Is there a digital equivalent to the Great Seal of the United States? Since antiquity (and, for ceremonial purposes, up to the present day), physical documents have had seals affixed. In principle, these seals enable anyone, if he knows what he's looking for, to judge the authenticity of a document without consulting the authority that issued it. (In practice, there are ways of producing convincing forgeries, but leave that to one side for a moment.)
Is it possible, by affixing a string of characters, or suchlike, to a PDF, to achieve the same thing digitally?
I can think of a way of doing something similar:

Come up with a private key and a hashing function.
Hash the private key together with the timestamp for when the document was issued.
Affix the above hash, together with the timestamp, to the document.
When someone wants to authenticate a document, check that hashing the private key together with the timestamp produces the same hash as on the document.

However, the major difference between the above method and a physical Great Seal (e.g., The Great Seal of the United States) is the need to consult someone who has the private key. This check would have to come from a central authority since making the private key public would allow free reign to any forger.
Is there a way, then, of doing something like the above, but without the need for a central authority? If no, is it logically impossible, or is it an open problem?
Is there a way, then, of doing something like the above, but without the need for a central authority? 


Why, yes, we do have a way to solve that problem - we refer to it as a 'digital signature'

Here's how it works:


Select a signature algorithm, and come up with a public/private key pair for a signature algorithm
With the private key, generate a signature of the document (possibly with the time stamp and anything else you find relevant).  Affix the above hash, together with the timestamp, to the document.
When someone wants to authenticate a document, simply get the public key, and check that the signature validation algorithm, together with the document and the timestamp, states that the signature verifies.


The fundamental property of a signature algorithm is that, while it is easy to sign any document you want with the private key, you cannot sign any document (where "sign" means generate a signature that verifies with the signature validation algorithm and the public key).

Common signature algorithms in use today are RSA, ECDSA and EdDSA

Of course, one step I didn't go into is "how, in step 3, does someone get the public key" and "how do they know that the public key they get is actually the signer's, and not some attackers"?  These are valid questions, with good answers; but I think I'll leave the exploration of that for another question...