  The discrete logarithm problem (DLP) is the basis for several cryptographic
primitives. Since Shor's work, it has been known that the DLP can be solved by
combining a polynomial-size quantum circuit and a polynomial-time classical
post-processing algorithm. Evaluating and predicting the instance size that
quantum devices can solve is an emerging research topic. In this paper, we
propose a quantitative measure based on the success probability of the
post-processing algorithm to determine whether an experiment on a quantum
device (or a classical simulator) succeeded. We also propose a procedure to
modify bit strings observed from a Shor circuit to increase the success
probability of a lattice-based post-processing algorithm. We report preliminary
experiments conducted on IBM-Quantum quantum computers and near-future
predictions based on noisy-device simulations. We conducted our experiments
with the ibm_kawasaki device and discovered that the simplest circuit (7
qubits) from a 2-bit DLP instance achieves a sufficiently high success
probability to proclaim the experiment successful. Experiments on another
circuit from a slightly harder 2-bit DLP instance, on the other hand, did not
succeed, and we determined that reducing the noise level by half is required to
achieve a successful experiment. Finally, we give a near-term prediction based
on required noise levels to solve some selected small DLP and integer factoring
instances.
