A signature algorithm combined with a secret key allows someone to generate a digital signature for the contents of a message. A signature algorithm combined with a public key allows someone to verify the digital signature for a message. Hash algorithms are often called message digest algorithms. Hash algorithms are also one way functions, and a good hash algorithm is one in which very similar inputs produce dramatically different outputs. Hence, if even a single bit is altered or corrupted in transit, the hash value will be different.
All discussion now hinges on how keys are used for asymmetric algorithms.
To start, we have to introduce the notion of a public key certificate. Unfortunately, we now enter the realm of sociology rather than technology. The only theoretical limitation on an identity is that you have to be able to represent it digitally. Fortunately, the guarantees are a bit simpler to describe they are digital signatures from other public keys that vouch for the veracity of the binding. It turns out that our two contending technologies have different answers to that question.