What does it mean to say breaking RSA generically is equivalent to factoring? I am giving a one hour presentation on the RSA crypto-system as one of the requirements for Masters degree. I just want to get some facts straight here. I was told casually by a professor that RSA is equivalent to factoring, but I am having a hard time verifying this with resources online. 
So, is breaking RSA generically equivalent to factoring as in the title of this paper? https://eprint.iacr.org/2008/260.pdf
What does it mean to use the word "generically" to describe the difficulty of breaking RSA.
Any other resources would be helpful.
 A: I am turning my comment into an answer.
RSA is secure under the RSA Assumption, which basically states that RSA is secure. This assumption is not known to be the same as the assumption that factorization is hard. While others have pointed out that factorization is sufficient to break RSA, it has not been proven that factorization is necessary to break RSA.
Supporting the position that breaking RSA is easier than factoring, if factoring is easy then the RSA secret key may be obtained from the public key, but breaking RSA encryption only requires distinguishing between the encryptions of two adversarially-chosen plaintexts, which is intuitively an easier task. Similarly, breaking RSA signing does not require obtaining the secret key, but instead forging the signature of an adversarially-chosen message, which is again intuitively easier. Of course I can't offer any proofs because the problem is open.

As for the meaning of the word "generically", it appears to mean "only by performing operations in the algebraic structure being analyzed" as opposed to exploiting additional structure, such as the bit representation of numbers. Note that proving the equivalence of factoring and the RSA assumption "generically" is a more modest result than proving their equivalence universally because the computational models that they consider have been restricted to just perform certain operations, and it's possible that by allowing other operations, the two problems are no longer equivalent.
A: I think the idea is this: the RSA algorithm is powerful because it relies on the fact that for large numbers, say numbers with 200 digits, it is very difficult to factor them or even find out if they are composite. If we were somehow able to figure a way to factor large numbers relatively easily, then the RSA algorithm would no longer be that powerful, thus it would be "broken".
Edit: As for what "generically" means, it might explain that in the paper but my guess is that the author is using the word "generic" to mean that this is characteristic of the RSA algorithm, i.e. that is why we can say it is equivalent to factoring.
A: The general concept behind cryptography is the requirement that one operation takes a small amount of time, while undoing the same operation without a priori knowledge must take a large amount of time.
The entire premise of RSA is the idea that the time complexity to multiply two numbers together is much lower than factoring a number into two numbers. Thus, we might be able to encrypt a message in a few seconds, but it might takes years to decrypt the message without the key. If we could find a faster way to factor a number, then the entire premise behind RSA would be invalidated.
Obviously there is more to it than that, but it is the underlying premise, hence 'generically'.
A: The assumption in RSA is the hardness of computing $\phi(n)$ without knowing $p$ or $q$.
Given an algorithm which takes $n$ and computes $\phi(n)$, we can factor $n$ using $\phi(n)$:


*

*$n = pq$

*$\phi(n) = (p-1)(q-1) = pq-p-q+1 = n-\frac{n}{q}-q+1$

*$q^2-(n-\phi(n)+1)q+n = 0$

*$q = \frac{(n-\phi(n)+1)+\sqrt{(n-\phi(n)+1)^2-4n}}{2}$

*$p = \frac{n}{q}$
