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Let <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> be a random variable distributed according to some appropriate distribution over the set of products of two primes, such that factoring <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is believed to be hard. The RSA assumption states that, given an <inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> chosen uniformly at random from <inline-formula> <tex-math notation="LaTeX">$ {\mathbb {Z}}_{N}$ </tex-math></inline-formula> and an <inline-formula> <tex-math notation="LaTeX">$e \in {\mathbb {N}} \setminus \{1\}$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$\gcd (e, \phi (N)) = 1$ </tex-math></inline-formula>, it is computationally hard to find an <inline-formula> <tex-math notation="LaTeX">$x\in {\mathbb {Z}} _{N}$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$x^{e} - a \equiv 0 \pmod N$ </tex-math></inline-formula>. When complexity-theoretic (relative) lower bounds for certain cryptographic problems in a general model of computation seem to elude discovery, a common practice in cryptography is to give proofs of computational security in meaningful restricted models of computation. An example of such a restricted model that is interesting in cryptography is the generic group model that has been used for proving lower bounds for the discrete logarithm problem and other related problems. A generic model captures that an algorithm does not exploit the bit representation of the elements other than for testing equality. In this paper, we prove that the problem of factoring <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> can be efficiently reduced to solving the RSA problem on <inline-formula> <tex-math notation="LaTeX">$ {\mathbb {Z}}_{N}$ </tex-math></inline-formula> in the generic ring model of computation, where an algorithm can perform ring operations, inverse ring operations, and test equality. This provides evidence toward the soundness of the RSA encryption and digital signature scheme, in particular showing that under the factoring assumption, they are not vulnerable to certain kinds of cryptanalytic attacks.

Breaking RSA Generically Is Equivalent to Factoring