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

We propose a novel paradigm for defining security of cryptographic protocols, called universally composable security. The salient property of universally composable definitions of security is that they guarantee security even when a secure protocol is composed of an arbitrary set of protocols, or more generally when the protocol is used as a component of an arbitrary system. This is an essential property for maintaining security of cryptographic protocols in complex and unpredictable environments such as the Internet. In particular, universally composable definitions guarantee security even when an unbounded number of protocol instances are executed concurrently in an adversarially controlled manner, they guarantee non-malleability with respect to arbitrary protocols, and more. We show how to formulate universally composable definitions of security for practically any cryptographic task. Furthermore, we demonstrate that practically any such definition can be realized using known techniques, as long as only a minority of the participants are corrupted. We then proceed to formulate universally composable definitions of a wide array of cryptographic tasks, including authenticated and secure communication, key-exchange, public-key encryption, signature, commitment, oblivious transfer, zero knowledge and more. We also make initial steps towards studying the realizability of the proposed definitions in various settings.

/pdf/universally-composable-security-a-new-paradigm-for-3kyhnjofve.pdf

Universally composable security: a new paradigm for cryptographic protocols

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

/pdf/the-knowledge-complexity-of-interactive-proof-systems-31apre0ecf.pdf

The knowledge complexity of interactive proof-systems

We show how to construct a variety of "trapdoor" cryptographic tools assuming the worst-case hardness of standard lattice problems (such as approximating the length of the shortest nonzero vector to within certain polynomial factors). Our contributions include a new notion of trapdoor function with preimage sampling, simple and efficient "hash-and-sign" digital signature schemes, and identity-based encryption. A core technical component of our constructions is an efficient algorithm that, given a basis of an arbitrary lattice, samples lattice points from a discrete Gaussian probability distribution whose standard deviation is essentially the length of the longest Gram-Schmidt vector of the basis. A crucial security property is that the output distribution of the algorithm is oblivious to the particular geometry of the given basis.

https://www.mit.edu/~vinodv/papers/trapcvp.pdf

Trapdoors for hard lattices and new cryptographic constructions

Trapdoors for Hard Lattices and New Cryptographic Constructions.

We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems Our constructions are asymptotically efficient and highly parallelizable in a practical sense, ie, they can be computed by simple, relatively small low-depth arithmetic or boolean circuits (eg, in NC1 or even TC0) In addition, they are the first low-depth PRFs that have no known attack by efficient quantum algorithms Central to our results is a new "derandomization" technique for the learning with errors (LWE) problem which, in effect, generates the error terms deterministically

/pdf/pseudorandom-functions-and-lattices-1641i4t2l8.pdf

Pseudorandom functions and lattices

We propose SWIFFT, a collection of compression functions that are highly parallelizable and admit very efficient implementations on modern microprocessors. The main technique underlying our functions is a novel use of the Fast Fourier Transform(FFT) to achieve "diffusion," together with a linear combination to achieve compression and "confusion." We provide a detailed security analysis of concrete instantiations, and give a high-performance software implementation that exploits the inherent parallelism of the FFT algorithm. The throughput of our implementation is competitive with that of SHA-256, with additional parallelism yet to be exploited.

Our functions are set apart from prior proposals (having comparable efficiency) by a supporting asymptotic security proof: it can be formally proved that finding a collision in a randomly-chosen function from the family (with noticeable probability) is at least as hard as finding short vectors in cyclic/ideal lattices in the worst case.

/pdf/swifft-a-modest-proposal-for-fft-hashing-3ewngkgsnb.pdf

SWIFFT: A Modest Proposal for FFT Hashing

We show that every language in NP has a (black-box) concurrent zero-knowledge proof system using O/spl tilde/(log n) rounds of interaction. The number of rounds in our protocol is optimal, in the sense that any language outside BPP requires at least /spl Omega//spl tilde/(log n) rounds of interaction in order to be proved in black-box concurrent zero-knowledge. The zero-knowledge property of our main protocol is proved under the assumption that there exists a collection of claw free functions. Assuming only the existence of one-way functions, we show the existence of O/spl tilde/(log n)-round concurrent zero-knowledge arguments for all languages in NP.

/pdf/concurrent-zero-knowledge-with-logarithmic-round-complexity-13j0a45dog.pdf

Concurrent zero knowledge with logarithmic round-complexity

The generalized knapsack function is defined as f a (x)= Σ i a i x i , where a = (a 1 ,...,a m ) consists of m elements from some ring R, and x = (x i ,...,x m ) consists of m coefficients from a specified subset S C R. Micciancio (FOCS 2002) proposed a specific choice of the ring R and subset S for which inverting this function (for random a, x) is at least as hard as solving certain worst-case problems on cyclic lattices. We show that for a different choice of S ⊂ R, the generalized knapsack function is in fact collision-resistant, assuming it is infeasible to approximate the shortest vector in n-dimensional cyclic lattices up to factors 0(n). For slightly larger factors, we even get collision-resistance for any m > 2. This yields very efficient collision-resistant hash functions having key size and time complexity almost linear in the security parameter n. We also show that altering S is necessary, in the sense that Micciancio's original function is not collision-resistant (nor even universal one-way). Our results exploit an intimate connection between the linear algebra of n-dimensional cyclic lattices and the ring Z[α]/(α n -1), and crucially depend on the factorization of a n - 1 into irreducible cyclotomic polynomials. We also establish a new bound on the discrete Gaussian distribution over general lattices, employing techniques introduced by Micciancio and Regev (FOCS 2004) and also used by Micciancio in his study of compact knapsacks.

Alon Rosen

Papers

Pseudorandom functions and lattices

SWIFFT: A Modest Proposal for FFT Hashing

Concurrent zero knowledge with logarithmic round-complexity

Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices

Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices