I have a basic understanding of encryption and I got back to the topic because of an interesting site that encrypts financial data using homomorphic encryption (HE) and I would be happy for any input from the community here.
They say that as addition and multiplication in HE is "preserved" one can apply machine learning algorithms (they usually use all operations - not only polynomial ones).
My question: if data is enrypted then the data that was originally on the real line is usually mapped to the algebraic structure of a ring. Thus if we get those elements of the ring, we have to perform the operations that are defined on this ring. Finally we can not (!) apply the usual real number operations that the ML algorithms consist of.
Is this tue for HE? Is it true for order-preserving symmetric encryption?
An example as EDIT as I am not a crypto-pro at all: say I am given the following data: $$ (0.2,0.1,0.5,0); (0.1,0.2,0.3,1); (0.02,0.7,0.33,1) $$ and several rows thousands of them (and in my application more columns). In this example the first 3 entries are inputs and the 4th one is the target. All I know is that the inputs were decrypted (either HE or order-preserving symmetric) and I see that each column has exactly 1001 unique values (which makes me think that the data is not real numbers but data on some grid or finite ring). If I interpret the inputs as real numbers and perform the usual ML-algorithms (logistic regression and more complex ones). Is this mathematically sound or am I doing complete nonsense (because the data is not real numbers but rather objects in an algebraic strucutre that does not allow for the usual $+$,$\times\ldots$?