Lattice-Based Fully Dynamic Multi-Key FHE with Short Ciphertexts

We present a multi-key fully homomorphic encryption scheme that supports an unbounded number of homomorphic operations for an unbounded number of parties. Namely, it allows to perform arbitrarily many computational steps on inputs encrypted by an a-priori unbounded (polynomial) number of parties. Inputs from new parties can be introduced into the computation dynamically, so the final set of parties needs not be known ahead of time. Furthermore, the length of the ciphertexts, as well as the space complexity of an atomic homomorphic operation, grow only linearly with the current number of parties. Prior works either supported only an a-priori bounded number of parties (Lopez-Alt, Tromer and Vaikuntanthan, STOC \u2712), or only supported single-hop evaluation where all inputs need to be known before the computation starts (Clear and McGoldrick, Crypto \u2715, Mukherjee and Wichs, Eurocrypt \u2716). In all aforementioned works, the ciphertext length grew at least quadratically with the number of parties. Technically, our starting point is the LWE-based approach of previous works. Our result is achieved via a careful use of Gentry\u27s bootstrapping technique, tailored to the specific scheme. Our hardness assumption is that the scheme of Mukherjee and Wichs is circular secure (and thus bootstrappable). A leveled scheme can be achieved under standard LWE

Efficient Multi-key FHE with short extended ciphertexts and less public parameters

Multi-Key Full Homomorphic Encryption (MKFHE) can perform arbitrary operations on encrypted data under different public keys (users), and the final ciphertext can be jointly decrypted by all involved users. Therefore, MKFHE has natural advantages and application value in security multi-party computation (MPC). The MKFHE scheme based on Brakerski-Gentry-Vaikuntanathan (BGV) inherits the advantages of BGV FHE scheme in aspects of encrypting a ring element, the ciphertext/plaintext ratio, and supporting the Chinese Remainder Theorem (CRT)-based ciphertexts packing technique. However some weaknesses also exist such as large ciphertexts and keys, and complicated process of generating evaluation keys. In this paper, we present an efficient BGV-type MKFHE scheme. Firstly, we construct a nested ciphertext extension for BGV and separable ciphertext extension for Gentry-Sahai-Waters (GSW), which can reduce the size of the extended ciphertexts about a half. Secondly, we apply the hybrid homomorphic multiplication between RBGV ciphertext and RGSW ciphertext to the generation process of evaluation keys, which can significantly reduce the amount of input/output ciphertexts and improve the efficiency. Finally, we construct a directed decryption protocol which allows the evaluated ciphertext to be decrypted by any target user, thereby enhancing the ability of data owner to control their own plaintext, and abolish the limitation in current MKFHE schemes that the evaluated ciphertext can only be decrypted by users involved in homomorphic evaluation

PPP-Completeness with Connections to Cryptography

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups

Polynomials Whose Secret Shares Multiplication Preserves Degree for 2-CNF Circuits Over a Dynamic Set of Secrets

One of the most interesting research topics in cryptography is finding efficient homomorphic encryption schemes, preferably information-theoretically secure, which are not based on unproven computational hardness assumptions. The most significant breakthrough in this field was made by Craig Gentry in 2009, and since then, there were various developments. We suggest here an information-theoretically secure secret sharing scheme that efficiently supports one homomorphic multiplication of secrets in addition to homomorphic additions of, practically, any number of such multiplied secrets. In particular, our scheme enables sharing a dynamic set of secrets amongst N participants, using polynomials of degree N-1. Quadratic functions and 2-CNF circuits over the set of secrets can then be homomorphically evaluated, while no information is revealed to any single participant, both before and after the computation. Our scheme is statistically secure against coalitions of less than N-1 participants. One possible application of our scheme is a secure homomorphic evaluation of multi-variate quadratic functions and 2-CNF circuits

Homomorphic Encryption for Multiple Users with Less Communications

Keeping privacy for every entity in outsourced computation is always a crucial issue. For efficient secure computation, homomorphic encryption (HE) can be one of nice solutions. Especially, multikey homomorphic encryption (MKHE) which allows homomorphic evaluation on encrypted data under different keys can be one of the simplest solutions for a secure computation which handles multiple users\u27 data. However, the current main problem of MKHE is that the dimension of its evaluated ciphertext relies on the number of users. To solve this problem, there are several variants of multikey homomorphic encryption schemes to keep the size of ciphertext constant for a fixed number of users. However, users interact one another before computation to provide their inputs, which increases setup complexity. Moreover, all the existing MKHE schemes and their variants have unique benefits which cannot be easily achieved at the same time in one scheme. In other words, each type of scheme has a suitable computational scenario to put its best performance. In this paper, we suggest more efficient evaluation key generation algorithms (relinearization key and bootstrapping key) for the existing variants of MKHE schemes which have no ciphertext expansion for a fixed number of users. Our method only requires a very simple and minor pre-processing; distributing public keys, which is not counted as a round at all in many other applications. Regarding bootstrapping, we firstly provide an efficient bootstrapping for multiple users which is the same as the base single-key scheme thanks to our simplified key generation method without a communication. As a result, participants have less communication, computation, and memory cost in online phase. Moreover, we provide a practical conversion algorithm between the two types of schemes in order to \emph{efficiently} utilize both schemes\u27 advantages together in more various applications. We also provide detailed comparison among similar results so that users can choose a suitable scheme for their homomorphic encryption based application scenarios

On the IND-CCA1 Security of FHE Schemes

Fully homomorphic encryption (FHE) is a powerful tool in cryptography that allows one to perform arbitrary computations on encrypted material without having to decrypt it first. There are numerous FHE schemes, all of which are expanded from somewhat homomorphic encryption (SHE) schemes, and some of which are considered viable in practice. However, while these FHE schemes are semantically (IND-CPA) secure, the question of their IND-CCA1 security is much less studied, and we therefore provide an overview of the IND-CCA1 security of all acknowledged FHE schemes in this paper. To give this overview, we grouped the SHE schemes into broad categories based on their similarities and underlying hardness problems. For each category, we show that the SHE schemes are susceptible to either known adaptive key recovery attacks, a natural extension of known attacks, or our proposed attacks. Finally, we discuss the known techniques to achieve IND-CCA1-secure FHE and SHE schemes. We concluded that none of the proposed schemes were IND-CCA1-secure and that the known general constructions all had their shortcomings.publishedVersio

Asymptotically Faster Multi-Key Homomorphic Encryption from Homomorphic Gadget Decomposition

Homomorphic Encryption (HE) is a cryptosytem that allows us to perform an arbitrary computation on encrypted data. The standard HE, however, has a disadvantage in that the authority is concentrated in the secret key owner since computations can only be performed on ciphertexts encrypted under the same secret key. To resolve this issue, research is underway on Multi-Key Homomorphic Encryption (MKHE), which is a variant of HE supporting computations on ciphertexts possibly encrypted under different keys. Despite its ability to provide privacy for multiple parties, existing MKHE schemes suffer from poor performance due to the cost of multiplication which grows at least quadratically with the number of keys involved. In this paper, we revisit the work of Chen et al. (ACM CCS 2019) on MKHE schemes from CKKS and BFV and significantly improve their performance. Specifically, we redesign the multi-key multiplication algorithm and achieve an asymptotically optimal complexity that grows linearly with the number of keys. Our construction relies on a new notion of gadget decomposition, which we call homomorphic gadget decomposition, where arithmetic operations can be performed over the decomposed vectors with guarantee of its functionality. Finally, we implement our MKHE schemes and demonstrate their benchmarks. For example, our multi-key CKKS multiplication takes only 0.5, 1.0, and 1.9 seconds compared to 1.6, 5.9, and 23.0 seconds of the previous work when 8, 16, and 32 keys are involved, respectively

Key lifting : Multi-key Fully Homomorphic Encryption in plain model without noise flooding

Multi-key Fully Homomorphic Encryption (\MK), based on the Learning With Error assumption (\LWE), usually lifts ciphertexts of different users to new ciphertexts under a common public key to enable homomorphic evaluation. The efficiency of the current Multi-key Fully Homomorphic Encryption (\MK) scheme is mainly restricted by two aspects: Expensive ciphertext expansion operation: In a boolean circuit with input length $N$, multiplication depth $L$, security parameter $\lambda$, the number of additional encryptions introduced to achieve ciphertext expansion is $O(N\lambda^6L^4)$. Noise flooding technology resulting in a large modulus $q$ : In order to prove the security of the scheme, the noise flooding technology introduced in the encryption and distributed decryption stages will lead to a huge modulus $q = 2^{O(\lambda L)}B_\chi$, which corrodes the whole scheme and leads to sub-exponential approximation factors $\gamma = \tilde{O}(n\cdot 2^{\sqrt{nL}})$. This paper solves the first problem by presenting a framework called Key-Lifting Multi-key Fully Homomorphic Encryption (\KL). With this \emph{key lifting} procedure, the number of encryptions for a local user is reduced to $O(N)$, similar to single-key fully homomorphic encryption (\FHE). For the second problem, based on R\\u27{e}nyi divergence, we propose an optimized proof method that removes the noise flooding technology in the encryption phase. Additionally, in the distributed decryption phase, we prove that the asymmetric nature of the DGSW ciphertext ensures that the noise after decryption does not leak the noise in the initial ciphertext, as long as the depth of the circuit is sufficient. Thus, our initial ciphertext remains semantically secure even without noise flooding, provided the encryption scheme is leakage-resilient. This approach significantly reduces the size of the modulus $q$ (with $\log q = O(L)$) and the computational overhead of the entire scheme

A Practical TFHE-Based Multi-Key Homomorphic Encryption with Linear Complexity and Low Noise Growth

Fully Homomorphic Encryption enables arbitrary computations over encrypted data and it has a multitude of applications, e.g., secure cloud computing in healthcare or finance. Multi-Key Homomorphic Encryption (MKHE) further allows to process encrypted data from multiple sources: the data can be encrypted with keys owned by different parties. In this paper, we propose a new variant of MKHE instantiated with the TFHE scheme. Compared to previous attempts by Chen et al. and by Kwak et al., our scheme achieves computation runtime that is linear in the number of involved parties and it outperforms the faster scheme by a factor of 4.5-6.9x, at the cost of a slightly extended pre-computation. In addition, for our scheme, we propose and practically evaluate parameters for up to 128 parties, which enjoy the same estimated security as parameters suggested for the previous schemes (100 bits). It is also worth noting that our scheme—unlike the previous schemes—did not experience any error in any of our nine experiments, each running 1 000 trials