International Association
for Cryptologic Research

Learning parity with noise (LPN) has been widely studied and used in cryptography. It was recently brought to new prosperity since Boyle et al. (CCS'18), putting LPN to a central role in designing secure multi-party computation, zero-knowledge proofs, private set intersection, and many other protocols. In this paper, we thoroughly studied security of LPN problems in this particular context. We found that some important aspects are long ignored and many conclusions from classical LPN cryptanalysis do not apply to this new setting, due to the low noise rates, extremely high dimensions, various types (in addition to $\FF_2$) and noise distributions. For LPN over a field, we give a parameterized reduction from exact-noise LPN to regular-noise LPN. Compared to the recent result by Feneuil, Joux and Rivain (Crypto'22), we significantly reduce the security loss by paying only a small additive price in dimension and number of samples. We analyze the security of LPN over a ring $\ZZ_{2^\lambda}$. Existing protocols based on LPN over integer rings use parameters as if they are over fields, but we found an attack that effectively reduces the weight of a noise by half compared to LPN over fields. Consequently, prior works that use LPN over $\ZZ_{2^\lambda}$ overestimate up to 40 bits of security. We provide a complete picture of the hardness of LPN over integer rings by showing: 1) the equivalence between its search and decisional versions; 2) an efficient reduction from LPN over $\FF_{2}$ to LPN over $\ZZ_{2^\lambda}$; and 3) generalization of our results to any integer ring. Finally, we provide an all-in-one estimator tool for the bit security of LPN parameters in the context of PCG, incorporating the recent advanced attacks.

GGM tree is widely used in designing correlated oblivious transfer (COT), subfield vector oblivious linear evaluation (sVOLE), distributed point function (DPF), and distributed comparison function (DCF). Often, the computation and communication cost associated with GGM tree is the major cost in these protocols. In this paper, we propose a suite of optimizations that can reduce this cost by half. - Halving the cost of COT and sVOLE. Our basic protocol introduces extra correlation in each level of a GGM tree used by the state-of-the-art COT protocol. As a result, it reduces the number of AES calls and the communication by half. Extending the idea to sVOLE, we are able to achieve similar improvement with either halved computation or halved communication. - Halving the cost of DPF and DCF. We propose improved two-party protocols for distributed generation of DPF/DCF keys. Our tree structures behind these protocols lead to more efficient full-domain evaluation and halve the communication and the round complexity of the state-of-the-art DPF/DCF protocols. All protocols are provably secure in the random-permutation model and can be accelerated based on fixed-key AES-NI. We also improve the state-of-the-art schemes of puncturable pseudorandom function (PPRF), DPF, and DCF, which are of independent interest in dealer-available scenarios.

Actively secure two-party computation (2PC) is one of the canonical building blocks in modern cryptography. One main goal for designing actively secure 2PC protocols is to reduce the communication overhead, compared to semi-honest 2PC protocols. In this paper, we propose a new actively secure constant-round 2PC protocol with one-way communication of $2\kappa+5$ bits per AND gate (for $\kappa$-bit computational security and any statistical security), essentially matching the one-way communication of semi-honest half-gates protocol. This is achieved by two new techniques: - The recent compression technique by Dittmer et al. (Crypto 2022) shows that a relaxed preprocessing is sufficient for authenticated garbling that does not reveal masked wire values to the garbler. We introduce a new form of authenticated bits and propose a new technique of generating authenticated AND triples to reduce the one-way communication of preprocessing from $5\rho+1$ bits to $2$ bits per AND gate for $\rho$-bit statistical security. - Unfortunately, the above compressing technique is only compatible with a less compact authenticated garbled circuit of size $2\kappa+3\rho$ bits per AND gate. We designed a new authenticated garbling that does not use information theoretic MACs but rather dual execution without leakage to authenticate wire values in the circuit. This allows us to use a more compact half-gates based authenticated garbled circuit of size $2\kappa+1$ bits per AND gate, and meanwhile keep compatible with the compression technique. Our new technique can achieve one-way communication of $2\kappa+5$ bits per AND gate. Our technique of yielding authenticated AND triples can also be used to optimize the two-way communication (i.e., the total communication) by combining it with the authenticated garbled circuits by Dittmer et al., which results in an actively secure 2PC protocol with two-way communication of $2\kappa+3\rho+4$ bits per AND gate.

In this paper, we study zero-knowledge (ZK) proofs for circuit satisfiability that can prove to $n$ verifiers at a time efficiently. The proofs are secure against the collusion of a prover and a subset of $t$ verifiers. We refer to such ZK proofs as multi-verifier zero-knowledge (MVZK) proofs and focus on the case that a majority of verifiers are honest (i.e., $t<n/2$). We construct efficient MVZK protocols in the random oracle model where the prover sends one message to each verifier, while the verifiers only exchange one round of messages. When the threshold of corrupted verifiers $t<n/2$, the prover sends $1/2+o(1)$ field elements per multiplication gate to every verifier; when $t<n(1/2-\epsilon)$ for some constant $0<\epsilon<1/2$, we can further reduce the communication to $O(1/n)$ field elements per multiplication gate per verifier. Our MVZK protocols demonstrate particularly high scalability: the proofs are streamable and only require a memory proportional to what is needed to evaluate the circuit in the clear.

Non-interactive zero knowledge (NIZK) enables a prover, to prove that a statement in an NP language is valid, given an accepting witness, without leaking any information about the witness. We study universally composable (UC) NIZKs which are secure against adaptive corruption of parties and provides adaptive soundness, i.e. the statement is adaptively chosen by a malicious prover based on the setup string distribution. The only known adaptively secure NIZK protocols either fail to achieve full adaptive soundness or rely on non-falsifiable knowledge assumptions. We construct the first NIZK protocols which are triply adaptive - secure against adaptive corruptions, guarantees adaptive soundness and satisfies adaptive zero knowledge, from falsifiable assumptions. We do so using the following methodology: - We define a new ideal functionality, denoted as F_NICOM, for non-interactive commitment schemes in the UC framework. - We define and construct Sigma protocols which satisfy triply adaptive security in the F_NICOM model. - By relying on correlation intractable (CI) hash functions, we compile a triply adaptively secure Sigma protocol (in F_NICOM model) into a triply adaptive UC-NIZK argument in the F_NICOM+common reference string (crs) model. In addition to CI hash functions, our compiler requires standard cryptographic primitives - non-interactive equivocal commitments and public key encryption with obliviously samplable ciphertexts, for implementing F_NICOM in the crs model. We instantiate our framework by demonstrating that most statically secure Sigma protocols can be proven to be triply adaptively secure in the F_NICOM model, hence, bridging the gap between static and adaptive security for NIZKs. Our NIZK arguments can be concretely based on assumptions, like LWE, or LPN and DDH.

Motivated by the recent trend towards low multiplicative complexity blockciphers (e.g., Zorro, CHES 2013; LowMC, EUROCRYPT 2015; HADES, EUROCRYPT 2020; MALICIOUS, CRYPTO 2020), we study their underlying structure partial SPNs, i.e., Substitution-Permutation Networks (SPNs) with parts of the substitution layer replaced by an identity mapping, and put forward the first provable security analysis for such partial SPNs built upon dedicated linear layers. For different instances of partial SPNs using MDS linear layers, we establish strong pseudorandom security as well as practical provable security against impossible differential attacks. By extending the well-established MDS code-based idea, we also propose the first principled design of linear layers that ensures optimal differential propagation. Our results formally confirm the conjecture that partial SPNs achieve the same security as normal SPNs while consuming less non-linearity, in a well-established framework.

Oblivious Transfer (OT) is an important building block for multi-party computation (MPC). Since OT requires expensive public-key operations, efficiency-conscious MPC protocols use an OT extension (OTE) mechanism [Beaver 96, Ishai et al. 03] to provide the functionality of many independent OT instances with the same sender and receiver, using only symmetric-key operations plus few instances of some base OT protocol. Consequently there is significant interest in constructing OTE friendly protocols, namely protocols that, when used as base-OT for OTE, result in extended OT that are both round-efficient and cost-efficient. We present the most efficient OTE-friendly protocol to date. Specifically: Our base protocol incurs only 3 exponentiations per instance. Our base protocol results in a 3 round extended OT protocol. The extended protocol is UC secure in the Observable Random Oracle Model (ROM) under the CDH assumption. For comparison, the state of the art for base OTs that result in 3-round OTE are proven only in the programmable ROM, and require 4 exponentiations under Interactive DDH or 6 exponentiations under DDH [Masney-Rindal 19]. We also implement our protocol and benchmark it against the Simplest OT protocol [Chou and Orlandi, Latincrypt 2015], which is the most efficient and widely used OT protocol but not known to suffice for OTE. The computation cost is roughly the same in both cases. Interestingly, our base OT is also 3 rounds. However, we slightly modify the extension mechanism (which normally adds a round) so as to preserve the number of rounds in our case.

We study the concrete security of high-performance implementations of half-gates garbling, which all rely on (hardware-accelerated) AES. We find that current instantiations using k-bit wire labels can be completely broken—in the sense that the circuit evaluator learns all the inputs of the circuit garbler—in time O(2k/C), where C is the total number of (non-free) gates that are garbled, possibly across multiple independent executions. The attack can be applied to existing circuit-garbling libraries using k = 80 when C ≈ $10^9$, and would require 267 machine-months and cost about $3500 to implement on the Google Cloud Platform. Since the attack can be entirely parallelized, the attack could be carried out in about a month using ≈ 250 machines. With this as our motivation, we seek a way to instantiate the hash function in the half-gates scheme so as to achieve better concrete security. We present a construction based on AES that achieves optimal security in the single-instance setting (when only a single circuit is garbled). We also show how to modify the half-gates scheme so that its concrete security does not degrade in the multi-instance setting. Our modified scheme is as efficient as prior work in networks with up to 2 Gbps bandwidth.

We construct the most efficient two-round adaptively secure bit-OT in the Common Random String (CRS) model. The scheme is UC secure under the Decisional Diffie-Hellman (DDH) assumption. It incurs O(1) exponentiations and sends O(1) group elements, whereas the state of the art requires O(k^2) exponentiations and communicates poly(k) bits, where k is the computational security parameter. Along the way, we obtain several other efficient UC-secure OT protocols under DDH : - The most efficient yet two-round adaptive string-OT protocol assuming global programmable random oracle. Furthermore, the protocol can be made non-interactive in the simultaneous message setting, assuming random inputs for the sender. - The first two-round string-OT with amortized constant exponentiations and communication overhead which is secure in the global observable random oracle model. - The first two-round receiver equivocal string-OT in the CRS model that incurs constant computation and communication overhead. We also obtain the first non-interactive adaptive string UC-commitment in the CRS model which incurs a sublinear communication overhead in the security parameter. Specically, we commit to polylog(k) bits while communicating O(k) bits. Moreover, it is additively homomorphic. We can also extend our results to the single CRS model where multiple sessions share the same CRS. As a corollary, we obtain a two-round adaptively secure MPC protocol in this model.

The notion of covert security for secure two-party computation serves as a compromise between the traditional semi-honest and malicious security definitions. Roughly, covert security ensures that cheating behavior is detected by the honest party with reasonable probability (say, 1/2). It provides more realistic guarantees than semi-honest security with significantly less overhead than is required by malicious security.The rationale for covert security is that it dissuades cheating by parties that care about their reputation and do not want to risk being caught. But a much stronger disincentive is obtained if the honest party can generate a publicly verifiable certificate when cheating is detected. While the corresponding notion of publicly verifiable covert (PVC) security has been explored, existing PVC protocols are complex and less efficient than the best covert protocols, and have impractically large certificates.We propose a novel PVC protocol that significantly improves on prior work. Our protocol uses only “off-the-shelf” primitives (in particular, it avoids signed oblivious transfer) and, for deterrence factor 1/2, has only 20–40% overhead compared to state-of-the-art semi-honest protocols. Our protocol also has, for the first time, constant-size certificates of cheating (e.g., 354 bytes long at the 128-bit security level).As our protocol offers strong security guarantees with low overhead, we suggest that it is the best choice for many practical applications of secure two-party computation.

We present a new efficient protocol for computing private set union (PSU). Here two semi-honest parties, each holding a dataset of known size (or of a known upper bound), wish to compute the union of their sets without revealing anything else to either party. Our protocol is in the OT hybrid model. Beyond OT extension, it is fully based on symmetric-key primitives. We motivate the PSU primitive by its direct application to network security and other areas.At the technical core of our PSU construction is the reverse private membership test (RPMT) protocol. In RPMT, the sender with input $$x^*$$ interacts with a receiver holding a set X. As a result, the receiver learns (only) the bit indicating whether $$x^* \in X$$, while the sender learns nothing about the set X. (Previous similar protocols provide output to the opposite party, hence the term “reverse” private membership.) We believe our RPMT abstraction and constructions may be a building block in other applications as well.We demonstrate the practicality of our proposed protocol with an implementation. For input sets of size $$2^{20}$$ and using a single thread, our protocol requires 238 s to securely compute the set union, regardless of the bit length of the items. Our protocol is amenable to parallelization. Increasing the number of threads from 1 to 32, our protocol requires only 13.1 s, a factor of $$18.25{\times }$$ improvement.To the best of our knowledge, ours is the first protocol that reports on large-size experiments, makes code available, and avoids extensive use of computationally expensive public-key operations. (No PSU code is publicly available for prior work, and the only prior symmetric-key-based work reports on small experiments and focuses on the simpler 3-party, 1-corruption setting.) Our work improves reported PSU state of the art by factor up to $$7,600{\times }$$ for large instances.

Wang et al. (CCS 2017) recently proposed a protocol for malicious secure two-party computation that represents the state-of-the-art with regard to concrete efficiency in both the single-execution and amortized settings, with or without preprocessing. We show here several optimizations of their protocol that result in a significant improvement in the overall communication and running time. Specifically:We show how to make the “authenticated garbling” at the heart of their protocol compatible with the half-gate optimization of Zahur et al. (Eurocrypt 2015). We also show how to avoid sending an information-theoretic MAC for each garbled row. These two optimizations give up to a 2.6$$\times $$× improvement in communication, and make the communication of the online phase essentially equivalent to that of state-of-the-art semi-honest secure computation.We show various optimizations to their protocol for generating AND triples that, overall, result in a 1.5$$\times $$× improvement in the communication and a 2$$\times $$× improvement in the computation for that step.

We construct new four-party protocols for secure computation that are secure against a single malicious corruption. Our protocols can perform computations over a binary ring, and require sending just 1.5 ring elements per party, per gate. In the special case of Boolean circuits, this amounts to sending 1.5 bits per party, per gate. One of our protocols is robust, yet requires almost no additional communication. Our key technique can be viewed as a variant of the “dual execution” approach, but, because we rely on four parties instead of two, we can avoid any leakage, achieving the standard notion of security.

We show a protocol for two-server oblivious RAM (ORAM) that is simpler and more efficient than the best prior work. Our construction combines any tree-based ORAM with an extension of a two-server private information retrieval scheme by Boyle et al., and is able to avoid recursion and thus use only one round of interaction. In addition, our scheme has a very cheap initialization phase, making it well suited for RAM-based secure computation. Although our scheme requires the servers to perform a linear scan over the entire data, the cryptographic computation involved consists only of block-cipher evaluations.A practical instantiation of our protocol has excellent concrete parameters: for storing an N-element array of arbitrary size data blocks with statistical security parameter $$\lambda $$, the servers each store 4N encrypted blocks, the client stores $$\lambda +2\log N$$ blocks, and the total communication per logical access is roughly $$10 \log N$$ encrypted blocks.