Is there any cryptosystem that works on floating point real number?

Question

I see that all cryptosystems are work on finite field. But i consider that have any cryptosystem that works on real number (do not consider the round step and it also can be an estimate cipher)

Answer 1

One of the big problems in using floating-point as part of an encryption algorithm is round-off error. For more details, look at What Every Computer Scientist Should Know About Floating-Point Arithmetic and Round-off error (Wikipedia).

The basic problem is that in floating-point math, not only does round-off error occur, but you can't depend on different computers doing it consistently. Here is an example where the same source program gives different results (on the same system) depending on whether x87-based floating point (which stores numbers in a 80-bit format) was used rather than the SSE instructions (which use a more traditional 64-bit double precision float).

So to make a floating-point algorithm work, you would have to find a way of avoiding round-off error, while integers and bitstrings avoid that problem entirely.

Answer 2

I see that all cryptosystems are work on finite field.

No. Modern [1] cryptosystems usually work on bitstrings; you give them a bitstring (possibly limited to multiples of 8 in length) as plaintext, and they give you back a bitstring of ciphertext.

They may use finite fields internally, or they might not - depends on the internals of the cipher (which users generally shouldn't have to care about)

But i consider that have any cryptosystem that works on real number

That's easy; express your real number as a bitstring, and pass the bitstring to the modern cryptosystem - problem solved.

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[1]: I say 'modern' to distinguish them from historic (pre-computer) ciphers, which tended to be more typically character based.