There are a few possible interpretations of "breaking RSA." The most damaging would be for an attacker to discover the private key corresponding to a given public key? this would enable the attacker both to read all messages encrypted with the public key and to forge signatures. The obvious way to do this attack is to factor the public modulus, n, into its two prime factors, p and q.
From p, q, and e, the public exponent, the attacker can easily get d, the private exponent. The hard part is factoring n? the security of RSA depends on factoring being difficult. In fact, the task of recovering the private key is equivalent to the task of factoring the modulus: you can use d to factor n, as well as use the factorization of n to find d. It should be noted that hardware improvements alone will not weaken RSA, as long as appropriate key lengths are used? in fact, hardware improvements should increase the security of RSA.   _{Your Comment}