FHE Circuit Privacy Almost for Free

International audienceCircuit privacy is an important property for many applications of fully homomorphic encryption. Prior approaches for achieving circuit privacy rely on superpolynomial noise flooding or on bootstrapping. In this work, we present a conceptually different approach to circuit privacy based on a novel characterization of the noise growth amidst homomorphic evaluation. In particular, we show that a variant of the GSW FHE for branching programs already achieves circuit privacy; this immediately yields a circuit-private FHE for NC1 circuits under the standard LWE assumption with polynomial modulus-to-noise ratio. Our analysis relies on a variant of the discrete Gaussian leftover hash lemma which states that e G −1 (v) + small noise does not depend on v. We believe that this result is of independent interest

PEPSI: Practically Efficient Private Set Intersection in the Unbalanced Setting

Two parties with private data sets can find shared elements using a Private Set Intersection (PSI) protocol without revealing any information beyond the intersection. Circuit PSI protocols privately compute an arbitrary function of the intersection - such as its cardinality, and are often employed in an unbalanced setting where one party has more data than the other. Existing protocols are either computationally inefficient or require extensive server-client communication on the order of the larger set. We introduce Practically Efficient PSI or PEPSI, a non-interactive solution where only the client sends its encrypted data. PEPSI can process an intersection of 1024 client items with a million server items in under a second, using less than 5 MB of communication. Our work is over 4 orders of magnitude faster than an existing non-interactive circuit PSI protocol and requires only 10% of the communication. It is also up to 20 times faster than the work of Ion et al., which computes a limited set of functions and has communication costs proportional to the larger set. Our work is the first to demonstrate that non-interactive circuit PSI can be practically applied in an unbalanced setting

Integral Matrix Gram Root and Lattice Gaussian Sampling Without Floats

Many advanced lattice based cryptosystems require to sample lattice points from Gaussian distributions. One challenge for this task is that all current algorithms resort to floating-point arithmetic (FPA) at some point, which has numerous drawbacks in practice: it requires numerical stability analysis, extra storage for high-precision, lazy/backtracking techniques for efficiency, and may suffer from weak determinism which can completely break certain schemes. In this paper, we give techniques to implement Gaussian sampling over general lattices without using FPA. To this end, we revisit the approach of Peikert, using perturbation sampling. Peikert’s approach uses continuous Gaussian sampling and some decomposition Σ= A At of the target covariance matrix Σ. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix A with real (not integer) entries. Our idea, in a nutshell, is to replace this decomposition by an integral one. While there is in general no integer solution if we restrict A to being a square matrix, we show that such a decomposition can be efficiently found by allowing A to be wider (say n × 9n). This can be viewed as an extension of Lagrange’s four-square theorem to matrices. In addition, we adapt our integral decomposition algorithm to the ring setting: for power-of-2 cyclotomics, we can exploit the tower of rings structure for improved complexity and compactness

Functional Commitments for All Functions, with Transparent Setup and from SIS

A *functional commitment* scheme enables a user to concisely commit to a function from a specified family, then later concisely and verifiably reveal values of the function at desired inputs. Useful special cases, which have seen applications across cryptography, include vector commitments and polynomial commitments. To date, functional commitments have been constructed (under falsifiable assumptions) only for functions that are essentially *linear*, with one recent exception that works for arbitrarily complex functions. However, that scheme operates in a strong and non-standard model, requiring an online, trusted authority to generate special keys for any opened function inputs. In this work, we give the first functional commitment scheme for nonlinear functions---indeed, for *all functions* of any bounded complexity---under a standard setup and a falsifiable assumption. Specifically, the setup is ``transparent,\u27\u27 requiring only public randomness (and not any trusted entity), and the assumption is the hardness of the standard Short Integer Solution (SIS) lattice problem. Our construction also has other attractive features, including: *stateless updates* via generic composability; excellent *asymptotic efficiency* for the verifier, and also for the committer in important special cases like vector and polynomial commitments, via preprocessing; and *post-quantum security*, since it is based on SIS

Non-interactive zero-knowledge arguments for QMA, with preprocessing

A non-interactive zero-knowledge (NIZK) proof system for a language L∈NP allows a prover (who is provided with an instance x∈L, and a witness w for x) to compute a classical certificate π for the claim that x∈L such that π has the following properties: 1) π can be verified efficiently, and 2) π does not reveal any information about w, besides the fact that it exists (i.e. that x∈L). NIZK proof systems have recently been shown to exist for all languages in NP in the common reference string (CRS) model and under the learning with errors (LWE) assumption. We initiate the study of NIZK arguments for languages in QMA. Our first main result is the following: if LWE is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover who is successful in our protocol. We also show that any language in QMA has an (interactive) proof of quantum knowledge

Unbalanced Private Set Intersection from Homomorphic Encryption and Nested Cuckoo Hashing

Private Set Intersection (PSI) is a well-studied secure two-party computation problem in which a client and a server want to compute the intersection of their input sets without revealing additional information to the other party. With this work, we present nested Cuckoo hashing, a novel hashing approach that can be combined with additively homomorphic encryption (AHE) to construct an efficient PSI protocol for unbalanced input sets. We formally prove the security of our protocol against semi-honest adversaries in the standard model. Our protocol yields client computation and communication complexity that is sublinear in the server’s set size and is thus of interest to clients with limited resources. The implementation and empirical evaluation of our protocol using the exponential ElGamal and BGV/BFV encryption schemes attests to state-of-the-art practical performance

Simple Threshold (Fully Homomorphic) Encryption From LWE With Polynomial Modulus

The learning with errors (LWE) assumption is a powerful tool for building encryption schemes with useful properties, such as plausible resistance to quantum computers, or support for homomorphic computations. Despite this, essentially the only method of achieving threshold decryption in schemes based on LWE requires a modulus that is superpolynomial in the security parameter, leading to a large overhead in ciphertext sizes and computation time. In this work, we propose a (fully homomorphic) encryption scheme that supports a simple $t$-out-of-$n$ threshold decryption protocol while allowing for a polynomial modulus. The main idea is to use the Rényi divergence (as opposed to the statistical distance as in previous works) as a measure of distribution closeness. This comes with some technical obstacles, due to the difficulty of using the Rényi divergence in decisional security notions such as standard semantic security. We overcome this by constructing a threshold scheme with a weaker notion of one-way security and then showing how to transform any one-way threshold scheme into one guaranteeing indistinguishability-based security