  We provide a high-level cost comparison between Regev's quantum algorithm
with Eker{\aa}-G\"artner's extensions on the one hand, and existing
state-of-the-art quantum algorithms for factoring and computing discrete
logarithms on the other. This when targeting cryptographically relevant problem
instances, and when accounting for the space-saving optimizations of Ragavan
and Vaikuntanathan that apply to Regev's algorithm, and optimizations such as
windowing that apply to the existing algorithms.
  Our conclusion is that Regev's algorithm without the space-saving
optimizations may achieve a per-run advantage, but not an overall advantage, if
non-computational quantum memory is cheap. Regev's algorithm with the
space-saving optimizations does not achieve an advantage, since it uses more
computational memory, whilst also performing more work, per run and overall,
compared to the existing state-of-the-art algorithms. As such, further
optimizations are required for it to achieve an advantage for cryptographically
relevant problem instances.
