Shor's algorithm

Quantum computing will be a threat to the cryptography world in the next years, as certified by the NIST call for proposals to select quantum-resistant algorithms. An analysis of the circuit needed to implement Shor’s algorithm, which can be used to break public-key cryptosystems like RSA and elliptic curves, is provided in this work, explaining all the necessary elements to understand it. Additionally, we implemented different realizations of the oracle using Qiskit, writing programs to break RSA and elliptic curve cryptography, and tested them in IBM quantum devices and simulators. The final objective is to estimate the distance between the needed capabilities to actually be able to break the security mechanisms used nowadays and the current state of the art of the quantum hardware world, considering the time that would be needed to update these mechanisms with quantum-resistant ones.