Security parameter

Abstract: We present a novel approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), without Gentry's bootstrapping procedure.Specifically, we offer a choice of FHE schemes based on the learning with error (LWE) or ring-LWE (RLWE) problems that have 2λ security against known attacks. For RLWE, we have:• A leveled FHE scheme that can evaluate L-level arithmetic circuits with O(λ · L3) per-gate computation -- i.e., computation quasi-linear in the security parameter. Security is based on RLWE for an approximation factor exponential in L. This construction does not use the bootstrapping procedure.• A leveled FHE scheme that uses bootstrapping as an optimization, where the per-gate computation (which includes the bootstrapping procedure) is O(λ2), independent of L. Security is based on the hardness of RLWE for quasi-polynomial factors (as opposed to the sub-exponential factors needed in previous schemes).We obtain similar results to the above for LWE, but with worse performance.Based on the Ring LWE assumption, we introduce a number of further optimizations to our schemes. As an example, for circuits of large width -- e.g., where a constant fraction of levels have width at least λ -- we can reduce the per-gate computation of the bootstrapped version to O(λ), independent of L, by batching the bootstrapping operation. Previous FHE schemes all required Ω(λ3.5) computation per gate.At the core of our construction is a much more effective approach for managing the noise level of lattice-based ciphertexts as homomorphic operations are performed, using some new techniques recently introduced by Brakerski and Vaikuntanathan (FOCS 2011).

Abstract: We present a fully homomorphic encryption scheme that is based solely on the(standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worst-case hardness of ``short vector problems'' on arbitrary lattices. Our construction improves on previous works in two aspects:\begin{enumerate}\item We show that ``somewhat homomorphic'' encryption can be based on LWE, using a new {\em re-linearization} technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings. \item We deviate from the "squashing paradigm'' used in all previous works. We introduce a new {\em dimension-modulus reduction} technique, which shortens the cipher texts and reduces the decryption complexity of our scheme, {\em without introducing additional assumptions}. \end{enumerate}Our scheme has very short cipher texts and we therefore use it to construct an asymptotically efficient LWE-based single-server private information retrieval (PIR) protocol. The communication complexity of our protocol (in the public-key model) is $k \cdot \polylog(k)+\log \dbs$ bits per single-bit query (here, $k$ is a security parameter).

Abstract: We propose a general multiparty computation protocol secure against an active adversary corrupting up to $$n-1$$ of the n players. The protocol may be used to compute securely arithmetic circuits over any finite field $$\mathbb {F}_{p^k}$$. Our protocol consists of a preprocessing phase that is both independent of the function to be computed and of the inputs, and a much more efficient online phase where the actual computation takes place. The online phase is unconditionally secure and has total computational and communication complexity linear in n, the number of players, where earlier work was quadratic in n. Moreover, the work done by each player is only a small constant factor larger than what one would need to compute the circuit in the clear. We show this is optimal for computation in large fields. In practice, for 3 players, a secure 64-bit multiplication can be done in 0.05 ms. Our preprocessing is based on a somewhat homomorphic cryptosystem. We extend a scheme by Brakerski et al., so that we can perform distributed decryption and handle many values in parallel in one ciphertext. The computational complexity of our preprocessing phase is dominated by the public-key operations, we need $$On^2/s$$ operations per secure multiplication where s is a parameter that increases with the security parameter of the cryptosystem. Earlier work in this model needed $$\varOmega n^2$$ operations. In practice, the preprocessing prepares a secure 64-bit multiplication for 3 players in about 13 ms.

Abstract: We introduce and formalize the notion of Verifiable Computation, which enables a computationally weak client to "outsource" the computation of a function F on various dynamically-chosen inputs x1, ...,xk to one or more workers. The workers return the result of the function evaluation, e.g., yi = F(xi), as well as a proof that the computation of F was carried out correctly on the given value xi. The primary constraint is that the verification of the proof should require substantially less computational effort than computing F(i) from scratch. We present a protocol that allows the worker to return a computationally-sound, non-interactive proof that can be verified in O(mċpoly(λ)) time, where m is the bit-length of the output of F, and λ is a security parameter. The protocol requires a one-time pre-processing stage by the client which takes O(|C|ċpoly(λ)) time, where C is the smallest known Boolean circuit computing F. Unlike previous work in this area, our scheme also provides (at no additional cost) input and output privacy for the client, meaning that the workers do not learn any information about the xi or yi values.