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Help me award some contest winners! -- securely

Okay, I have 72 submissions to my photo backup survey. And I promised prizes, so I need to give out prizes. But you shouldn't trust me since I might be evil. Of course I'd like to preserve the privacy of individuals, yet allow anyone to verify that I didn't give someone a prize. Help!

Some things I've thought of:
- Create 1-72 buckets.
- Use the NIST random beacon to pick random bits at a specified, agreed upon time in the future. https://beacon.nist.gov/home
- Use reservoir sampling to use those random bits to pick 5 items from the 72 total. https://en.wikipedia.org/wiki/Reservoir_sampling

Now the question is can we mask the participants in the survey yet have everyone know with accuracy that the above is on the up-and-up.

- I could just publish a signed document with the number->ldap mappings, but that reveals everyone who participated.
- I could send each participant their participant number. However then how would anyone know that I had awarded the prizes?

Might need to read up on secure election protocols to make this happen.

Suggestions welcome!

https://beacon.nist.gov/home