In a computational setting it is clear that we cannot do computation over the real numbers, as we couldn't generate, much less store them, so we have to instead deal with numbers that approximate them. If a cryptosystem has some component defined over the real numbers, or uses "real valued" functions such as $\sin$, then does using such systems over the discretized space compromise the theoretical security of the system?
I am particularly thinking of chaos based cryptosystems for example, where encryption consists of iterating a map over an interval of the real line. Are such systems inherently less secure because they cannot be actually implemented over the reals, and instead are approximations? Can such approximations be used to cryptanalyze the systems?