International Association
for Cryptologic Research

Pseudorandom Correlation Functions (PCFs) allow two parties, given correlated evaluation keys, to locally generate arbitrarily many pseudorandom correlated strings, e.g. Oblivious Transfer (OT) correlations, which can then be used by the two parties to jointly run secure computation protocols. In this work, we provide a novel and simple approach for constructing PCFs for OT correlation, by relying on constrained pseudorandom functions for a class of constraints containing a weak pseudorandom function (wPRF). We then show that tweaking the Naor-Reingold pseudorandom function and relying on low-complexity pseudorandom functions allow us to instantiate our paradigm. We further extend our ideas to obtain efficient public-key PCFs, which allow the distribution of correlated keys between parties to be non-interactive: each party can generate a pair of public/secret keys, and any pair of parties can locally derive their correlated evaluation key by combining their secret key with the other party’s public key. In addition to these theoretical contributions, we detail various optimizations and provide concrete instantiations of our paradigm relying on the Boneh-Ishai-Passelègue-Sahai-Wu wPRF and the Goldreich-Applebaum-Raykov wPRF. Putting everything together, we obtain public-key PCFs with a throughput of 15k-40k OT/s, which is of a similar order of magnitude to the state-of-the-art interactive PCFs and about 4 orders of magnitude faster than state-of-the-art public-key PCFs. As a side result, we also show that public-key PCFs can serve as a building block to construct reusable designated-verifier non-interactive zero-knowledge proofs (DV-NIZK) for NP. Combined with our instantiations, this yields simple and efficient reusable DV-NIZKs for NP in pairing-free groups.

We study constructions of threshold fully homomorphic encryption with small partial decryption shares. In this context, we discuss in details the technicalities for achieving full-fledged threshold FHE, and bring light to limitations regarding prior works, inclucing an attack against the recent construction from Boudgoust and Scholl [ASIACRYPT 2023]. In light of our observations, we generalize the definition of threshold fully homomorphic encryption by adding an algorithm which allows to sanitize evaluated ciphertexts before they are decrypted by parties. In this setting, we are able to propose a construction which offers small partial decryption shares and avoids exponential noise flooding during partial decryption. In addition, we also propose an alternative protocol based on circuit-private (non-threshold) FHE and threshold public-key encryption for private delegation of computation on joint data, but requires an additional round of communication compared to our threshold FHE construction.

Abstract. We describe an adaptation of Schnorr’s signature to the lattice setting, which relies on Gaussian convolution rather than flooding or rejection sampling as previous approaches. It does not involve any abort, can be proved secure in the ROM and QROM using existing analyses of the Fiat-Shamir transfom, and enjoys smaller signature sizes (both asymptotically and for concrete security levels).

We propose and analyze a simple strategy for constructing 1-key constrained pseudorandom functions (CPRFs) from homomorphic secret sharing. In the process, we obtain the following contributions: first, we identify desirable properties for the underlying HSS scheme for our strategy to work. Second, we show that (most of) recent existing HSS schemes satisfy these properties, leading to instantiations of CPRFs for various constraints and from various assumptions. Notably, we obtain the first (1-key selectively secure, private) CPRFs for inner-product and (1-key selectively secure) CPRFs for NC 1 from the DCR assumption, and more. Last, we revisit two applications of HSS equipped with these additional properties to secure computation: we obtain secure computation in the silent preprocessing model with one party being able to precompute its whole preprocessing material before even knowing the other party, and we construct one-sided statistically secure computation with sublinear communication for restricted forms of computation.

Lyubashevky’s signatures are based on the Fiat-Shamir with Aborts paradigm. It transforms an interactive identification protocol that has a non-negligible probability of aborting into a signature by repeating executions until a loop iteration does not trigger an abort. Interaction is removed by replacing the challenge of the verifier by the evaluation of a hash function, modeled as a random oracle in the analysis. The access to the random oracle is classical (ROM), resp. quantum (QROM), if one is interested in security against classical, resp. quantum, adversaries. Most analyses in the literature consider a setting with a bounded number of aborts (i.e., signing fails if no signature is output within a prescribed number of loop iterations), while practical instantiations (e.g., Dilithium) run until a signature is output (i.e., loop iterations are unbounded). In this work, we emphasize that combining random oracles with loop iterations induces numerous technicalities for analyzing correctness, runtime, and security of the resulting schemes, both in the bounded and unbounded case. As a first contribution, we put light on errors in all existing analyses. We then provide two detailed analyses in the QROM for the bounded case, adapted from Kiltz et al [EUROCRYPT’18] and Grilo et al [ASIACRYPT’21]. In the process, we prove the underlying Σ-protocol to achieve a stronger zero-knowledge property than usually considered for Σ-protocols with aborts, which enables a corrected analysis. A further contribution is a detailed analysis in the case of unbounded aborts, the latter inducing several additional subtleties.

Updatable public key encryption has recently been introduced as a solution to achieve forward-security in the context of secure group messaging without hurting efficiency, but so far, no efficient lattice-based instantiation of this primitive is known. In this work, we construct the first LWE-based UPKE scheme with polynomial modulus-to-noise rate, which is CPA-secure in the standard model. At the core of our security analysis is a generalized reduction from the standard LWE problem to (a stronger version of) the Extended LWE problem. We further extend our construction to achieve stronger security notions by proposing two generic transforms. Our first transform allows to obtain CCA security in the random oracle model and adapts the Fujisaki-Okamoto transform to the UPKE setting. Our second transform allows to achieve security against malicious updates by adding a NIZK argument in the update mechanism. In the process, we also introduce the notion of Updatable Key Encapsulation Mechanism (UKEM), as the updatable variant of KEMs. Overall, we obtain a CCA-secure UKEM in the random oracle model whose ciphertext sizes are of the same order of magnitude as that of CRYSTALS-Kyber.

Lyubashevsky’s signatures are based on the Fiat-Shamir with aborts paradigm, whose central ingredient is the use of rejection sampling to transform secret-dependent signature samples into samples from (or close to) a secret-independent target distribution. Several choices for the underlying distributions and for the rejection sampling strategy can be considered. In this work, we study Lyubashevsky’s signatures through the lens of rejection sampling, and aim to minimize signature size given signing runtime requirements. Several of our results concern rejection sampling itself and could have other applications. We prove lower bounds for compactness of signatures given signing run- time requirements, and for expected runtime of perfect rejection sampling strategies. We also propose a Rényi-divergence-based analysis of Lyuba- shevsky’s signatures which allows for larger deviations from the target distribution, and show hyperball uniforms to be a good choice of distri- butions: they asymptotically reach our compactness lower bounds and offer interesting features for practical deployment. Finally, we propose a different rejection sampling strategy which circumvents the expected runtime lower bound and provides a worst-case runtime guarantee.

Vector commitments allow a user to commit to a vector of length n using a constant-size commitment while being able to locally open the commitment to individual vector coordinates. Importantly, the size of position-wise openings should be independent of the dimension n. Gorbunov, Reyzin, Wee, and Zhang recently proposed PointProofs (CCS 2020), a vector commitment scheme that supports non-interactive aggregation of proofs across multiple commitments, allowing to drastically reduce the cost of block propagation in blockchain smart contracts. Gorbunov et al. provide a security analysis combining the algebraic group model and the random oracle model, under the weak n-bilinear Diffie- Hellman Exponent assumption (n-wBDHE) assumption. In this work, we propose a novel analysis that does not rely on the algebraic group model. We prove the security in the random oracle model under the n- Diffie-Hellman Exponent (n-DHE) assumption, which is implied by the n-wBDHE assumption considered by Gorbunov et al. We further note that we do not modify their scheme (and thus preserve its efficiency) nor introduce any additional assumption. Instead, we prove the security of the scheme as it is via a strictly improved analysis.

Non-interactive zero-knowledge proofs (NIZKs) are important primitives in cryptography. A major challenge since the early works on NIZKs has been to construct NIZKs with a statistical zero-knowledge guarantee against unbounded verifiers. In the common reference string (CRS) model, such "statistical NIZK arguments" are currently known from k-Lin in a pairing-group and from LWE. In the (reusable) designated-verifier model (DV-NIZK), where a trusted setup algorithm generates a reusable verification key for checking proofs, we also have a construction from DCR. If we relax our requirements to computational zero-knowledge, we additionally have NIZKs from factoring and CDH in a pairing group in the CRS model, and from nearly all assumptions that imply public-key encryption (e.g., CDH, LPN, LWE) in the designated-verifier model. Thus, there still remains a gap in our understanding of statistical NIZKs in both the CRS and the designated-verifier models. In this work, we develop new techniques for constructing statistical NIZK arguments. First, we construct statistical DV-NIZK arguments from the k-Lin assumption in pairing-free groups, the QR assumption, and the DCR assumption. These are the first constructions in pairing-free groups and from QR that satisfy statistical zero-knowledge. All of our constructions are secure even if the verification key is chosen maliciously (i.e., they are "malicious-designated-verifier" NIZKs), and moreover, they satisfy a "dual-mode" property where the CRS can be sampled from two computationally indistinguishable distributions: one distribution yields statistical DV-NIZK arguments while the other yields computational DV-NIZK proofs. We then show how to adapt our k-Lin construction in a pairing group to obtain new publicly-verifiable statistical NIZK arguments from pairings with a qualitatively weaker assumption than existing constructions of pairing-based statistical NIZKs. Our constructions follow the classic paradigm of Feige, Lapidot, and Shamir (FLS). While the FLS framework has traditionally been used to construct computational (DV)-NIZK proofs, we newly show that the same framework can be leveraged to construct dual-mode (DV)-NIZKs.

The Naor-Yung paradigm is a well-known technique that constructs IND-CCA2-secure encryption schemes by means of non-interactive zero-knowledge proofs satisfying a notion of simulation-soundness. Until recently, it was an open problem to instantiate it under the sole Learning-With-Errors (LWE) assumption without relying on random oracles. While the recent results of Canetti et al. (STOC'19) and Peikert-Shiehian (Crypto'19) provide a solution to this problem by applying the Fiat-Shamir transform in the standard model, the resulting constructions are extremely inefficient as they proceed via a reduction to an NP-complete problem. In this paper, we give a direct, non-generic method for instantiating Naor-Yung under the LWE assumption outside the random oracle model. Specifically, we give a direct construction of an unbounded simulation-sound NIZK argument system which, for carefully chosen parameters, makes it possible to express the equality of plaintexts encrypted under different keys in Regev's cryptosystem. We also give a variant of our argument that provides tight security. As an application, we obtain an LWE-based public-key encryption scheme for which we can prove (tight) key-dependent message security under chosen-ciphertext attacks in the standard model.

In the last decade, several works have focused on finding the best way to model the leakage in order to obtain provably secure implementations. One of the most realistic models is the noisy leakage model, introduced in [PR13, DDF14] together with secure constructions. These works suffer from various limitations, in particular the use of ideal leak-free gates in [PR13] and an important loss (in the size of the field) in the reduction in [DDF14].In this work, we provide new strategies to prove the security of masked implementations and start by unifying the different noisiness metrics used in prior works by relating all of them to a standard notion in information theory: the pointwise mutual information. Based on this new interpretation, we define two new natural metrics and analyze the security of known compilers with respect to these metrics. In particular, we prove (1) a tighter bound for reducing the noisy leakage models to the probing model using our first new metric, (2) better bounds for amplification-based security proofs using the second metric.To support that the improvements we obtain are not only a consequence of the use of alternative metrics, we show that for concrete representation of leakage (e.g., “Hamming weight + Gaussian noise”), our approach significantly improves the parameters compared to prior works. Finally, using the Rényi divergence, we quantify concretely the advantage of an adversary in attacking a block cipher depending on the number of leakage acquisitions available to it.

Due to the vast number of successful related-key attacks against existing block-ciphers, related-key security has become a common design goal for such primitives. In these attacks, the adversary is not only capable of seeing the output of a function on inputs of its choice, but also on related keys. At Crypto 2010, Bellare and Cash proposed the first construction of a pseudorandom function that could provably withstand such attacks based on standard assumptions. Their construction, as well as several others that appeared more recently, have in common the fact that they only consider linear or polynomial functions of the secret key over complex groups. In reality, however, most related-key attacks have a simpler form, such as the XOR of the key with a known value. To address this problem, we propose the first construction of RKA-secure pseudorandom function for XOR relations. Our construction relies on multilinear maps and, hence, can only be seen as a feasibility result. Nevertheless, we remark that it can be instantiated under two of the existing multilinear-map candidates since it does not reveal any encodings of zero. To achieve this goal, we rely on several techniques that were used in the context of program obfuscation, but we also introduce new ones to address challenges that are specific to the related-key-security setting.

Functional encryption lies at the frontiers of the current research in cryptography; some variants have been shown sufficiently powerful to yield indistinguishability obfuscation (IO), while other variants have been constructed from standard assumptions such as LWE. Indeed, most variants have been classified as belonging to either the former or the latter category. However, one mystery that has remained is the case of secret-key functional encryption with an unbounded number of keys and ciphertexts. On the one hand, this primitive is not known to imply anything outside of minicrypt, the land of secret-key cryptography, but, on the other hand, we do no know how to construct it without the heavy hammers in obfustopia. In this work, we show that (subexponentially secure) secret-key functional encryption is powerful enough to construct indistinguishability obfuscation if we additionally assume the existence of (subexponentially secure) plain public-key encryption. In other words, secret-key functional encryption provides a bridge from cryptomania to obfustopia. On the technical side, our result relies on two main components. As our first contribution, we show how to use secret-key functional encryption to get “exponentially efficient indistinguishability obfuscation” (XIO), a notion recently introduced by Lin et al. (PKC’16) as a relaxation of IO. Lin et al. show how to use XIO and the LWE assumption to build IO. As our second contribution, we improve on this result by replacing its reliance on the LWE assumption with any plain public-key encryption scheme. Lastly, we ask whether secret-key functional encryption can be used to construct public-key encryption itself and therefore take us all the way from minicrypt to obfustopia. A result of Asharov and Segev (FOCS’15) shows that this is not the case under black-box constructions, even for exponentially secure functional encryption. We show, through a non-black-box construction, that subexponentially secure-key functional encryption indeed leads to public-key encryption. The resulting public-key encryption scheme, however, is at most quasi-polynomially secure, which is insufficient to take us to obfustopia.

Pseudorandom functions (PRFs) are one of the fundamental building blocks in cryptography. Traditionally, there have been two main approaches for PRF design: the “practitioner’s approach” of building concretely-efficient constructions based on known heuristics and prior experience, and the “theoretician’s approach” of proposing constructions and reducing their security to a previously-studied hardness assumption. While both approaches have their merits, the resulting PRF candidates vary greatly in terms of concrete efficiency and design complexity.In this work, we depart from these traditional approaches by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits—specifically, depth-2$$\mathsf {ACC}^0$$ circuits. Concretely, our main weak PRF candidate is a “piecewise-linear” function that first applies a secret mod-2 linear mapping to the input, and then a public mod-3 linear mapping to the result. We also put forward a similar depth-3 strong PRF candidate.The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory (e.g., learnability of $$\mathsf {ACC}^0$$ and width-3 branching programs, interpolation and property testing for sparse polynomials, and new natural proof barriers for showing super-linear circuit lower bounds). On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multiparty computation (MPC). Using our PRF candidates, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity (often both) compared to protocols obtained by combining standard MPC protocols with PRFs like AES, LowMC, or Rasta (the latter two are specialized MPC-friendly PRFs).Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We conclude by showing that our main weak PRF candidate can plausibly be boosted to an encoded-input PRF by leveraging standard error-correcting codes.