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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 don't really tell the precise method they use. In the blog they mention the Fan and Vercauteren scheme and on the other hand they mention order-preserving symmetric encryption.

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 $+$,$\times\ldots$?

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    $\begingroup$ My guess is that the blog writer really doesn't understand encryption. Yes, homomorphic encryption can do computation on encrypted data, however unless you have the private key, the result of the computation looks random. It's hard to see how homomorphic encryption could be used to compute something that the user's computer couldn't do itself... $\endgroup$ – poncho Jul 25 '16 at 13:54
  • $\begingroup$ @poncho Thanks for your remark. Thus if I am given the encrypted data and no key (neither public nor private). Can I do anything useful with the data? Does HE usually map to the real numbers so I can do real number calculations or does it map to a ring where I can only do the ring operations? If the 2nd is true then I can not go to my shelf of ML alogrithms and apply them - right? $\endgroup$ – Richard Jul 25 '16 at 14:26
  • $\begingroup$ You may be able to determine the lengths of the ciphertexts. ​ For small T, you can get an encryption of the result of a T-time operation on the plaintexts, which may be useful if someone else has the private key, even if you don't. ​ (The previous sentence is in fact the point of Fully Homomorphic Encryption.) ​ ​ ​ ​ $\endgroup$ – user991 Jul 25 '16 at 14:51
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    $\begingroup$ Well, the general 'proof that FHE can do anything' is typically 'encode everything in binary' (and is, each encrypted value is either $E(0)$ or $E(1)$, and so an actual data point in the interval [-0.3, 0.3] might be encrypted as 32 bits, in either a fixed point or a floating point notation (whichever's more convienent). Adding two such values would involve simulating the addition circuit (we can do AND by multiplying, and NOT by subtracting the bit from an encrypted 1). This technique is expensive, but we can obviously use it to compute anything that can be done in constant time. $\endgroup$ – poncho Jul 25 '16 at 15:22
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    $\begingroup$ Real number are typically represented as floating-point numbers in the programs, so you don't have real values anyway when you are running a machine learning algorithm. If you are able to encode such floating-point values into the plaintext messages used by the FHE schemes, then you can run ML algorithms homomorphically. The problem is that using FHE to do that would be very very inefficient... $\endgroup$ – Hilder Vítor Lima Pereira Jul 25 '16 at 17:10
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Homomorphic Encryption on Reals
In theory, homomorphic encryption can be done on real numbers. This answer describes two options you have when dealing with real numbers or operations that will result in real numbers. Kristin Lauter is doing some of the cutting edge research in this area. In a recent paper, CryptoNets: Applying Neural Networks to Encrypted Data with High Throughput and Accuracy, the authors (which includes Kristin), state:

One thing to note is that the encryption scheme does not support floating-point numbers. Instead, we use fixed precision real numbers by converting them to integers by proper scaling, although there are also other ways to do this (Dowlin et al., 2015).

They go on to say:

The neural network takes as its input a vector of real numbers and, through a series of additions, multiplications, and other real functions, it computes its outputs, which are also real numbers. However, the homomorphic encryption scheme works over the ring $R_n^t := \mathbb{Z}_t [x] /(x^n + 1)$. This means that some conversion process between real numbers and elements of $R_n^t$ is needed.

So they are not operating on real numbers directly. Instead, they encode (using scaling as mentioned in the other answer I linked to). And then decode after decryption. III.B of the Dowlin paper discusses encoding methods for real numbers. Real numbers are encountered everywhere in data analytics, and so I expect there to be continued research into how best to operate on reals using homomorphic encryption. In other words, it is an active area of research and not something that you can just quickly pull a tool out of your hat and off you go. Some thought and care must be taken in making sure you aren't invalidating the analysis techniques by using some of the encoding methods that are out there.

Order-Preserving Symmetric Encryption
The claim in the blog post is

Simpler schemes like order-preserving symmetric encryption also allow strong security in certain settings, and are easy to use with out of the box machine learning tools.

The author doesn't appear to go into much more detail than that, which leaves me wondering what they really mean. Order-preserving symmetric encryption is all about encrypting data in such a way that a natural ordering of the plaintexts is preserved. This allows you to, given only ciphertexts, do comparisons. This leads to useful operations. Specifically, the paper the post links to mentions range queries on encrypted numbers. The 2004 SIGMOD paper by Agrawal describes how their OPE scheme can be directly applied to IEEE 754 single precision floating point numbers.

Not being a machine learning expert, I can't tell you how useful only being able to do comparisons, operations like min and max, and range queries would be for machine learning. I guess if your machine learning tools require only range queries and comparisons, OPE could be used with "out of the box machine learning tools", but something tells me that would leave much to be desired in terms of data analysis.

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  • $\begingroup$ I have been informed today that the encryption is just order preserving. ... thus only max, min .. as you say. Thanks for the clear answer! $\endgroup$ – Richard Jul 28 '16 at 13:14
  • $\begingroup$ @Richard - can you confirm what you meant by your comment ? Can you confirm that the only operations that can be done are max and min on the sample data that they give ? My bigger question is - what classes of ML algorithms can be applied . I know decision trees can.. but what about things like SVM, etc $\endgroup$ – Sandeep Feb 14 '17 at 7:58
  • $\begingroup$ @Sandeep if the encryption were only order preserving then only min and max were ok. The case I am working on must be something different. There is a lot of noise but besides that all well known algorithms can be applied. $\endgroup$ – Richard Feb 16 '17 at 12:57

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