We propose a fully homomorphic encryption scheme -- i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result -- that, to construct an encryption scheme that permits evaluation of arbitrary circuits, it suffices to construct an encryption scheme that can evaluate (slightly augmented versions of) its own decryption circuit; we call a scheme that can evaluate its (augmented) decryption circuit bootstrappable.Next, we describe a public key encryption scheme using ideal lattices that is almost bootstrappable.Lattice-based cryptosystems typically have decryption algorithms with low circuit complexity, often dominated by an inner product computation that is in NC1. Also, ideal lattices provide both additive and multiplicative homomorphisms (modulo a public-key ideal in a polynomial ring that is represented as a lattice), as needed to evaluate general circuits.Unfortunately, our initial scheme is not quite bootstrappable -- i.e., the depth that the scheme can correctly evaluate can be logarithmic in the lattice dimension, just like the depth of the decryption circuit, but the latter is greater than the former. In the final step, we show how to modify the scheme to reduce the depth of the decryption circuit, and thereby obtain a bootstrappable encryption scheme, without reducing the depth that the scheme can evaluate. Abstractly, we accomplish this by enabling the encrypter to start the decryption process, leaving less work for the decrypter, much like the server leaves less work for the decrypter in a server-aided cryptosystem.

/pdf/fully-homomorphic-encryption-using-ideal-lattices-3oirjo0huq.pdf

Fully homomorphic encryption using ideal lattices

A fruitful direction for future data mining research will be the development of techniques that incorporate privacy concerns. Specifically, we address the following question. Since the primary task in data mining is the development of models about aggregated data, can we develop accurate models without access to precise information in individual data records? We consider the concrete case of building a decision-tree classifier from training data in which the values of individual records have been perturbed. The resulting data records look very different from the original records and the distribution of data values is also very different from the original distribution. While it is not possible to accurately estimate original values in individual data records, we propose a novel reconstruction procedure to accurately estimate the distribution of original data values. By using these reconstructed distributions, we are able to build classifiers whose accuracy is comparable to the accuracy of classifiers built with the original data.

Privacy-preserving data mining

Let ψ be a 2-DNF formula on boolean variables x1,...,xn ∈ {0,1}. We present a homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn. In other words, given the encryption of the bits x1,...,xn, anyone can create the encryption of ψ(x1,...,xn). More generally, we can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set. We present a number of applications of the system:

In a database of size n, the total communication in the basic step of the Kushilevitz-Ostrovsky PIR protocol is reduced from $\sqrt{n}$ to $\sqrt[3]{n}$.

An efficient election system based on homomorphic encryption where voters do not need to include non-interactive zero knowledge proofs that their ballots are valid. The election system is proved secure without random oracles but still efficient.

A protocol for universally verifiable computation.

/pdf/evaluating-2-dnf-formulas-on-ciphertexts-24t0jxv6hg.pdf

Evaluating 2-DNF formulas on ciphertexts

We propose a generalisation of Paillier's probabilistic public key system, in which the expansion factor is reduced and which allows to adjust the block length of the scheme even after the public key has been fixed, without loosing the homomorphic property. We show that the generalisation is as secure as Paillier's original system.

We construct a threshold variant of the generalised scheme as well as zero-knowledge protocols to show that a given ciphertext encrypts one of a set of given plaintexts, and protocols to verify multiplicative relations on plaintexts.

We then show how these building blocks can be used for applying the scheme to efficient electronic voting.This reduces dramatically the work needed to compute the final result of an election, compared to the previously best known schemes.W e show how the basic scheme for a yes/no vote can be easily adapted to casting a vote for up to t out of L candidates. The same basic building blocks can also be adapted to provide receipt-free elections, under appropriate physical assumptions. The scheme for 1 out of L elections can be optimised such that for a certain range of parameter values, a ballot has size only O(log L) bits.

/pdf/a-generalisation-a-simplification-and-some-applications-of-4npavulwcp.pdf

A Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System

We consider the problem of computing the intersection of private datasets of two parties, where the datasets contain lists of elements taken from a large domain. This problem has many applications for online collaboration. We present protocols, based on the use of homomorphic encryption and balanced hashing, for both semi-honest and malicious environments. For lists of length k, we obtain O(k) communication overhead and O(k ln ln k) computation. The protocol for the semi-honest environment is secure in the standard model, while the protocol for the malicious environment is secure in the random oracle model. We also consider the problem of approximating the size of the intersection, show a linear lower-bound for the communication overhead of solving this problem, and provide a suitable secure protocol. Lastly, we investigate other variants of the matching problem, including extending the protocol to the multi-party setting as well as considering the problem of approximate matching.

/pdf/efficient-private-matching-and-set-intersection-2y0mki6s1a.pdf

Efficient Private Matching and Set Intersection

We propose a generalisation of Paillier's probabilistic public key system, in which the expansion factor is reduced and which allows to adjust the block length of the scheme even after the public key has been fixed, without losing the homomorphic property. We show that the generalisation is as secure as Paillier's original system. We construct a threshold variant of the generalised scheme as well as zero-knowledge protocols to show that a given ciphertext encrypts one of a set of given plaintexts, and protocols to verify multiplicative relations on plaintexts. We then show how these building blocks can be used for applying the scheme to efficient electronic voting. This reduces dramatically the work needed to compute the final result of an election, compared to the previously best known schemes. We show how the basic scheme for a yes/no vote can be easily adapted to casting a vote for up to t out of L candidates. The same basic building blocks can also be adapted to provide receipt-free elections, under appropriate physical assumptions. The scheme for 1 out of L elections can be optimised such that for a certain range of parameter values, a ballot has size only O(log L) bits.

/pdf/a-generalisation-a-simplification-and-some-applications-of-3okun55vdd.pdf

A Generalisation, a Simplification and some Applications of Paillier’s Probabilistic Public-Key System

We propose a public-key cryptosystem which is derived from the Paillier cryptosystem. The scheme inherits the attractive homomorphic properties of Paillier encryption. In addition, we achieve two new properties: First, all users can use the same modulus when generating key pairs, this allows more efficient proofs of relations between different encryptions. Second, we can construct a threshold decryption protocol for our scheme that is length flexible, i.e., it can handle efficiently messages of arbitrary length, even though the public key and the secret key shares held by decryption servers are of fixed size. We show how to apply this cryptosystem to build: 1) a self-tallying election scheme with perfect ballot secrecy. This is a small voting system where the result can be computed from the submitted votes without the need for decryption servers. The votes are kept secret unless the cryptosystem can be broken, regardless of the number of cheating parties. This is in contrast to other known schemes that usually require a number of decryption servers, the majority of which must be honest. 2) a length-flexible mix-net which is universally verifiable, where the size of keys and ciphertexts do not depend on the number of mix servers, and is robust against a corrupt minority. Mix-nets can provide anonymity by shuffling messages to provide a random permutation of input ciphertexts to the output plaintexts such that no one knows which plaintexts relate to which ciphertexts. The mix-net inherits several nice properties from the underlying cryptosystem, thus making it useful for a setting with small messages or high computational power, low-band width and that anyone can verify that the mix have been done correctly.

/pdf/a-length-flexible-threshold-cryptosystem-with-applications-2rj0jpla8i.pdf

A Length-Flexible Threshold Cryptosystem with Applications

We present various trade offs for voting schemes which, compared to known solutions, allow voters to do less work at the expense of more work done by the tallying servers running the election One such scheme produces ballots of essentially minimal size while keeping the work load on the tally servers on a practical level Another type of trade off leads to a voting scheme that remains secure, even if an adversary can monitor all client machines used by voters to participate This comes at the price of introducing an additional party who is trusted to carry out registration of voters correctly

/pdf/client-server-tradeoffs-for-online-elections-2tpvbt1tvx.pdf

Client/Server Tradeoffs for Online Elections

We propose a cryptosystem that has an inherent key escrow mechanism. This leads us to propose a session based public verifiable key escrow system that greatly improves the amount of key material the escrow servers has to keep in order to decrypt an encryption. In our scheme the servers will only have a single secret sharing, as opposed to a single key from every escrowed player. This is done while still having the properties: 1) public verifiable: the user proves to everyone that the encryption can indeed be escrowed, and 2) no secret leakage: no matter how many decryptions a law enforcement agency is presented, it will gain no more information on the users private key, than it couldn’t have calculated itself.

Scalable Key-Escrow

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