Bootstrapping is a technique, originally due to Gentry (STOC 2009), for "refreshing" ciphertexts of a somewhat homomorphic encryption scheme so that they can support further homomorphic operations. To date, bootstrapping remains the only known way of obtaining fully homomorphic encryption for arbitrary unbounded computations. Over the past few years, several works have dramatically improved the efficiency of bootstrapping and the hardness assumptions needed to implement it. Recently, Brakerski and Vaikuntanathan (ITCS 2014) reached the major milestone of a bootstrapping algorithm based on Learning With Errors for polynomial approximation factors. Their method uses the Gentry-Sahai-Waters (GSW) cryptosystem (CRYPTO 2013) in conjunction with Barrington's "circuit sequentialization" theorem (STOC 1986). This approach, however, results in very large polynomial runtimes and approximation factors. (The approximation factors can be improved, but at even greater costs in runtime and space.) In this work we give a new bootstrapping algorithm whose runtime and associated approximation factor are both small polynomials. Unlike most previous methods, ours implements an elementary and efficient arithmetic procedure, thereby avoiding the inefficiencies inherent to the use of boolean circuits and Barrington's Theorem. For 2 λ security under conventional lattice assumptions, our method requires only a quasi-linear O(λ) number of homomorphic operations on GSW ciphertexts, which is optimal (up to polylogarithmic factors) for schemes that encrypt just one bit per ciphertext. As a contribution of independent interest, we also give a technically simpler variant of the GSW system and a tighter error analysis for its homomorphic operations.
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Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such identification is not intended to imply recommendation or endorsement by NIST, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose. There may be references in this publication to other publications currently under development by NIST in accordance with its assigned statutory responsibilities. The information in this publication, including concepts and methodologies, may be used by federal agencies even before the completion of such companion publications. Thus, until each publication is completed, current requirements, guidelines, and procedures, where they exist, remain operative. For planning and transition purposes, federal agencies may wish to closely follow the development of these new publications by NIST. Organizations are encouraged to review all draft publications during public comment periods and provide feedback to NIST. Many NIST cybersecurity publications, other than the ones noted above, are available at https://csrc.nist.gov/publications.
Abstract. We initiate the study of security for key-dependent messages (KDM), sometimes also known as "circular" or "clique" security, in the setting of identity-based encryption (IBE). Circular/KDM security requires that ciphertexts preserve secrecy even when they encrypt messages that may depend on the secret keys, and arises in natural usage scenarios for IBE.We construct an IBE system that is circular secure for affine functions of users' secret keys, based on the learning with errors (LWE) problem (and hence on worst-case lattice problems). The scheme is secure in the standard model, under a natural extension of a selectiveidentity attack. Our three main technical contributions are (1) showing the circular/KDM-security of a "dual"-style LWE public-key cryptosystem, (2) proving the hardness of a version of the "extended LWE" problem due to O'Neill, Peikert and Waters (CRYPTO'11), and (3) building an IBE scheme around the dual-style system using a novel lattice-based "all-but-d" trapdoor function.
Gentry's "bootstrapping" technique (STOC 2009) constructs a fully homomorphic encryption (FHE) scheme from a "somewhat homomorphic" one that is powerful enough to evaluate its own decryption function. To date, it remains the only known way of obtaining unbounded FHE. Unfortunately, bootstrapping is computationally very expensive, despite the great deal of effort that has been spent on improving its efficiency. The current state of the art, due to Gentry, Halevi, and Smart (PKC 2012), is able to bootstrap "packed" ciphertexts (which encrypt up to a linear number of bits) in time only quasilinearÕ(λ) = λ • log O(1) λ in the security parameter. While this performance is asymptotically optimal up to logarithmic factors, the practical import is less clear: the procedure composes multiple layers of expensive and complex operations, to the point where it appears very difficult to implement, and its concrete runtime appears worse than those of prior methods (all of which have quadratic or larger asymptotic runtimes). In this work we give simple, practical, and entirely algebraic algorithms for bootstrapping in quasilinear time, for both "packed" and "non-packed" ciphertexts. Our methods are easy to implement (especially in the non-packed case), and we believe that they will be substantially more efficient in practice than all prior realizations of bootstrapping. One of our main techniques is a substantial enhancement of the "ring-switching" procedure of Gentry et al. (SCN 2012), which we extend to support switching between two rings where neither is a subring of the other. Using this procedure, we give a natural method for homomorphically evaluating a broad class of structured linear transformations, including one that lets us evaluate the decryption function efficiently.
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