Is RSA provably secure in the sense of Douglas Stinson's ``provable security''? Here's a quote from Douglas Stinson:


  “[i]f a cryptosystem can be ‘broken’ in some specific way, then it
  would be possible to efficiently solve some well-studied problem that
  is thought to be difficult. For example, it may be possible to prove a
  statement of the type “a given cryptosystem is secure if a given
  integer n cannot be factored.” [...] [B]ut it must be understood that
  this approach only provides a proof of security relative to some other
  problem, not an absolute proof of security. This is a similar
  situation to proving that a problem is NP-complete [...].”


Source: Stinson, D. R. (2006). Cryptography, Theory and Practice. Chapman
and Hall, CRC, 3rd edition. Chapter 2, section 2.1, page 45.

Does this apply to RSA?  Suppose I find the private exponent by some other way other than factoring $n$.  Then I would have broken RSA without factoring.  

So although Stinson's example mentions factoring, we can't immediately think of RSA here because there's no proof that factoring is the only way to recover a private key.  What do you say?
Every cryptosystem is "provably secure" under at least one hardness assumption: the assumption that it cannot be broken. Hence, the only question which matters is whether a cryptosystem is provably secure under a well-known and well-studied assumption.

This is kind of the case for RSA, but in a somewhat unsatisfying way: the IND-CPA security of RSA with appropriate padding can be reduced to the RSA problem (extracting $e$-th roots mod $n$), which is a well-studied assumption... But only because this is the one underlying the RSA cryptosystem, and we have been using the cryptosystem a lot in the past decades.

Something much more satisfying would be to reduce it to a problem which had long been studied before, in different context; the most natural such candidate is indeed factoring. Unfortunately, we do not know of a reduction from RSA to the hardness of factoring integers - in fact, we even have proofs that it will be hard to find such a reduction (in a well defined sense).

We have, however, many other cryptosystems which can be reduced to factoring; the earliest example of this kind is the cryptosystem of Rabin.