This question was originally posted here on Crypto StackExchange. As suggested by an answer I am posting it here to help get a better perspective on the math side.
Public-key cryptography was not invented until the 1970's. Apart from the idea not existing earlier (as talked about here), is there any reason it could not have been used earlier? For example, are there forms that are easy enough to perform by hand but complicated enough to not be solved (easily) by hand?
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My hypothesis is a bit different. From Euclid's ''Elements'' enough number theory (gcd, lcm) has been available to introduce (with some research) the theory of the RSA algorithm. The necessity to use cryptography can be seen from the Caesar cipher. Some handcrafting could have led to some larger primes and encrypting/decrypting via modular arithmetic would have not been a big deal due to successive multiplication and reduction.
I think that a system like RSA would have been impractical in pre-computer days. How to generate large enough primes? There were tables for the smaller primes, but your opponent has the same tables, so then trial division would be a threat...
Also, modular exponentiation is no party to do by hand either. And encoding a message to a number is awkard too. I posit that it's impractical to do by hand for parameters that would have been considered safe.