I stumbled upon this while digging up something... so here comes what I know of this.
The solution to this is legal and not technical. There are technological means in place that you mention in fact: digital signatures, or any kind of signature of some form. They just need to be legally binding to make the process paperless.
For example, In Sweden you have such scenario with Skatteverket. The documentation you receive from Skatteverket is not really (well) signed. You get on paper a declaration that could have been forged or if you chose the digital declaration (and soon will be the only way) you get it via a private company (!!!). How do you verify authenticity? There is nothing on the document that you can actually consider a signature...
You have to go to the source and ask for the document again.
In Spain, for example, they have solved this a bit more nicely I would say... first, no private companies storing public documents :), then, every document that a public institution issues to you or receives from you gets a unique code for integrity and authenticity purposes. With that code, and by means of a service in the source, you input it and get in return either a code not valid or the document that corresponds to such code, so the authenticity is implied when the original document is given by the source that issued it with the code and the integrity part you do when comparing with the document you got the code from.
This last approach of Spanish institutions is the one that Swedish universities take when issuing a grades report as they include a verification code for anyone who receives that report and doesn't trust it...
In both cases... you rely on some trusted authority, and the legal system to allow for a paperless process (or even a combined version). It is just a matter of universities (and possibly government) to pass the corresponding laws to allow a full digitalisation of Ladok...
I'm sure this has been considered in Ladok. There is no reason to trust better an archive of paper than a hard-disk with files. The later can be replicated in different places (and cannot be altered, or at least alterations can be detected), the former cannot be replicated that easily (besides can be burnt).
But again, this is a matter of trust and legal means to allow it. Privacy-wise, technology can do better for this than an archive of paper (a human should be willing to register that another one is checking some report from Ladok, while the act of accessing a document is automatically registered if it was in a machine; and again, it is about trusting one vs the other... :)