This question is certainly somewhat opinion-based, but hopefully not hopelessly so.

The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) is obviously breaking the public-key cryptography systems that currently encrypt Internet traffic.

But if we could eventually implement Shor's algorithm on a large-scale quantum computer, would it have any *other* useful applications, whether for pure math research or out in the "real world"?

This question is inspired by an essay by Boaz Barak, in which he mentions that he doesn't know of any reason why efficient factoring is inherently interesting other than for cryptography.

Two scope clarifications:

- I know that the quantum Fourier transform at the heart of Shor's algorithm has other potential practical applications, e.g. for solving large linear systems via the HHL algorithm, but I'm specifically wondering about period finding and factoring, and not general ultra-fast Fourier transforms.
- I know that the
*existence*of an efficient quantum factoring algorithm has huge implications for computational complexity theory: it provides perhaps the strongest evidence we have (a) that**BPP**!=**BQP**, (b) against the extended Church-Turing thesis, and (c) depending on your philosophical beliefs, perhaps against the real-world feasibility of building a fault-tolerant quantum computer. But I'm wondering about actually*executing*such an algorithm, not about its existence or properties.