I have at least three, distinct motivations for having a simple way to use asymmetric key encryption by hand or without modern computers.
First, I was recently demonstrating to a 12-year-old how encryption works. I created some basic cyphers and had him figure them out. (I recommend this; it's quite fun.) Wanting to up the ante, I looked for ways to send messages back and forth using public-private key encryption, and I failed with flying colors.
Second reason for wanting this is that it would really help me understand how asymmetric encryption works. At the moment, I can't figure out a way to do solve this on my own, and I feel that betrays my lack of understanding in the subject.
Finally, (this one may be a stretch) I'm writing a fantasy story, and I'd like to do something unique with the form of exchange. One thought was to have a memorized key that the owner held, which she would use to sign her (paper) payment of 30 units for a loaf of bread, or whatever, combining her key with the shopkeeper's public key. (UPDATE: Note that this gets into digital signatures, rather than asymmetric key encryption. I'm hoping they are related, but if they're not, forget it.)
Any ideas how to accomplish this? If it's simply not feasible, I would appreciate an explanation as to why.
There are two similar questions on this site, but neither of them address asymmetric key encryption, unless I'm more misinformed than I thought: