I expect that such a study cannot, in fact, be effectively done.
The reason is that academic re-invention of a closely held secret will be virtually impossible to reliably document except in particular anecdotal high-profile cases, such as RSA.
- Most secret information probably does not become public at all: I suspect that most obsolete trade secrets in companies eventually simply end up in the trash or shredder, because why would anybody bother doing anything else?
- For any secret that does become publicly available, there generally will be a long delay before that occurs. At that point, the relationship between two independently invented decades-old technologies will be difficult to detect by anybody not specifically motivated, since they likely would have very different terminology and specifics.
As such, I would expect that it is impossible to do any general study on the frequency with which academic research is a re-invention of secret work elsewhere.