Revisiting Shor's quantum algorithm for computing general discrete logarithms

• URL: http://arxiv.org/abs/1905.09084v4
• Date: Fri, 13 Sep 2024 17:23:47 GMT
• Title: Revisiting Shor's quantum algorithm for computing general discrete logarithms
• Authors: Martin Ekerå,
• Abstract summary: We show that Shor's algorithm for computing general discrete logarithms achieves an expected success probability of approximately 60% to 82% in a single run. By slightly increasing the number of group operations that are evaluated quantumly, we show how the algorithm can be further modified to achieve a success probability thatally exceeds 99% in a single run.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We heuristically show that Shor's algorithm for computing general discrete logarithms achieves an expected success probability of approximately 60% to 82% in a single run when modified to enable efficient implementation with the semi-classical Fourier transform. By slightly increasing the number of group operations that are evaluated quantumly and performing a single limited search in the classical post-processing, or by performing two limited searches in the post-processing, we show how the algorithm can be further modified to achieve a success probability that heuristically exceeds 99% in a single run. We provide concrete heuristic estimates of the success probability of the modified algorithm, as a function of the group order $r$, the size of the search space in the classical post-processing, and the additional number of group operations evaluated quantumly. In the limit as $r \rightarrow \infty$, we heuristically show that the success probability tends to one. In analogy with our earlier works, we show how the modified quantum algorithm may be heuristically simulated classically when the logarithm $d$ and $r$ are both known. Furthermore, we heuristically show how slightly better tradeoffs may be achieved, compared to our earlier works, if $r$ is known when computing $d$. We generalize our heuristic to cover some of our earlier works, and compare it to the non-heuristic analyses in those works.

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