Elliptic Curve Cryptography (ECC) and RSA are fundamental protocols in public-key cryptosystems, both relying on challenging mathematical problems: the generalized discrete logarithm problem for ECC and integer factorization for RSA. These problems are currently considered intractable for classical algorithms within polynomial time. However, with the emergence of quantum computing and algorithms like Shor’s algorithm, the security of these protocols is under threat. In this paper, we implemented Shor’s algorithm to factorize the number 91 on the IBM Qiskit platform using three different approaches: first, on real quantum hardware (127 qubits); second, on a quantum simulator; and third, on a simulated quantum device (fake quantum device). The results of the research highlight the challenges posed by noise and decoherence in real quantum computers, the accuracy of noise-free simulations, and the effectiveness of noise modeling using a simulated quantum device. The findings emphasize both the potential and the limitations of quantum computers in analyzing the security of cryptographic protocols.