Operation sequence authentication shared storage
Yes. $\:$ Since "Eva sees everything written on Bob's PC", the process needs to involve something

that's not written on Bob's PC. $\:$ They could use either some sort-of MAC that can be computed

with pencil-and paper, or a signature scheme whose private key is held on a hardware token.

I don't have any particular such MACs in mind, although my immediate idea would

be to lay the message out on a grid, interpret characters as modular integers,

and let the tag consist of sums of characters from certain small subsets of the grid.

Also, it is very rare for using "Number encrypted by private key"

and decrypting values with public keys to be secure.

Almost always, one should be using Number signed by private key

and verifying with the public key instead.