What tools are typically used to develop these algorithms? The site has https://password-hashing.net/faq.html#qd which mentions attempts to formally define the security of a good algorithm though a quick scan of the two papers indicates the definitions are mathematical properties described in English. However, when it comes to implementations is there a generally accepted language/framework in which correctness can b…
There are really two reasons for this: (1) optimization is a really big deal; some impls go so far as to have asm for specific architectures or use architecture-specific features like SSE2 and (2) there really isn't very much code for a cryptographic primitive, typically https://github.com/khovratovich/Argon2/tree/master/Argon2i/r...).
Once a simple reference implementation is hand-checked, it can be used to generate comprehensive test vectors for other implementations.
This isn't my area of expertise, but I believe there's some research in applied crypto for implementation verification. I know https://github.com/GaloisInc/cryptol is a language that's supposed to help in that regard, but I don't know offhand who all uses that.