Abstract Homomorphic encryption () protects data in-use, but can be computationally expensive. To avoid the costly bootstrapping procedure that refreshes ciphertexts, some works have explored client-aided outsourcing protocols, where the client intermittently refreshes ciphertexts for a server that is performing homomorphic computations. But is this approach secure against malicious servers? We present a -secure encryption scheme that is completely insecure in this setting. We define a new notion of security, called , that we prove is sufficient. Additionally, we show: Homomorphic encryption schemes that have a certain type of circuit privacy—for example, schemes in which ciphertexts can be “sanitized"—are -secure. In particular, assuming certain existing schemes are -secure, they are also -secure. For certain encryption schemes, like Brakerski-Vaikuntanathan, that have a property that we call oblivious secret key extraction, -security implies circular security—i.e., that it is secure to provide an encryption of the secret key in a form usable for bootstrapping (to construct fully homomorphic encryption).

In the era of cloud computing and machine learning, data has become a highly valuable resource. Recent history has shown that the benefits brought forth by this data driven culture come at a cost of potential data leakage. Such breaches have a devastating impact on individuals and industry, and lead the community to seek privacy preserving solutions. A promising approach is to utilize Fully Homomorphic Encryption ( \( \mathsf {FHE } \) ) to enable machine learning over encrypted data, thus providing resiliency against information leakage. However, computing over encrypted data incurs a high computational overhead, thus requiring the redesign of algorithms, in an “ \( \mathsf {FHE } \) -friendly” manner, to maintain their practicality. In this work we focus on the ever-popular tree based methods, and propose a new privacy-preserving solution to training and prediction for trees over data encrypted with homomorphic encryption. Our solution employs a low-degree approximation for the step-function together with a lightweight interactive protocol, to replace components of the vanilla algorithm that are costly over encrypted data. Our protocols for decision trees achieve practical usability demonstrated on standard UCI datasets encrypted with fully homomorphic encryption. In addition, the communication complexity of our protocols is independent of the tree size and dataset size in prediction and training, respectively, which significantly improves on prior works. 1

In their seminal work on nonmalleable cryptography, Dolev, Dwork, and Naor showed how to construct a nonmalleable commitment with logarithmically-many “rounds''/``slots,” the idea being that any adversary may successfully maul in some slots but would fail in at least one. Since then new ideas have been introduced, ultimately resulting in constant-round protocols based on any one-way function. Yet, in spite of this remarkable progress, each of the known constructions of nonmalleable commitments leaves something to be desired. In this paper we propose a new technique that allows us to construct a nonmalleable protocol with only a single slot and to improve in at least one aspect over each of the previously proposed protocols. Two direct byproducts of our new ideas are a four-round nonmalleable commitment and a four-round nonmalleable zero-knowledge argument, the latter matching the round-complexity of the best known zero-knowledge argument (without the nonmalleability requirement). The protocols are based on the existence of one-way functions and admit very efficient instantiations via standard homomorphic commitments and sigma protocols. Our analysis relies on algebraic reasoning, and makes use of error correcting codes in order to ensure that committers' tags differ in many coordinates. One way of viewing our construction is as a method for combining many atomic subprotocols in a way that simultaneously amplifies soundness and nonmalleability, thus requiring much weaker guarantees to begin with, and resulting in a protocol which is much trimmer in complexity compared to the existing ones.

The notion of non-malleability in cryptography refers to the setting where the adversary is a man-in-the-middle (MIM) who takes part in two or more protocol executions and tries to use information obtained in one, to violate the security of another. Despite two decades of research, non-malleable commitments (NMCs) have remained too inefficient to be implemented in practice, without some sort of trusted setup. In this work, we give a fast implementation of NMC in the plain model, based on the DDH assumption being hard over elliptic curve groups. Our main theoretical result is a new NMC scheme which can be thought of as a "high dimensional" generalization of the one in the recent work of [GRRV14]. Central to our efficiency improvements is a method of constraining challenges sent by the receiver. This new approach enables us to obtain dramatically improved parameters over those suggested in [GRRV14]. In particular, our work opens the door to implementations based on Elliptic Curves.

In their seminal work on non-malleable cryptography, Dolev, Dwork and Naor, showed how to construct a non-malleable commitment with logarithmically-many "rounds"/"slots", the idea being that any adversary may successfully maul in some slots but would fail in at least one. Since then new ideas have been introduced, ultimately resulting in constant-round protocols based on any one-way function. Yet, in spite of this remarkable progress, each of the known constructions of non-malleable commitments leaves something to be desired. In this paper we propose a new technique that allows us to construct a non-malleable protocol with only a single "slot", and to improve in at least one aspect over each of the previously proposed protocols. Two direct byproducts of our new ideas are a four round non-malleable commitment and a four round non-malleable zero-knowledge argument, the latter matching the round complexity of the best known zero-knowledge argument (without the non-malleability requirement). The protocols are based on the existence of one-way functions and admit very efficient instantiations via standard homomorphic commitments and sigma protocols. Our analysis relies on algebraic reasoning, and makes use of error correcting codes in order to ensure that committers' tags differ in many coordinates. One way of viewing our construction is as a method for combining many atomic sub-protocols in a way that simultaneously amplifies soundness and non-malleability, thus requiring much weaker guarantees to begin with, and resulting in a protocol which is much trimmer in complexity compared to the existing ones.

Rational proofs, recently introduced by Azar and Micali (STOC 2012), are a variant of interactive proofs in which the prover is neither honest nor malicious, but rather rational. The advantage of rational proofs over their classical counterparts is that they allow for extremely low communication and verification time. Azar and Micali demonstrated their potential by giving a one message rational proof for #SAT, in which the verifier runs in time O(n), where $n$ denotes the instance size. In a follow-up work (EC 2013), Azar and Micali proposed "super-efficient" and interactive versions of rational proofs and argued that they capture precisely the class TC0 of constant-depth, polynomial-size circuits with threshold gates.