What Shor's Algorithm Forces—and What It Does Not

David Zierler, Ryan Dahn

David Lloyd Selke

Loops that enforce a correct output and that restart with a changed parameter may emulate a brute force search, even against the design intent. A Python program is presented analogous to Shor's Algorithm but with random number generation replacing the math. It factors integers. Shor's Algorithm devices may operate similarly to the Python program, not in being random, but in being classical.

Arnav Jain

The advent of quantum computing poses a fundamental threat to classical cryptographic systems, particularly RSA encryption, which relies on the computational limit of prime factorization. This paper presents a comparative analysis of four integer factorization approaches-a selfcurated brute-force classical implementation, optimized Trial Division, Pollard's Rho algorithm, and Shor's algorithmevaluated across a dataset of increasing semiprime inputs representative of scaled-down RSA encryption keys. Through Python-based implementations and simulations, we empirically measure and plot runtime complexity for classical methods while modeling theoretical runtimes for Shor's algorithm, given hardware limitations. Our findings reveal the exponential scaling limitations of classical algorithms, and contrast them with the polynomial-time behavior of Shor's algorithm. Despite being simulated, Shor's consistently flatter runtime trendline demonstrated its theoretical superiority and the critical urgency of post-quantum cryptography. This study not only visualizes the performance divide between classical and quantum paradigms but also underscores the transformational potential of quantum computing in modern cryptanalysis.

Hiu Yung Wong

Wei Liu, Shuming Chen, Jian Zhang +3 more

It is widely believed that Shor’s factoring algorithm provides a driving force to boost the quantum computing research. However, a serious obstacle to its binary implementation is the large number of quantum gates. Non-binary quantum computing is an efficient way to reduce the required number of elemental gates. Here, we propose optimization schemes for Shor’s algorithm implementation and take a ternary version for factorizing 21 as an example. The optimized factorization is achieved by a two-qutrit quantum circuit, which consists of only two single qutrit gates and one ternary controlled-NOT gate. This two-qutrit quantum circuit is then encoded into the nine lower vibrational states of an ion trapped in a weakly anharmonic potential. Optimal control theory (OCT) is employed to derive the manipulation electric field for transferring the encoded states. The ternary Shor’s algorithm can be implemented in one single step. Numerical simulation results show that the accuracy of the state transformations is about 0.9919.

Belal Ehsan Baaquie, L. C. Kwek