. Unlike traditional algorithms, such as the quadratic sieve, Pollard Rho [23] or General Sieve of Number Fields [24], which operate on fixed scales, the multiscale sieve might employ a hierarchical or layered approach to examine the \(N\) number and might uncover structural weaknesses or patterns that make factoring easier. The implications of this are profound. Firstly, if the sieve demonstrates superior performance in factorization, it could challenge the current assumptions about the security of RSA. This could lead to the rethinking of what constitutes an adequate key length in RSA encryption, as, to date, keys are typically 2048 bits or longer to ensure security. However, if factorization becomes easier, much longer keys might be necessary, increasing the computational load for encryption and decryption processes. Secondly, it could reveal vulnerabilities in the ways in which prime numbers \(p\) and \(q\) are generated. If certain properties of these prime numbers make them more susceptible to factoring when analyzed with a multiscale approach, it could reveal flaws in the random number generation processes used in cryptographic applications. In a broader sense, the application of the multiscale sieve could have far-reaching consequences  
Figure 6: Growth in memory usage in relation to input size. beyond RSA alone; many cryptographic protocols are based on the assumption that certain mathematical problems, such as factoring, are computationally unattainable at large scales. Furthermore, in this way, the algorithm can be run for such applications on massive arrays of graphic processing units (GPUs) or even simple supercomputers (SSCs). It would be interesting to integrate our algorithm into a system such as MuReQua [25], thus involving more users in the search for new prime numbers, with a coin reward system. Integration takes place through the use of an oracle, which functions as a tool for the handling of information and the storage of the prime numbers found. It is also responsible for checking the correctness of the proposed prime numbers and sorting them by creating a structured and easily accessible record. This sorting function is particularly important to ensure the optimization of future searches within the network, facilitating the retrieval and analysis of data by other users or applications integrated in the system. Due to the quantum and multiscale nature of the MuReQua blockchain, the validation and storage process takes place with high efficiency, ensuring data security and integrity. The implementation of this feature not only encourages user participation through gamification, but also contributes to the strengthening of the network itself.