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I would be interested how the performance of running a homomorphic circuit compares to running the same algorithm without encryption

Depends on the scheme and your security parameters. The best known ones we have are probably 5-6 orders of magnitude slower than plaintext in the best case (obviously the worst case is much worse).

Also, how much slower both in terms of throughput and latency. Low latency Low throughput is not so bad for some applications as would high latency high throughput for others.

I went down this rabbit hole a week or two ago. From my understanding, it is a few orders of magnitude slower.

More than a few. Also, all the benchmark algorithms I've seen in papers have been trivial so this is very much still an active area of research. The idea is tantalizing though, and I'm looking forward to new breakthroughs in this space.

if my memory serves, the difference is an overhead of O(1), with a C that is large enough to prevent it from being practical

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