Using Linearly-Homomorphic Encryption to Evaluate Degree-2 Functions on Encrypted Data

This post is about a talk in the final session of CCS 2015 that was presented by Dario Fiore. The presented work is a joint work with Dario Catalano and the full version of their paper can be found here.

Homomorphic Encryption (HE) provides a solution to the problem of outsourcing computations privately to a cloud. There are a number of Somewhat Homomorphic Encryption (SHE) schemes that support many additions but only a limited number of multiplications. Some examples of such schemes in the chronological order are Goldwasser-Micali, El Gamal, Okamoto-Uchiyama, Paillier, Boneh-Goh-Nissim, and all the underlying SHE schemes of Fully Homomorphic Encryption (FHE) schemes since the construction of Gentry. Though SHE schemes can evaluate only a limited set of circuits compared to FHE schemes, it is still interesting to consider them for applications mainly for efficiency reason. Those SHE schemes that support only a single operation on ciphertexts – addition or multiplication – is called linearly homomorphic. That is, (additive) linearly-homomorphic encryption schemes can evaluate only linear functions on ciphertexts. (Here, by  a “linear function” we mean that the function can be expressed as a polynomial of degree one in the input variables with the scalars coming from a ring.)

The main problem addressed in the current work is to enable evaluation of degree-two functions using linearly-homomorphic encryption schemes. To this end, the authors propose a generic transformation that extends an (additively-written) linearly-homomorphic scheme to support a single multiplication operation so that it will now be able to evaluate a subset of functions of degree two.  The only requirement of the underlying linearly-homomorphic scheme is that it should be public space, which means that the message space must be a publicly known finite commutative ring and that it is possible to efficiently sample a random element from this ring. This property seems to hold for nearly all of the known linearly-homomorphic encryption schemes or can be easily adapted to become so.

Compactness is one of the three main requirements of a (fully) HE scheme that says that the complexity of decryption must be independent of the complexity of the circuit evaluating the given function. The other two requirements are semantic security and circuit privacy for the underlying plaintext messages. The latter means that it must not be possible to learn anything about the messages in the ciphertexts input to a function. It is shown that the proposed transformation preserves semantic security and the transformed scheme will be levelled circuit private if the underlying scheme is circuit private. The transformed scheme will be able to compactly evaluate a subset of degree-two functions (represented as polynomials) $mathcal{F}_2^* $ consisting of the following polynomials $f$
[
f(m_1,…,m_t) = P(m_1,…,m_t) + sum_{i=1}^L Q_i(m_1,…,m_t) * R_i(m_1,…,m_t),
]
where $P$, $Q_i$, $R_i$ are linear polynomials.  It turns out that the subclass $mathcal{F}_2^*$ of quadratic polynomials is relevant for applications, for example, SPDZ protocol for multi-party computation requires a HE scheme for $mathcal{F}_2^*$ with $L=1$. Though there are many prior works that aim to construct efficient SHE schemes, they do not achieve full compactness even for this subclass of quadratic functions. In the current work, this limitation for the quadratic functions outside $mathcal{F}_2^*$ is overcome by working in a different model that uses two servers that do not communicate nor collude. A transformation of the underlying HE scheme is given in this model that is similar to the previous transformation, using which it is now possible to evaluate any quadratic polynomial compactly. Interestingly, the former (single-server) transformation preserves other properties such as zero-knowledge proofs of plaintext knowledge, threshold decryption, and multi-key homomorphicity. Also, the above results are generalised to obtain a $d$-homomorphic to $2d$-homomorphic transformation, and to obtain a two-server protocol to evaluate all degree-three polynomials from BGN HE scheme. 

The authors have compared implementations of the Paillier scheme and the Joyce-Libert scheme when transformed to support evaluation of degree-two functions, with that of the HElib implementation of the BGV FHE scheme instantiated with parameters to support a single multiplication, and also with an implementation of the BGN SHE scheme. Upto values of $L = 10$, the transformed Joyce-Libert scheme has comparable decryption time, but with ciphertexts 25 times shorter, and the encryption and the homomorphic operations at least 30 times faster.

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