It wouldn't really break RSA or other algorithms, it would just push the security parameters on everything way up until you can't verify the solutions in <1 second. In particular, encoding a single code word would always require at least 1 second of time, so cryptographic algorithms would become slow.

If I were a jerk, I could publish a prime number as my RSA public key. Then anyone who tries to use stable time loops to find my private key would find themselves or their computers disrupted by bizarre coincidences (and as you've mentioned, those coincidences probably aren't a good thing for the most part).