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PhilGoetz comments on Existing Absurd Technologies - Less Wrong Discussion
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Public-key encryption is very weird; most cryptographers did not believe it was possible before it was done. But anyone who understands public-key encryption enough to appreciate this point is probably not in your target non-rationalist audience.
I've found that most people who are initially dubious of the idea of public key encryption will at least agree that it is plausible once you demonstrate the asymmetry of inverse operations like multiplying primes / factorising products.
Once you convince them that it's easy to multiply primes but hard to factorise products, you can say "there is an algorithm where you can encrypt a message using the product, but to decrypt it you need the original primes" and (in my experience) they will usually find this sufficiently plausible to overcome the initial skepticism.
If they still doubt it of course you can go into more detail but I've found this was enough to convince lay-people at least of the possibility of public-key encryption.