We present distributed key generation and decryption protocols for an additively homomorphic cryptosystem based on class groups, improving on a similar system proposed by Braun, Damgård, and Orlandi at CRYPTO ‘23. Our key generation is similarly constant round but achieves lower communication complexity than the previous work. This improvement is in part the result of relaxing the reconstruction property required of the underlying integer verifiable secret sharing scheme. This eliminates the reliance on potentially costly proofs of knowledge in unknown order groups. We present a new method to batch zero-knowledge proofs in unknown order groups which strengthens these improvements. We also present a protocol which is proven secure against adaptive adversaries in the single inconsistent player (SIP) model. Our protocols are secure in the universal composability (UC) framework and provide guaranteed output delivery. We demonstrate the relative efficiency of our techniques by presenting the running times and communication costs associated with our implementation of the statically secure protocol and provide a direct comparison with alternate state of the art constructions.
We present distributed key generation and decryption protocols for an additively homomorphic cryptosystem based on class groups, improving on a similar system proposed by Braun, Damgård, and Orlandi at CRYPTO ‘23. Our key generation is similarly constant round but achieves lower communication complexity than the previous work. This improvement is in part the result of relaxing the reconstruction property required of the underlying integer verifiable secret sharing scheme. This eliminates the reliance on potentially costly proofs of knowledge in unknown order groups. We present a new method to batch zero-knowledge proofs in unknown order groups which strengthens these improvements. We also present a protocol which is proven secure against adaptive adversaries in the single inconsistent player (SIP) model. Our protocols are secure in the universal composability (UC) framework and provide guaranteed output delivery. We demonstrate the relative efficiency of our techniques by presenting the running times and communication costs associated with our implementation of the statically secure protocol and provide a direct comparison with alternate state of the art constructions.
We present distributed key generation and decryption protocols for an additively homomorphic cryptosystem based on class groups, improving on a similar system proposed by Braun, Damgård, and Orlandi at CRYPTO ‘23. Our key generation is similarly constant round but achieves lower communication complexity than the previous work. This improvement is in part the result of relaxing the reconstruction property required of the underlying integer verifiable secret sharing scheme. This eliminates the reliance on potentially costly proofs of knowledge in unknown order groups. We present a new method to batch zero-knowledge proofs in unknown order groups which strengthens these improvements. We also present a protocol which is proven secure against adaptive adversaries in the single inconsistent player (SIP) model. Our protocols are secure in the universal composability (UC) framework and provide guaranteed output delivery. We demonstrate the relative efficiency of our techniques by presenting the running times and communication costs associated with our implementation of the statically secure protocol and provide a direct comparison with alternate state of the art constructions.
We construct the first actively-secure threshold version of the cryptosystem based on class groups from the so-called CL framework (Castagnos and Laguillaumie, 2015). We show how to use our threshold scheme to achieve general universally composable (UC) secure multiparty computation (MPC) with only transparent set-up, i.e., with no secret trapdoors involved. On the way to our goal, we design new zero-knowledge (ZK) protocols with constant communication complexity for proving multiplicative relations between encrypted values. This allows us to use the ZK proofs to achieve MPC with active security with only a constant factor overhead. Finally, we adapt our protocol for the so called “You-Only-Speak-Once” (YOSO) setting, which is a very promising recent approach for performing MPC over a blockchain. This is possible because our key generation protocol is simpler and requires significantly less interaction compared to previous approaches: in particular, our new key generation protocol allows the adversary to bias the public key, but we show that this has no impact on the security of the resulting cryptosystem.
Cramer, Damgard, and Nielsen [CDN01] show how to construct an efficient secure multiparty computation scheme using a threshold homomorphic encryption scheme that has four properties i) a honest-verifier zero-knowledge proof of knowledge of encrypted values, ii) proving multiplications correct iii) threshold decryption and iv) trusted shared key setup. Naor and Nissim [NN01a] show how to construct secure multi-party protocols for a function f whose communication is proportional to the communication required to evaluate f without security, albeit at the cost of computation that might be exponential in the description of f . Gentry [Gen09a] shows how to combine both ideas with fully homomorphic encryption in order to construct secure multi-party protocol that allows evaluation of a function f using communication that is independent of the circuit description of f and computation that is polynomial in | f |. This paper addresses the major drawback’s of Gentry’s approach: we eliminate the use of non-black box methods that are inherent in Naor and Nissim’s compiler. To do this we show how to modify the fully homomorphic encryption construction of van Dijk et al. [vDGHV10] to be threshold fully homomorphic encryption schemes. We directly construct (information theoretically) secure protocols for sharing the secret key for our threshold scheme (thereby removing the setup assumptions) and for jointly decrypting a bit. All of these constructions are constant round and we thoroughly analyze their complexity; they address requirements (iii) and (iv). The fact that the encryption scheme is fully homomorphic addresses requirement (ii). To address the need for an honest-verifier zero-knowledge proof of knowledge of encrypted values, we instead argue that a weaker solution suffices. We provide a 2-round blackbox protocol that allows us to prove knowledge of encrypted bits. Our protocol is not zeroknowledge, but it provably does not release any information about the bit being discussed, and this is sufficient to prove the correctness of a simulation in a method similar to Cramer et al. Altogether, we construct the first black-box secure multi-party computation protocol that allows evaluation of a function f using communication that is independent of the circuit description of f .
We present a new compact verifiable secret sharing scheme, based on this we present the first construction of a homomorphic UC commitment scheme that requires only cheap symmetric cryptography, except for a small number of seed OTs. To commit to a k -bit string, the amortized communication cost is O ( k ) bits. Assuming a sufficiently efficient pseudorandom generator, the computational complexity is O ( k ) for the verifier and O ( k 1 + ε ) for the committer (where ε < 1 is a constant). In an alternative variant of the construction, all complexities are O ( k · polylog ( k )). Our commitment scheme extends to vectors over any finite field and is additively homomorphic. By sending one extra message, the prover can allow the verifier to also check multiplicative relations on committed strings, as well as verifying that committed vectors a , b satisfy a = φ ( b ) for a linear function φ . These properties allow us to non-interactively implement any one-sided functionality where only one party has input (this includes UC secure zero-knowledge proofs of knowledge). We also present a perfectly secure implementation of any multiparty functionality, based directly on our VSS. The communication required is proportional to a circuit implementing the functionality, up to a logarithmic factor. For a large natural class of circuits the overhead is even constant. We also improve earlier results by Ranellucci et al. on the amount of correlated randomness required for string commitments with individual opening of bits. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract. We present a new constant round additively homomorphic commitment scheme with (amortized) computational and communication complexity linear in the size of the string committed to. Our scheme is based on the non-homomorphic commitment scheme of Cascudo et al. presented at PKC 2015. However, we manage to add the additive homomorphic property, while at the same time reducing the constants. In fact, when opening a large enough batch of commitments we achieve an amortized communication complexity converging to the length of the message committed to, i.e., we achieve close to rate 1 as the commitment protocol by Garay et al. from Eurocrypt 2014. A main technical improvement over the scheme mentioned above, and other schemes based on using error correcting codes for UC commitment, we develop a new technique which allows to based the extraction property on erasure decoding as opposed to error correction. This allows to use a code with significantly smaller minimal distance and allows to use codes without efficient decoding. Our scheme only relies on standard assumptions. Specifically we require a pseudorandom number generator, a linear error correcting code and an ideal oblivious transfer functionality. Based on this we prove our scheme secure in the Universal Composability (UC) framework against a static and malicious adversary corrupting any number of parties. On a practical note, our scheme improves significantly on the non-homomorphic scheme of Cascudo et al. Based on their observations in regards to efficiency of using linear error correcting codes for commitments we conjecture that our commitment scheme might in practice be more efficient than all existing constructions of UC commitment, even non-homomorphic constructions and even constructions in the random oracle model. In particular, the amortized price of computing one of our commitments is less than that of evaluating a hash function once.
Abstract. Fully homomorphic encryption (FHE) enables secure computation over the encrypted data of a single party. We explore how to extend this to multiple parties, using threshold fully homomorphic encryption (TFHE). In such scheme, the parties jointly generate a common FHE public key along with a secret key that is shared among them; they can later cooperatively decrypt ciphertexts without learning anything but the plaintext. We show how to instantiate this approach efficiently, by extending the recent FHE schemes of Brakerski, Gentry and Vaikuntanathan (CRYPTO ’11, FOCS ’11, ITCS ’12) based on the (ring) learning with errors assumption. Our main tool is to exploit the property that such schemes are additively homomorphic over their keys. Using TFHE, we construct simple multiparty computation protocols secure against fully malicious attackers, tolerating any number of corruptions, and providing security in the universal composability framework. Our protocols have the following properties: Low interaction: 3 rounds of interaction given a common random string, or 2 rounds with a public-key infrastructure. Low communication: independent of the function being computed (proportional to just input and output sizes). Cloud-assisted computation: the bulk of the computation can be efficiently outsourced to an external entity (e.g. a cloud service) so that the computation of all other parties is independent of the complexity of the evaluated function. 1