Modifying Shor's algorithm to compute short discrete logarithms.

Martin Ekerå, Johan Håstad

Martin Ekerå

We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.

Martin Ekerå, Joel Gärtner

Burton S. Kaliski

Martin Ekerå

Abstract We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.

Cédric Pilatte

Abstract In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multidimensional version of Shor’s algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb {Z}/N\mathbb {Z})^{\times }$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.