A Quantum "Magic Box" for the Discrete Logarithm Problem

In their foundational paper on pseudorandom bit generation, Blum and Micali showed that the discrete logarithm problem could be solved efficiently given a “magic box” oracle that computes the most significant bit of the discrete logarithm with a slight advantage over guessing. This magic box can be realized on a quantum computer with a new, simplified variant of Shor’s algorithm. The resulting combination of Blum and Micali’s reduction and this new quantum magic box offers an intriguing hybrid approach to solving the discrete logarithm problem with a quantum computer. Because the only requirement on the quantum portion of the algorithm is that it provide an approximate estimate of a single bit of the discrete logarithm, the new algorithm may be easier to implement, more resilient to errors, and more amenable to optimization than previous approaches. Further analysis is needed to quantify the extent of these benefits in practice. The result applies to the discrete logarithm problem over both finite fields and elliptic curves.