International Association for Cryptologic Research

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Papers from PKC 2024

Year
Venue
Title
2024
PKC
A Refined Hardness Estimation of LWE in Two-step Mode
Recently, researchers have proposed many LWE estimators, such as lattice-estimator (Albrecht et al, Asiacrypt 2017) and leaky-LWE-Estimator (Dachman-Soled et al, Crypto 2020), while the latter has already been used in estimating the security level of Kyber and Dilithium using only BKZ. However, we prove in this paper that solving LWE by combining a lattice reduction step (by LLL or BKZ) and a target vector searching step (by enumeration or sieving), which we call a Two-step mode, is more efficient than using only BKZ. Moreover, we give a refined LWE estimator in Two-step mode by analyzing the relationship between the probability distribution of the target vector and the solving success rate in a Two-step mode LWE solving algorithm. While the latest Two-step estimator for LWE, which is the “primal-bdd” mode in lattice-estimator1, does not take into account some up-to-date results and lacks a thorough theoretical analysis. Under the same gate-count model, our estimation for NIST PQC standards drops by 2.1∼3.4 bits (2.2∼4.6 bits while considering more flexible blocksize and jump strategy) compared with leaky-LWE-Estimator. Furthermore, we also give a conservative estimation for LWE from the Two-step solving algorithm. Compared with the Core-SVP model, which is used in previous conservative estimations, our estimation relies on weaker assumptions and outputs higher evaluation results than the Core-SVP model. For NIST PQC standards, our conservative estimation is 4.17∼8.11 bits higher than the Core-SVP estimation. Hence our estimator can give a closer estimation for both upper bound and lower bound of LWE hardness.
2024
PKC
A Simpler and More Efficient Reduction of DLOG to CDH for Abelian Group Actions
Abelian group actions appear in several areas of cryptography, especially isogeny-based post-quantum cryptography. A natural problem is to relate the analogues of the computational Diffie-Hellman (CDH) and discrete logarithm (DLOG) problems for abelian group actions. Galbraith, Panny, Smith and Vercauteren (Mathematical Cryptology '21) gave a quantum reduction of DLOG to CDH, assuming a CDH oracle with perfect correctness. Montgomery and Zhandry (Asiacrypt '22, best paper award) showed how to convert an unreliable CDH circuit into one that is correct with overwhelming probability. However, while a theoretical breakthrough, their reduction is quite inefficient: if the CDH oracle is correct with probability $q$ then their algorithm to amplify the success requires on the order of $1/q^{21}$ calls to the CDH oracle. We revisit this line of work and give a much simpler and tighter algorithm. Our method only takes on the order of $1/q^{4}$ CDH oracle calls and is much conceptually simpler than the Montgonery-Zhandry reduction. Our algorithm is also fully black-box, whereas the Montgomery-Zhandry algorithm is slightly non-black-box. Our main tool is a thresholding technique that replaces the comparison of distributions in Montgomery-Zhandry with testing equality of thresholded sets. We also give evidence that $1/q^{2}$ calls to the CDH oracle (or perhaps even more) is necessary, showing that it will be potentially difficult to substantially improve our method further.
2024
PKC
An algorithm for efficient detection of (N,N)-splittings and its application to the isogeny problem in dimension 2
We develop an efficient algorithm to detect whether a superspecial genus 2 Jacobian is optimally (N,N)-split for each integer N <=11. Incorporating this algorithm into the best-known attack against the superspecial isogeny problem in dimension 2 gives rise to significant cryptanalytic improvements. Our implementation shows that when the underlying prime p is 100 bits, the attack is sped up by a factor 25x; when the underlying prime is 200 bits, the attack is sped up by a factor 42x; and, when the underlying prime is 1000 bits, the attack is sped up by a factor 160x.
2024
PKC
Breaking Parallel ROS: Implication for Isogeny and Lattice-based Blind Signatures
Many of the three-round blind signatures based on identification protocols are only proven to be $\ell$-concurrently unforgeable for $\ell = \polylog(\secpar)$. It was only recently shown in a seminal work by Benhamouda et al.~(EUROCRYPT'21) that this is not just a limitation of the proof technique. They proposed an elegant polynomial time attack against the $\ell$-concurrently unforgeability of the classical blind Schnorr protocol for $\ell = \poly(\secpar)$. However, there are still many blind signatures following a similar recipe to blind Schnorr where the attack by Benhamouda et al. does not apply. This includes for instance the isogeny-based blind signature CSI-Otter by Katsumata et al (CRYPTO'23), the lattice-based blind signatures BLAZE+ by Alkeilani et al (ACISP'20) and BlindOR by Alkeilani et al (CANS'20). In this work, we provide a simple and novel attack on blind signatures based on identification protocols performing \emph{parallel repetition} to reduce the soundness error. Our attack translates to a polynomial time break for the $\ell$-concurrent unforgeability of CSI-Otter, BLAZE+, and BlindOR for $\ell = \poly(\secpar)$. More formally, we define an intermediate problem called Parallel Random inhomogeneities in an Overdetermined Solvable system of linear equations ($\pROS$) problem and show that an attack against $\pROS$ implies an attack to the above blind signatures. One takeaway of our finding is that while parallel repetition allows to exponentially reduce the soundness error of an identification protocol, this has minimal effect on the resulting blind signature.Our attack is concretely very efficient and for instance breaks 4-concurrent unforgeability of CSI-Otter in time roughly 2^{34} hash computations.
2024
PKC
Chosen-Ciphertext Secure Dual-Receiver Encryption in the Standard Model Based on Post-Quantum Assumptions
Dual-receiver encryption (DRE) is a special form of public key encryption (PKE) that allows a sender to encrypt a message for two recipients. Without further properties, the difference between DRE and PKE is only syntactical. One such important property is soundness, which requires that no ciphertext can be constructed such that the recipients decrypt to different plaintexts. Many applications rely on this property in order to realize more complex protocols or primitives. In addition many of these applications explicitly avoid the usage of the random oracle, which poses an additional requirement on a DRE construction. We show that all of the IND-CCA2 secure standard model DRE constructions based on post-quantum assumptions fall short of augmenting the constructions with soundness and describe attacks thereon. We then give an overview over all applications of IND-CCA2 secure DRE, group them into generic (i. e., applications using DRE as black-box) and non-generic applications and demonstrate that all generic ones require either soundness or public verifiability. Conclusively, we identify the gap of IND-CCA2 secure DRE constructions with soundness based on post-quantum assumptions in the standard model. In order to fill this gap we provide two direct IND-CCA2 secure DRE constructions based on the standard post-quantum assumptions, Normal Form Learning With Errors (NLWE) and Learning Paritiy with Noise (LPN).
2024
PKC
Compact Selective Opening Security From LWE
Selective opening (SO) security is a security notion for public-key encryption schemes that captures security against adaptive corruptions of senders. SO security comes in chosen-plaintext (SO-CPA) and chosen-ciphertext (SO-CCA) variants, neither of which is implied by standard security notions like IND-CPA or IND-CCA security. In this paper, we present the first SO-CCA secure encryption scheme that combines the following two properties: (1) it has a constant ciphertext expansion (i.e., ciphertexts are only larger than plaintexts by a constant factor), and (2) its security can be proven from a standard assumption. Previously, the only known SO-CCA secure encryption scheme achieving (1) was built from an ad-hoc assumption in the RSA regime. Our construction builds upon LWE, and in particular on a new and surprisingly simple construction of compact lossy trapdoor functions (LTFs). Our LTF can be converted into an “all-but-many LTF” (or ABM-LTF), which is known to be sufficient to obtain SO-CCA security. Along the way, we fix a technical problem in that previous ABM-LTF-based construction of SO-CCA security.
2024
PKC
Cryptanalysis of the Peregrine Lattice-Based Signature Scheme
The Peregrine signature scheme is one of the candidates in the ongoing Korean post-quantum cryptography competition. It is proposed as a high-speed variant of Falcon, which is a hash-and-sign signature scheme over NTRU lattices and one of the schemes selected by NIST for standardization. To this end, Peregrine replaces the lattice Gaussian sampler in the Falcon signing procedure with a new sampler based on the centered binomial distribution. While this modification offers significant advantages in terms of efficiency and implementation, it does not come with a provable guarantee that signatures do not leak information about the signing key. Unfortunately, lattice based signature schemes in the hash-and-sign paradigm that lack such a guarantee (such as GGH, NTRUSign or DRS) have generally proved insecure. In this paper, we show that Peregrine is no exception, by demonstrating a practical key recovery attack against it. We observe that the distribution of Peregrine signatures is a hidden transformation of some public distribution and still leaks information about the signing key. By adapting the parallelepiped-learning technique of Nguyen and Regev (Eurocrypt 2006), we show that the signing key can be recovered from a relatively small number of signatures. The learning technique alone yields an approximate version of the key, from which we can recover the exact key using a decoding technique due to Thomas Prest (PKC 2023). For the reference implementation (resp. the official specification version) of Peregrine-512, we fully recover the secret key with good probability in a few hours given around 25,000 (resp. 11 million) signature samples.
2024
PKC
Dynamic Collusion Functional Encryption and Multi-Authority Attribute-Based Encryption
Functional Encryption (FE) is a powerful notion of encryp- tion which enables computations and partial message recovery of en- crypted data. In FE, each decryption key is associated with a function f such that decryption recovers the function evaluation f(m) from an encryption of m. Informally, security states that a user with access to function keys skf1 , skf2 , . . . (and so on) can only learn f1(m), f2(m), . . . (and so on) but nothing more about the message. The system is said to be q-bounded collusion resistant if the security holds as long as an adversary gets access to at most q = q(�) decryption keys. However, until very recently, all these works studied bounded col- lusion resistance in a static model, where the collusion bound q was a global system parameter. While the static model has led to many great applications, it has major drawbacks. Recently, Agrawal et al. (Crypto 2021) and Garg et al. (Eurocrypt 2022) introduced the dynamic model for bounded collusion resistance, where the collusion bound q was a fluid parameter, not globally set, but chosen by each encryptor. The dynamic model enabled harnessing many virtues of the static model, while avoid- ing its various drawbacks. In this work, we provide a generic compiler to upgrade any FE scheme from the static model to the dynamic model. We also extend our techniques to multi-authority attribute-based en- cryption (MA-ABE). We show that bounded collusion MA-ABE sup- porting predicates that can be represented as an e�cient computational secret sharing (CSS) scheme can be built from minimal assumptions. Ef- ficient CSS schemes are known for access structures whose characteristic function can be computed by a polynomial-size monotone circuit under the existence of one-way functions [Yao89, unpublished]. Thus, our MA- ABE construction is the first MA-ABE scheme from standard assump- tions for predicates beyond polynomial-size monotone boolean formula. Our construction also satisfies full adaptive security in the Random Or- acle Model.
2024
PKC
Efficient KZG-based Univariate Sum-check and Lookup Argument
We propose a novel KZG-based sum-check scheme, dubbed $\mathsf{Losum}$, with \emph{optimal} efficiency. Particularly, its proving cost is \emph{one} multi-scalar-multiplication of size $k$---the number of non-zero entries in the vector, its verification cost is \emph{one} pairing plus one group scalar multiplication, and the proof consists of only \emph{one} group element. Using $\mathsf{Losum}$ as a component, we then construct a new lookup argument, named $\mathsf{Locq}$, which enjoys a smaller proof size and a lower verification cost compared to the state of the arts $\mathsf{cq}$, $\mathsf{cq}$+ and $\mathsf{cq}$++. Specifically, the proving cost of $\mathsf{Locq}$ is comparable to $\mathsf{cq}$, keeping the advantage that the proving cost is independent of the table size after preprocessing. For verification, $\mathsf{Locq}$ costs four pairings, while $\mathsf{cq}$, $\mathsf{cq}$+ and $\mathsf{cq}$++ require five, five and six pairings, respectively. For proof size, a $\mathsf{Locq}$ proof consists of four $\mathbb{G}_1$ elements and one $\mathbb{G}_2$ element; when instantiated with the BLS12-381 curve, the proof size of $\mathsf{Locq}$ is $2304$ bits, while $\mathsf{cq}$, $\mathsf{cq}$+ and $\mathsf{cq}$++ have $3840$, $3328$ and $2944$ bits, respectively. Moreover, $\mathsf{Locq}$ is zero-knowledge as $\mathsf{cq}$+ and $\mathsf{cq}$++, whereas $\mathsf{cq}$ is not. $\mathsf{Locq}$ is more efficient even compared to the non-zero-knowledge (and more efficient) versions of $\mathsf{cq}$+ and $\mathsf{cq}$++.
2024
PKC
Fast and Simple Point Operations on Edwards448 and E448
Since Edwards curves were introduced in elliptic curve cryptography, they have attracted a lot of attention. The twisted Edwards curves are defined by the equation $E_{a,d}: ax^2 + y^2 = 1 + d x^2y^2$. Twisted Edwards curve is the state-of-the-art for $a=-1$, and even for $a \ne -1$. E448 and Edwards448 are NIST standard curve in 2023 and TLS 1.3 standard curve in 2018. They both can be converted to $d=-1$, but can not be converted to $a=-1$ through isomorphism. The motivation of using a curve with $d=-1$ is that we want to improve the efficiency of E448, and Edwards448, especially to achieve a great saving in terms of the number of field multiplications ($\bfm M$) and field squarings ($\bfm S$). We propose new explicit formulas for point operations on these curves. Our full point addition only requires $8 \bfm M$, and mixed addition requires $7 \bfm M$. Our results applied on the Edward448 and E448 yield a clean and simple implementation and achieve a brand new speed record. The scalar multiplication on Edwards448 and E448 have the same cost of $\bfm M$ and $\bfm S$ as that on Edwards25519 per bit.
2024
PKC
Faster Amortized FHEW bootstrapping using Ring Automorphisms
Amortized bootstrapping offers a way to simultaneously refresh many ciphertexts of a fully homomorphic encryption scheme, at a total cost comparable to that of refreshing a single ciphertext. An amortization method for FHEW-style cryptosystems was first proposed by (Micciancio and Sorrell, ICALP 2018), who showed that the amortized cost of bootstrapping $n$ FHEW-style ciphertexts can be reduced from $\tilde O(n)$ basic cryptographic operations to just $\tilde O(n^{\epsilon})$, for any constant $\epsilon>0$. However, despite the promising asymptotic saving, the algorithm was rather inpractical due to a large constant (exponential in $1/\epsilon$) hidden in the asymptotic notation. In this work, we propose an alternative amortized boostrapping method with much smaller overhead, still achieving $O(n^\epsilon)$ asymptotic amortized cost, but with a hidden constant that is only linear in $1/\epsilon$, and with reduced noise growth. This is achieved following the general strategy of (Micciancio and Sorrell), but replacing their use of the Nussbaumer transform, with a much more practical Number Theoretic Transform, with multiplication by twiddle factors implemented using ring automorphisms. A key technical ingredient to do this is a new ``scheme switching'' technique proposed in this paper which may be of independent interest.
2024
PKC
Formalizing Hash-then-Sign Signatures
Many practical signature schemes follow the Hash-then-Sign (HtS) paradigm: Instead of signing messages directly, messages are first hashed and then their hash values are signed. Attractive properties of the HtS approach include that the core signing algorithm does not have to get involved with handling arbitrarily long message inputs, and that the tasks of hashing and signing can be performed by different entities. For instance, if a signing algorithm is implemented in a smartcard setting, then an HtS scheme can allow sending only the hash value to the smartcard, instead of the whole message. While the HtS paradigm was introduced decades ago, most signature schemes leverage it, and many applications rely on it, security analyses for HtS signature schemes are typically conducted only holistically for the hash+sign hybrid. However, the corresponding security models (e.g., EUF-CMA) don’t cover the fact that the separation of hashing and signing allows for more attacks than monolithic schemes. In particular, cases where an attacker can interact with a smartcard and request the creation of signatures on arbitrary hash values (for which it may or may not know the messages), remain unaddressed. This work initiates a study of HtS signatures in the framework of provable security: After defining a precise syntax, we develop security notions that cover the artifacts of the separation of hashing and signing. We show that signature schemes exist that are weak in the HtS sense yet secure in the classic sense, demonstrating the relevance of our work. We then study the HtS security of a number of widely-standardized signature schemes, including of ECDSA. Finally, we propose a generic method for the secure separation of hashing and signing for signature schemes that use a Merkle–Damgård hash function.
2024
PKC
Fully Dynamic Attribute-Based Signatures for Circuits from Codes
Attribute-Based Signature (ABS), introduced by Maji et al. (CT-RSA'11), is an advanced privacy-preserving signature primitive that has gained a lot of attention. Research on ABS can be categorized into three main themes: expanding the expressiveness of signing policies, enabling new functionalities, and providing more diversity in terms of computational assumptions. We contribute to the development of ABS in all three dimensions, by providing a fully dynamic ABS scheme for arbitrary circuits from codes. The scheme is the first ABS from code-based assumptions and also the first ABS system offering the \texttt{full dynamicity} functionality (i.e., attributes can be enrolled and revoked simultaneously). Moreover, the scheme features much shorter signature size than a lattice-based counterpart proposed by El Kaafarani and Katsumata (PKC'18). In the construction process, we put forward a new theoretical abstraction of Stern-like zero-knowledge (ZK) protocols, which are the major tools for privacy-preserving cryptography from codes. Our main insight here actually lies in the questions we ask about the fundamental principles of Stern-like protocols that have remained unchallenged since their conception by Stern at CRYPTO'93. We demonstrate that these long-established principles are not essential, and then provide a refined framework generalizing existing Stern-like techniques and enabling enhanced constructions.
2024
PKC
Improved Cryptanalysis of HFERP
In this paper we introduce a new attack on the multivariate encryption scheme HFERP, a big field scheme including an extra variable set, additional equations of the UOV or Rainbow shape as well as additional random polynomials. Our attack brings several parameter sets well below their claimed security levels. The attack combines novel methods applicable to multivariate schemes with multiple equation types with insights from the Simple Attack that broke Rainbow in early 2022, though interestingly the technique is applied in an orthogonal way. In addition to this attack, we apply support minors techniques on a MinRank instance drawing coefficients from the big field, which was effective against other multivariate big field schemes. This work demonstrates that there exist previously unknown impacts of the above works well beyond the scope in which they were derived.
2024
PKC
Laconic Branching Programs from the Diffie-Hellman Assumption
Laconic cryptography enables secure two-party computation (2PC) on unbalanced inputs with asymptotically-optimal communication in just two rounds of communication. In particular, the receiver (who sends the first-round message) holds a long input and the sender (who sends the second-round message) holds a short input, and the size of their communication to securely compute a function on their joint inputs only grows with the size of the sender's input and is independent of the receiver's input size. The work on laconic oblivious transfer (OT) [Cho et al. CRYPTO 2017] and laconic private set intersection (PSI) [Alamati et al. TCC 2021] shows how to achieve secure laconic computation for OT and PSI from the Diffie-Hellman assumption. In this work, we push the limits further and achieve laconic branching programs from the Diffie-Hellman assumption. In particular, the receiver holds a large branching program $P$ and the sender holds a short input $x$. We present a two-round 2PC protocol that allows the receiver to learn $x$ iff $P(x) =1$, and nothing else. The communication only grows with the size of $x$ and the depth of $P$, and does not further depend on the size of $P$.
2024
PKC
Lookup Arguments: Improvements, Extensions and Applications to Zero-Knowledge Decision Trees
Lookup arguments allow to prove that the elements of a committed vector come from a (bigger) committed table. They enable novel approaches to reduce the prover complexity of general-purpose zkSNARKs, implementing ``non-arithmetic operations" such as range checks, XOR and AND more efficiently. We extend the notion of lookup arguments along two directions and improve their efficiency: (1) we extend vector lookups to matrix lookups (where we can prove that a committed matrix is a submatrix of a committed table). (2) We consider the notion of zero-knowledge lookup argument that keeps the privacy of both the sub-vector/sub-matrix and the table. (3) We present new zero-knowledge lookup arguments, dubbed cq+, zkcq+ and cq++, more efficient than the state of the art, namely the recent work by Eagen, Fiore and Gabizon named cq. Finally, we give a novel application of zero-knowledge matrix lookup argument to the domain of zero-knowledge decision tree where the model provider releases a commitment to a decision tree and can prove zero-knowledge statistics over the committed data structure. Our scheme based on lookup arguments has succinct verification, prover's time complexity asymptotically better than the state of the art, and is secure in a strong security model where the commitment to the decision tree can be malicious.
2024
PKC
More Efficient Public-Key Cryptography with Leakage and Tamper Resilience
In this paper, we study the design of efficient signature and public-key encryption (PKE) schemes in the presence of both leakage and tampering attacks. Firstly, we formalize the strong leakage and tamper-resilient (sLTR) security model for signature, which provides strong existential unforgeability, and deals with bounded leakage and restricted tampering attacks, as a counterpart to the sLTR security introduced by Sun et al. (ACNS 2019) for PKE. Then, we present direct constructions of signature and chosen-ciphertext attack (CCA) secure PKE schemes in the sLTR model, based on the matrix decisional Diffie-Hellman (MDDH) assumptions (which covers the standard symmetric external DH (SXDH) and k-Linear assumptions) over asymmetric pairing groups. Our schemes avoid the use of heavy building blocks such as the true-simulation extractable non-interactive zero-knowledge proofs (tSE-NIZK) proposed by Dodis et al. (ASIACRYPT 2010), which are usually needed in constructing schemes with leakage and tamper-resilience. Especially, our SXDH-based signature and PKE schemes are more efficient than the existing schemes in the leakage and tamper-resilient setting: our signature scheme has only 4 group elements in the signature, which is about 5×~8× shorter, and our PKE scheme has only 6 group elements in the ciphertext, which is about 1.3×~3.3× shorter. Finally, we note that our signature scheme is the {\it first} one achieving strong existential unforgeability in the leakage and tamper-resilient setting, where strong existential unforgeability has important applications in building more complex primitives such as signcryption and authenticated key exchange.
2024
PKC
Multi-Hop Fine-Grained Proxy Re-Encryption
Proxy re-encryption (PRE) allows a proxy to transform a ciphertext intended for Alice (delegator) to another ciphertext intended for Bob (delegatee) without revealing the underlying message. Recently, a new variant of PRE, namely fine-grained PRE (FPRE), was proposed in [Zhou et al., Asiacrypt 2023]. Generally, FPRE is designed for a function family F: each re-encryption key rk_{A→B}^f is associated with a function f ∈ F, and with rk_{A→B}^f, a proxy can transform Alice's ciphertext encrypting m to Bob's ciphertext encrypting f(m). However, their scheme only supports single-hop re-encryption and achieves only CPA security. In this paper, we formalize {\it multi-hop} FPRE (mFPRE) that supports multi-hop re-encryptions in the fine-grained setting, and propose two mFPRE schemes achieving CPA security and stronger HRA security (security against honest re-encryption attacks), respectively. -- For multi-hop FPRE, we formally define its syntax and formalize a set of security notions including CPA security, HRA security, undirectionality and ciphertext unlinkablity. HRA security is stronger and more reasonable than CPA security, and ciphertext unlinkablity blurs the proxy relations among a chain of multi-hop re-encryptions, hence providing better privacy. We establish the relations between these security notions. -- Our mFPRE schemes support fine-grained re-encryptions for bounded linear functions and have security based on the learning-with-errors (LWE) assumption in the standard model. In particular, one of our schemes is HRA secure and enjoys all the aforementioned desirable securities. To achieve CPA security and HRA security for mFPRE, we extend the framework of [Jafargholi et al., Crypto 2017] and the technique of the [Fuchsbauer et al., PKC 2019].
2024
PKC
Multi-Signatures for Ad-hoc and Privacy-Preserving Group Signing
Multi-signatures allow to combine individual signatures from different signers on the same message into a short aggregated signature. Newer schemes further allow to aggregate the individual public keys, such that the combined signature gets verified against a short aggregated key. This makes them a versatile alternative to threshold or distributed signatures: the aggregated key can serve as group key, and signatures under that key can only be computed with the help of all signers. What makes multi-signatures even more attractive is their simple key management, as users can re-use the same secret key in several and ad-hoc formed groups. In that context, it will be desirable to not sacrifice privacy as soon as keys get re-used and ensure that users are not linkable across groups. In fact, when multi-signatures with key aggregation were proposed, it was claimed that aggregated keys hide the signers’ identities or even the fact that it is a combined key at all. In our work we show that none of the existing multi-signature schemes provide these privacy guarantees when keys get re-used in multiple groups. This is due to the fact that all known schemes deploy deterministic key aggregation. To overcome this limitation, we propose a new variant of multi-signatures with probabilistic yet verifiable key aggregation. We formally define the desirable privacy and unforgeability properties in the presence of key re-use. This also requires to adapt the unforgeability model to the group setting, and ensure that key re-use does not weaken the expected guarantees. We present a simple BLS-based scheme that securely realizes our strong privacy and security guarantees. We also formalize and investigate the privacy that is possible by deterministic schemes, and prove that existing schemes provide the advertised privacy features as long as one public key remains secret.
2024
PKC
Network-Agnostic Multi-Party Computation Revisited (Extended Abstract)
We study network-agnostic {\it secure multi-party computation} (MPC) in the presence of {\it computationally-bounded} adversaries. A network-agnostic protocol provides the best possible security guarantees, irrespective of the type of underlying communication network. Previous MPC protocols in this regime either assume a setup for a common reference string (CRS) and a threshold additively homomorphic encryption (Blum et al. CRYPTO 2020) or a plain public-key infrastructure (PKI) setup (Bacho et al. CRYPTO 2023). Both these MPC protocols perform circuit-evaluation over encrypted data and also deploy different forms of zero-knowledge (ZK) proofs, along with other computationally-expensive cryptographic machinery. We aim to build an MPC protocol based on circuit evaluation on secret-shared data, {\it avoiding} ZK proofs and other computationally-expensive cryptographic machinery and based on a {\it plain} PKI setup. To achieve our goal, we present the {\it first} network-agnostic {\it verifiable secret sharing} (VSS) protocol with the {\it optimal} threshold conditions, which is of independent interest. Previously, network-agnostic VSS is known either with {\it perfect} security (Appan et al. IEEE IT 2023) where the threshold conditions are {\it not} known to be optimal or with {\it statistical security} (Appan et al. TCC 2023) where the threshold conditions are optimal, but the parties need to perform {\it exponential} amount of computation and communication. Although our proposed MPC protocol incurs higher communication complexity compared to state-of-the-art network-agnostic MPC protocols, it offers valuable insights and motivates alternative directions for designing {\it computationally inexpensive} MPC protocols, based on a plain PKI setup, which has not been explored in the domain of network-agnostic MPC.
2024
PKC
New proof systems and an OPRF from CSIDH
Isogeny computations in CSIDH (Asiacrypt 2018) are described using a commutative group G acting on the set of supersingular elliptic curves. The commutativity property gives CSIDH enough flexibility to allow the creation of many cryptographic primitives and protocols. Nevertheless, these operations are limited and more complex applications have not yet been proposed. When calling the composition of two group elements of G, \emph{addition}, our goal in this work is to explore exponentiation, multiplication with public elements, and multiplication between secret elements of this group. We first introduce a two-party interactive protocol for multiplication of secret group elements. Then, we explore zero-knowledge proofs of these different arithmetic operations. We present two types of approaches, using either standard sigma protocols or the MPC-in-the-Head paradigm. Most of our proofs need a trusted setup, which can be removed in the MPC-in-the-Head setting using cut-and-choose techniques. We conclude this work by presenting an oblivious pseudorandom function based on our new framework, that is competitive with current state-of-the-art designs.
2024
PKC
Oblivious Accumulators
A cryptographic accumulator is a succinct set commitment scheme with efficient (non-)membership proofs that typically supports updates (additions and deletions) on the accumulated set. When elements are added to or deleted from the set, an update message is issued. The collection of all the update messages essentially leaks the underlying accumulated set which in certain applications is not desirable. In this work, we define oblivious accumulators, a set commitment with concise membership proofs that hides the elements and the set size from every entity: an outsider, a verifier or other element holders. We formalize this notion of privacy via two properties: element hiding and add-delete indistinguishability. We also define almost-oblivious accumulators, that only achieve a weaker notion of privacy called add-delete unlinkability. Such accumulators hide the elements but not the set size. We consider the trapdoorless, decentralized setting where different users can add and delete elements from the accumulator and compute membership proofs. We then give a generic construction of an oblivious accumulator based on key-value commitments (KVC). We also show a generic way to construct KVCs from an accumulator and a vector commitment scheme. Finally, we give lower bounds on the communication (size of update messages) required for oblivious accumulators and almost-oblivious accumulators.
2024
PKC
On Algebraic Embedding for Unstructured Lattices
Lattice-based cryptography, the study of cryptographic primitives whose security is based on the hardness of so-called lattice problems, has taken center stage in cryptographic research in recent years. It potentially offers favorable security features, even against quantum algorithms. One of the main obstacles for wide adoption of this type of cryptography is its unsatisfactory efficiency. To address this point, efficient lattice-based cryptography usually relies on the intractability of problems on lattices with additional algebraic structure (such as so-called ideal-lattices or module-lattices). It is an important open question to evaluate the hardness of such lattice problems, and their relation to the hardness of problems on unstructured lattices. It is a known fact that an unstructured lattice, which is simply an additive discrete group in Euclidean space, can be cast as an ideal-lattice in some \emph{order} of a number field (and thus, in a rather trivial sense, that ideals in orders are as general as unstructured lattices). However, it is not known whether this connection can be used to imply useful hardness results for structured lattices, or alternatively new algorithmic techniques for unstructured lattices. In this work we establish a gradient of hardness for the Order-LWE problem (a generalization of the well known Ring-LWE problem), as it varies over orders in a number field. Furthermore, we show that, in every number field, there are certain orders such that the corresponding Order-LWE problem is at least as hard as the (unstructured) LWE problem. So in general one should not hope to solve (any) Order-LWE more efficiently than LWE. However, we show that this connection holds in orders that are very ``skewed'' and hence, perhaps, irrelevant for improving efficiency in cryptographic applications. We further improve the hardness result for Order-LWE, to include \textit{all} ideal lattices, closing a gap left in prior work. This establishes a direct connection between problems on unstructured lattices and the structured problem of Order-LWE.
2024
PKC
On Information-Theoretic Secure Multiparty Computation with Local Repairability
In this work we consider the task of designing information-theoretic MPC protocols for which the state of a given party can be recovered from a small amount of parties, a property we refer to as \emph{local repairability}. This is useful when considering MPC over dynamic settings where parties leave and join a computation, a scenario that has gained notable attention in recent literature. Thanks to the results of (Cramer \emph{et al.}~EUROCRYPT'00), designing such protocols boils down to constructing a linear secret-sharing scheme (LSSS) with good locality, that is, each share is determined by only a small amount of other shares, that also satisfies the so-called multiplicativity property. Previous constructions that achieve locality (\emph{e.g.}~using locally recoverable codes---LRCs) do not enjoy multiplicativity, and LSSS that are multiplicative (\emph{e.g.}~Shamir's secret-sharing) do not satisfy locality. Our construction bridges this literature gap by showing the existence of an LSSS that achieves both properties simultaneously. Our results are obtained by making use of well known connection between error correcting codes and LSSS, in order to adapt the LRC construction by (Tamo \& Barg, IEEE Transactions on Information Theory 2014) to turn it into a LSSS. With enough care, such coding-theoretic construction yields our desired locality property, but it falls short at satisfying multiplicativity. In order to address this, we perform an extensive analysis of the privacy properties of our scheme in order to identify parameter regimes where our construction satisfies multiplicativity. Finally, since our LSSS satisfies locality, every share is determined by a small amount of shares. However, in an MPC context it is not enough to let the (small set of) parties to send their shares to the repaired party, since this may leak more information than the regenerated share. To obtain our final result regarding MPC with local repairability, we construct a lightweight MPC protocol that performs such repairing process without any leakage. We provide both a passively secure construction (for the \emph{plain} multiplicative regime) and an actively secure one (for \emph{strong} multiplicativity).
2024
PKC
On Instantiating Unleveled Fully-Homomorphic Signatures from Falsifiable Assumptions
We build the first unleveled fully homomorphic signature scheme in the standard model. Our scheme is not constrained by any a-priori bound on the depth of the functions that can be homomorphically evaluated, and relies on subexponentially-secure indistinguishability obfuscation, fully-homomorphic encryption and a non-interactive zero-knowledge (NIZK) proof system with composable zero-knowledge. Our scheme is also the first to satisfy the strong security notion of context-hiding for an unbounded number of levels, ensuring that signatures computed homomorphically do not leak the original messages from which they were computed. All building blocks are instantiable from falsifiable assumptions in the standard model, avoiding the need for knowledge assumptions. Conceptually, the main difficulty overcome by our techniques concerns bootstrapping, which is a crucial tool for obtaining unleveled fully homomorphic encryption (FHE). No equivalent technique exists for homomorphic signatures, which is why constructing unleveled fully homomorphic signature schemes has proven elusive until now.
2024
PKC
On Proving Equivalence Class Signatures Secure from Non-interactive Assumptions
Equivalence class signatures (EQS), introduced by Hanser and Slamanig (AC'14, J.Crypto'19), sign vectors of elements from a bilinear group. Their main feature is ``adaptivity'': given a signature on a vector, anyone can transform it to a (uniformly random) signature on any multiple of the vector. A signature thus authenticates equivalence classes and unforgeability is defined accordingly. EQS have been used to improve the efficiency of many cryptographic applications, notably (delegatable) anonymous credentials, (round-optimal) blind signatures, group signatures and anonymous tokens. EQS security implies strong anonymity (or blindness) guarantees for these schemes which holds against malicious signers without trust assumptions. Unforgeability of the original EQS construction is proven directly in the generic group model. While there are constructions from standard assumptions, these either achieve prohibitively weak security notions (PKC'18) or they require a common reference string (AC'19, PKC'22), which reintroduces trust assumptions avoided by EQS. In this work we ask whether EQS schemes that satisfy the original security model can be proved secure under standard (or even non-interactive) assumptions with standard techniques. Our answer is negative: assuming a reduction that, after running once an adversary breaking unforgeability, breaks a non-interactive computational assumption, we construct efficient meta-reductions that either break the assumption or break class-hiding, another security requirement for EQS.
2024
PKC
On Sigma Protocols and (packed) Black-Box Secret Sharing Schemes
$\Sigma$-protocols are a widely utilized, relatively simple and well understood type of zero-knowledge proofs. However, the well known Schnorr $\Sigma$-protocol for proving knowledge of discrete logarithm in a cyclic group of known prime order, and similar protocols working over this type of groups, are hard to generalize to dealing with other groups. In particular with hidden order groups, due to the inability of the knowledge extractor to invert elements modulo the order. In this paper, we introduce a universal construction of $\Sigma$-protocols designed to prove knowledge of preimages of group homomorphisms for any abelian finite group. In order to do this, we first establish a general construction of a $\Sigma$-protocol for $\mathfrak{R}$-module homomorphism given only a linear secret sharing scheme over the ring $\mathfrak{R}$, where zero knowledge and special soundness can be related to the privacy and reconstruction properties of the secret sharing scheme. Then, we introduce a new construction of 2-out-of-$n$ packed black-box secret sharing scheme capable of sharing $k$ elements of an arbitrary (abelian, finite) group where each share consists of $k+\log n-3$ group elements. From these two elements we obtain a generic ``batch'' $\Sigma$-protocol for proving knowledge of $k$ preimages of elements via the same group homomorphism, which communicates $k+\lambda-3$ elements of the group to achieve $2^{-\lambda}$ knowledge error. For the case of class groups, we show that our $\s$-protocol improves in several aspects on existing proofs for knowledge of discrete logarithm and other related statements that have been used in a number of works. Finally, we extend our constructions from group homomorphisms to the case of ZK-ready functions, introduced by Cramer and Damg\aa rd in Crypto 09, which in particular include the case of proofs of knowledge of plaintext (and randomness) for some linearly homomorphic encryption schemes such as Joye-Libert encryption. However, in the case of Joye-Libert, we show an even better alternative, using Shamir secret sharing over Galois rings, which achieves $2^{-k}$ knowledge soundness by communicating $k$ ciphertexts to prove $k$ statements.
2024
PKC
On Structure-Preserving Cryptography and Lattices
The Groth-Sahai proof system is a highly efficient pairing-based proof system for a specific class of group-based languages. Cryptographic primitives that are compatible with these languages (such that we can express, e.g., that a ciphertext contains a valid signature for a given message) are called "structure-preserving". The combination of structure-preserving primitives with Groth-Sahai proofs allows to prove complex statements that involve encryptions and signatures, and has proved useful in a variety of applications. However, so far, the concept of structure-preserving cryptography has been confined to the pairing setting. In this work, we propose the first framework for structure-preserving cryptography in the lattice setting. Concretely, we - define "structure-preserving sets" as an abstraction of (typically noisy) lattice-based languages, - formalize a notion of generalized structure-preserving encryption and signature schemes (capturing a number of existing lattice-based encryption and signature schemes), - construct a compatible zero-knowledge argument system that allows to argue about lattice-based structure-preserving primitives, - offer a lattice-based construction of verifiably encrypted signatures in our framework. Along the way, we also discover a new and efficient strongly secure lattice-based signature scheme. This scheme combines Rückert's lattice-based signature scheme with the lattice delegation strategy of Agrawal et al., which yields more compact and efficient signatures. We hope that our framework provides a first step towards a modular and versatile treatment of cryptographic primitives in the lattice setting.
2024
PKC
On the Possibility of a Backdoor in the Micali-Schnorr Generator
In this paper, we study both the implications and potential impact of backdoored parameters for two RSA-based pseudorandom number generators: the ISO-standardized Micali-Schnorr generator and a closely related design, the RSA PRG. We observe, contrary to common understanding, that the security of the Micali-Schnorr PRG is not tightly bound to the difficulty of inverting RSA. We show that the Micali-Schnorr construction remains secure even if one replaces RSA with a publicly evaluatable PRG, or a function modeled as an efficiently invertible random permutation. This implies that any cryptographic backdoor must somehow exploit the algebraic structure of RSA, rather than an attacker’s ability to invert RSA or the presence of secret keys. We exhibit two such backdoors in related constructions: a family of exploitable parameters for the RSA PRG, and a second vulnerable construction for a finite-field variant of Micali-Schnorr. We also observe that the parameters allowed by the ISO standard are incompletely specified, and allow insecure choices of exponent. Several of our backdoor constructions make use of lattice techniques, in particular multivariate versions of Coppersmith’s method for finding small solutions to polynomials modulo integers.
2024
PKC
One-shot Signatures: Applications and Design Directions (invited talk)
More than 50 years ago, Stephen Wiesner envisioned how the uncertainty principle could be harnessed to create oblivious transfer quantum channels and unforgeable quantum money. This seminal work lead to a number of developments widening the impact of quantum enhanced protocols in cryptography. Recently, following the blossoming of this research domain, one-shot signatures were introduced by Amos, Georgiou, Kiayias, and Zhandry (STOC 2020). This cryptographic primitive enables digital signatures with classical public-key verification and a quantum signing algorithm that self-destructs after being used once. This impossible property to achieve in the classical setting (barring hardware assumptions) has a number of far reaching applications that include key-evolving signatures without erasures, provably secret signing keys, secure proof-of-stake blockchains without erasing keys or economic penalties as well as non-interactive publicly verifiable proofs of quantumness and min-entropy. Known design approaches for one-shot signatures rely on the one side of so called win-win results regarding the ``collapsing'' features of hash functions and commitments in the quantum setting. Specifically, while being collapsing is a desirable property of such primitives from a post-quantum security perspective, a failure to collapse combined with retaining a degree of security, may enable useful quantum enhanced primitives including one-shot signatures. In this talk we overview applications and the currently known design approaches for one-shot signatures as well as point to directions for future research.
2024
PKC
Parameter-Hiding Order-Revealing Encryption without Pairings
Order-Revealing Encryption (ORE) provides a practical solution for conducting range queries over encrypted data. Achieving a desirable privacy-efficiency tradeoff in designing ORE schemes has posed a significant challenge. At Asiacrypt 2018, Cash et al. proposed Parameter-hiding ORE (pORE), which specifically targets scenarios where the data distribution shape is known, but the underlying parameters (such as mean and variance) need to be protected. However, existing pORE constructions rely on impractical bilinear maps, limiting their real-world applicability. In this work, we propose an alternative and efficient method for constructing pORE using identification schemes. By leveraging the map-invariance property of identification schemes, we eliminate the need for pairing computations during ciphertext comparison. Specifically, we instantiate our framework with the pairing-free Schnorr identification scheme and demonstrate that our proposed pORE scheme reduces ciphertext size by approximately 31.25\% and improves encryption and comparison efficiency by over two times compared to the current state-of-the-art pORE construction. Our work provides a more efficient alternative to existing pORE constructions and could be viewed as a step towards making pORE a viable choice for practical applications.
2024
PKC
Private Set Operations from Multi-Query Reverse Private Membership Test
Private set operations allow two parties to perform secure computation on their private sets, including intersection, union and functions of intersection/union. In this paper, we put forth a framework to perform private set operations. The technical core of our framework is the multi-query reverse private membership test (mqRPMT) protocol (Zhang et al., USENIX Security 2023). We present two constructions of mqRPMT from newly introduced cryptographic notions, one is based on commutative weak pseudorandom function (cwPRF), and the other is based on permuted oblivious pseudorandom function (pOPRF). Both cwPRF and pOPRF can be realized from the decisional Diffie-Hellman (DDH)-like assumptions in the random oracle model. We demonstrate the practicality of our framework with implementations. By plugging our cwPRF-based mqRPMT into the framework, we obtain various PSO protocols that are superior or competitive to the state-of-the-art protocols. For intersection functionality, our protocol is faster than the most efficient one for small sets. For cardinality functionality, our protocol achieves a $2.4-10.5\times$ speedup and a $10.9-14.8\times$ reduction in communication cost. For cardinality-with-sum functionality, our protocol achieves a $28.5-76.3\times$ speedup and $7.4\times$ reduction in communication cost. For union functionality, our protocol is the first one that achieves strictly linear complexity, and requires the lowest concrete computation and communication costs in all settings, achieving a $2.7-17\times$ speedup and about $2\times$ reduction in communication cost. Furthermore, our improvement on PSU also translates to related functionality, yielding the most efficient private-ID protocol to date.
2024
PKC
Probabilistic Hash-and-Sign with Retry in the Quantum Random Oracle Model
A hash-and-sign signature based on a preimage-sampleable function (Gentry et al., STOC 2008) is secure in the quantum random oracle model if the preimage-sampleable function is collision-resistant (Boneh et al., ASIACRYPT 2011) or one-way (Zhandry, CRYPTO 2012). However, trapdoor functions in code-based and multivariate-quadratic-based signatures are not preimage-sampleable functions; for example, underlying trapdoor functions of the Courtois-Finiasz-Sendrier, Unbalanced Oil and Vinegar (UOV), and Hidden Field Equations (HFE) signatures are not surjections. Thus, such signature schemes adopt probabilistic hash-and-sign with retry. While Sakumoto et al. in PQCRYPTO 2011 showed the security of this paradigm in the classical random oracle model, their proof contains an error. Also, there is currently no known security proof for the probabilistic hash-and-sign with retry in the quantum random oracle model. We correct the proof in the random oracle model and give the first security proof in the quantum random oracle model for the probabilistic hash-and-sign with retry, assuming that the underlying trapdoor function is non-invertible, that is, it is hard to find a preimage of a given random value in the range. Our reduction from the non-invertibility assumption is tighter than the existing ones that apply only to signature schemes based on preimage-sampleable functions. We apply the security proof to code-based and multivariate-quadratic-based signatures. Additionally, we extend the proof into the multi-key setting and propose a generic method that provides security reduction without any security loss in the number of keys.
2024
PKC
Public-key Encryption with Keyword Search in Multi-User, Multi-Challenge Setting under Adaptive Corruptions
In the past decade, much progress has been made on proposing encryption schemes with multi-user security. However, no known work aims at constructing a Public-key Encryption with Keyword Search (PEKS) scheme that is secure in multi-user setting. PEKS is a well-known primitive to solve the problem of searching over encrypted data. In this paper, we fill the gap. For more realistic multi-user scenario, we consider a strong security notion. Specifically, the adversary can adaptively corrupt some users' secret keys, and can adaptively request searchable ciphertexts of related keywords under different public keys as well as trapdoors of related keywords under different secret keys. We present two multi-user PEKS schemes both under simple assumptions in the standard model to achieve this strong security notion. \text{\qquad}Technically, our first scheme is a variation of the Lewko-Waters identity-based encryption scheme, and our second scheme is a variation of the Wee identity-based encryption scheme. However, we need to prove that the presented public key encryption schemes are secure in the multi-user, multi-challenge setting under adaptive corruptions. We modify the dual system encryption methodology to meet the goal. In particular, the security loss is constant.
2024
PKC
Quantum CCA-Secure PKE, Revisited
Security against chosen-ciphertext attacks (CCA) concerns privacy of messages even if the adversary has access to the decryption oracle. While the classical notion of CCA security seems to be strong enough to capture many attack scenarios, it falls short of preserving the privacy of messages in the presence of quantum decryption queries, i.e., when an adversary can query a superposition of ciphertexts. Boneh and Zhandry (CRYPTO 2013) defined the notion of quantum CCA (qCCA) security to guarantee privacy of messages in the presence of quantum decryption queries. However, their construction is based on an exotic cryptographic primitive (namely, identity-based encryption with security against quantum queries), for which only one instantiation is known. In this work, we comprehensively study qCCA security for public-key encryption (PKE) based on both generic cryptographic primitives and concrete mathematical assumptions, yielding the following results: * We show that key-dependent message secure encryption (along with PKE) is sufficient to realize qCCA-secure PKE. This yields the first construction of qCCA-secure PKE from the LPN assumption. * We prove that hash proof systems imply qCCA-secure PKE, which results in the first instantiation of PKE with qCCA security from (isogeny-based) group actions. * We extend the notion of adaptive TDFs (ATDFs) to the quantum setting by introducing quantum ATDFs, and we prove that quantum ATDFs are sufficient to realize qCCA-secure PKE. We also show how to instantiate quantum ATDFs from the LWE assumption. * We show that a single-bit qCCA-secure PKE is sufficient to realize a multi-bit qCCA-secure PKE by extending the completeness of bit encryption for CCA security to the quantum setting.
2024
PKC
R3PO: Reach-Restricted Reactive Program Obfuscation and its Applications
In recent breakthrough results, novel use of grabled circuits yielded constructions for several primitives like Identity-Based Encryption (IBE) and 2-round secure multi-party computation, based on standard assumptions in public-key cryptography. While the techniques in these different results have many common elements, these works did not offer a modular abstraction that could be used across them. Our main contribution is to introduce a novel notion of obfuscation, called Reach-Restricted Reactive-Program Obfuscation (R3PO) that captures the essence of these constructions, and exposes additional capabilities. We provide a powerful composition theorem whose proof fully encapsulates the use of garbled circuits in these works. As an illustration of the potential of R3PO, and as an important contribution of independent interest, we present a variant of Multi-Authority Attribute-Based Encryption (MA-ABE) that can be based on (single-authority) CP-ABE in a blackbox manner, using only standard cryptographic assumptions (e.g., DDH) in addition. This is in stark contrast to the existing constructions for MA-ABE, which rely on the random oracle model and supports only limited policy classes.
2024
PKC
Rate-1 Fully Local Somewhere Extractable Hashing from DDH
Somewhere statistically binding (SSB) hashing allows us to sample a special hashing key such that the digest statistically binds the input at $m$ secret locations. This hash function is said to be somewhere extractable (SE) if there is an additional trapdoor that allows the extraction of the input bits at the $m$ locations from the digest. Devadas, Goyal, Kalai, and Vaikuntanathan (FOCS 22) introduced a variant of somewhere extractable hashing called rate-1 fully local SE hash functions. The rate-1 requirement states that the size of the digest is $m + \polyn(\lambda)$ (where $\lambda$ is the security parameter). The fully local property requires that for any index $i$, there is a ``very short" opening showing that $i$-th bit of the hashed input is equal to $b$ for some $b \in \bin$. The size of this opening is required to be independent of $m$ and in particular, this means that its size is independent of the size of the digest. Devadas et al. gave such a construction from Learning with Errors (LWE). In this work, we give a construction of a rate-1 fully local somewhere extractable hash function from Decisional Diffie-Hellman (DDH) and BARGs. Under the same assumptions, we give constructions of rate-1 BARG and RAM SNARG with partial input soundness whose proof sizes are only matched by prior constructions based on LWE.
2024
PKC
Registered Attribute-Based Signature
This paper introduces the notion of registered attribute-based signature (registered ABS). Distinctly different from classical attribute-based signature (ABS), registered ABS allows any user to generate their own public/secret key pair and register it with the system. The key curator is critical to keep the system flowing, which is a fully transparent entity that does not retain secrets. Our results can be summarized as follows. -This paper provides the first definition of registered ABS, which has never been defined. -This paper presents the first generic fully secure registered ABS over the prime-order group from $k$-Lin assumption under the standard model, which supports various classes of predicate. -This paper gives the first concrete registered ABS scheme for arithmetic branching program (ABP), which achieves full security in the standard model. Technically, our registered ABS is inspired by the blueprint of Okamoto and Takashima[PKC'11]. We convert the prime-order registered attribute-based encryption (registered ABE) scheme of Zhu et al.[ASIACRYPT'23] via predicate encoding to registered ABS by employing the technique of re-randomization with specialized delegation, while we employ the different dual-system method considering the property of registration. Prior to our work, the work of solving the key-escrow issue was presented by Okamoto and Takashima[PKC'13] while their work considered the weak adversary in the random oracle model.
2024
PKC
ReSolveD: Shorter Signatures from Regular Syndrome Decoding and VOLE-in-the-Head
We present ReSolveD, a new candidate post-quantum signature scheme under the regular syndrome decoding (RSD) assumption for random linear codes, which is a well-established variant of the well-known syndrome decoding (SD) assumption. Our signature scheme is obtained by designing a new zero-knowledge proof for proving knowledge of a solution to the RSD problem in the recent VOLE-in-the-head framework using a sketching scheme to verify that a vector has weight exactly one. We achieve a signature size of 3.99 KB with a signing time of 27.3 ms and a verification time of 23.1 ms on a single core of a standard desktop for a 128-bit security level. Compared to the state-of-the-art code-based signature schemes, our signature scheme achieves 1.5X ~ 2X improvement in terms of the common “signature size + public-key size” metric, while keeping the computational efficiency competitive.
2024
PKC
Ring/Module Learning with Errors under Linear Leakage - Hardness and Applications
This paper studies the hardness of decision Module Learning with Errors (MLWE) under linear leakage, which has been used as a foundation to derive more efficient lattice-based zero-knowledge proofs in a recent paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21). Unlike in the plain LWE setting, it was unknown whether this problem remains provably hard in the module/ring setting. This work shows a reduction from the search MLWE to decision MLWE with linear leakage. Thus, the main problem remains hard asymptotically as long as the non-leakage version of MLWE is hard. Additionally, we also refine the paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21) by showing a more fine-grained tradeoff between efficiency and leakage. This can lead to further optimizations of lattice proofs under the paradigm.
2024
PKC
SCALLOP-HD: group action from 2-dimensional isogenies
We present SCALLOP-HD, a novel group action that builds upon the recent SCALLOP group action introduced by De Feo, Fouotsa, Kutas, Leroux, Merz, Panny and Wesolowski in 2023. While our group action uses the same action of the class group $\textnormal{Cl}(\mathfrak{O})$ on $\mathfrak{O}$-oriented curves where $\mathfrak{O} = \mathbb{Z}[f\sqrt{-d}]$ for a large prime $f$ and small $d$ as SCALLOP, we introduce a different orientation representation: The new representation embeds an endomorphism generating $\mathfrak{O}$ in a $2^e$-isogeny between abelian varieties of dimension $2$ with Kani's Lemma, and this representation comes with a simple algorithm to compute the class group action. Our new approach considerably simplifies the SCALLOP framework, potentially surpassing it in efficiency — a claim supported by preliminary implementation results in SageMath. Additionally, our approach streamlines parameter selection. The new representation allows us to select efficiently a class group $\textnormal{Cl}(\mathfrak{O})$ of smooth order, enabling polynomial-time generation of the lattice of relation, hence enhancing scalability in contrast to SCALLOP. To instantiate our SCALLOP-HD group action, we introduce a new technique to apply Kani's Lemma in dimension 2 with an isogeny diamond obtained from commuting endomorphisms. This method allows one to represent arbitrary endomorphisms with isogenies in dimension 2, and may be of independent interest.
2024
PKC
Selective Opening Security in the Quantum Random Oracle Model, Revisited
We prove that two variants of the Fujisaki-Okamoto (FO) transformations are selective opening secure (SO) against chosen-ciphertext attacks in the quantum random oracle model (QROM), assuming that the underlying public-key encryption scheme is one-way secure against chosen-plaintext attacks (OW-CPA). The two variants we consider are $\mathsf{FO}^{\not{\bot}}$ (Hofheinz, Hövelmanns, and Kiltz, TCC 2017) and $\mathsf{U}^{\not{\bot}}_\mathsf{m}$ (Jiang et al., CRYPTO 2018). This is the first correct proof in the QROM. The previous work of Sato and Shikata (IMACC 2019) showed the SO security of $\mathsf{FO}^{\not{\bot}}$ in the QROM. However, we identify a subtle gap in their work. To close this gap, we propose a new framework that allows us to adaptively reprogram a QRO with respect to multiple queries that are computationally hard to predict. This is a property that can be easily achieved by the classical ROM, but is very hard to achieve in the QROM. Hence, our framework brings the QROM closer to the classical ROM. Under our new framework, we construct the \emph{first tightly} SO secure PKE in the QROM using lossy encryption. Our final application is proving $\mathsf{FO}^{\not{\bot}}$ and $\mathsf{U}^{\not{\bot}}_\mathsf{m}$ are bi-selective opening (Bi-SO) secure in the QROM. This is a stronger SO security notion, where an adversary can additionally corrupt some users' secret keys.
2024
PKC
Short Code-based One-out-of-Many Proofs and Applications
In this work, we propose two novel succinct one-out-of-many proofs from coding theory, which can be seen as extensions of the Stern's framework and Veron's framework from proving knowledge of a preimage to proving knowledge of a preimage for one element in a set, respectively. The size of each proof is short and scales better with the size of the public set than the code-based accumulator in \cite{nguyen2019new}. Based on our new constructions, we further present a logarithmic-size ring signature scheme and a logarithmic-size group signature scheme. Our schemes feature short signature sizes, especially our group signature. To our best knowledge, it is the most compact code-based group signature scheme so far. At 128-bit security level, our group signature size is about 144 KB for a group with $2^{20}$ members while the group signature size of the previously most compact code-based group signature constructed by the above accumulator exceeds 3200 KB.
2024
PKC
Simulation-Extractable KZG Polynomial Commitments and Applications to HyperPlonk
HyperPlonk is a recent SNARK proposal (Eurocrypt'23) that features a linear-time prover and supports custom gates of larger degree than Plonk. For the time being, its instantiations are only proven to be knowledge-sound (meaning that soundness is only guaranteed when the prover runs in isolation) while many applications motivate the stronger notion of simulation-extractability (SE). Unfortunately, the most efficient SE compilers are not immediately applicable to multivariate polynomial interactive oracle proofs. To address this problem, we provide an instantiation of HyperPlonk for which we can prove simulation-extractability in a strong sense. As a crucial building block, we describe KZG-based commitments to multivariate polynomials that also provide simulation-extractability while remaining as efficient as malleable ones. Our proofs stand in the combined algebraic group and random oracle model and ensure straight-line extractability (i.e., without rewinding).
2024
PKC
SoK: Learning With Errors, Circular Security, and Fully Homomorphic Encryption
All known constructions of fully homomorphic encryption (FHE) schemes from the learning with errors (LWE) assumption require the encryption schemes to be circular secure. A long-standing open problem in the study of FHE schemes is to demonstrate evidence for their circular security. In this work, we systematize the flavors of circular security required for a number of FHE constructions, formulate circular security conjectures, show search-to-decision reductions for them, and pose several open problems.
2024
PKC
SoK: Public Key Encryption with Openings
When modelling how public key encryption can enable secure communication, we should acknowledge that secret information, such as private keys or the encryption’s randomness, could become compromised. Intuitively, one would expect unrelated communication to remain secure, yet formalizing this intuition has proven challenging. Several security notions have appeared that aim to capture said scenario, ranging from the multi-user setting with corruptions, via selective opening attacks (SOA), to non-committing encryption (NCE). Remarkably, how the different approaches compare has not yet been systematically explored. We provide a novel framework that maps each approach to an underlying philosophy of confidentiality: indistinguishability versus simulatability based, each with an a priori versus an a posteriori variant, leading to four distinct philosophies. In the absence of corruptions, these notions are largely equivalent; yet, in the presence of corruptions, they fall into a hierarchy of relative strengths, from IND-CPA and IND-CCA at the bottom, via indistinguishability SOA and simulatability SOA, to NCE at the top. We provide a concrete treatment for the four notions, discuss subtleties in their definitions and asymptotic interpretations and identify limitations of each. Furthermore, we re-cast the main implications of the hierarchy in a concrete security framework, summarize and contextualize other known relations, identify open problems, and close a few gaps.
2024
PKC
Succinct Verification of Compressed Sigma Protocols in the Updatable SRS setting
We propose protocols in the Compressed Sigma Protocol framework that achieve a succinct verifier. Towards this, we construct a new inner product argument and cast it in the Compressed Sigma Protocol (CSP) framework as a protocol for opening a committed linear form, achieving logarithmic verification. We then use our succinct-verifier CSP to construct a zero-knowledge argument for circuit satisfiability (under the discrete logarithm assumption in bilinear groups) in the updatable Structured Reference String (SRS) setting that achieves $O(\log n)$ proof size and $O(\log n)$ verification complexity. Our circuit zero-knowledge protocol has concretely better proof/prover/verifier complexity compared to the the state-of-the-art protocol in the updatable setting under the same assumption. Our techniques of achieving verifier-succinctness in the compression framework is of independent interest. We then show a commitment scheme for committing to group elements using a structured commitment key. We construct protocols to open a committed homomorphism on a committed vector with verifier succinctness in the designated verifier setting. This has applications in making the verifier in compressed sigma protocols for bilinear group arithmetic circuits, succinct.
2024
PKC
Tagged Chameleon Hash from Lattices and Application to Redactable Blockchain
Chameleon hash (CH) is a trapdoor hash function. Generally it is hard to find collisions, but with the help of a trapdoor, finding collisions becomes easy. CH plays an important role in converting a conventional blockchain to a redactable one. However, most of existing CH schemes are too weak to support redactable blockchains. The currently known CH schemes serving for redactable blockchains have the best security of so-called ``full collision resistance (f-CR)'', but they are built either in the random oracle model or rely on heavy tools like the simulation-sound extractable non-interactive zero-knowledge (SSE-NIZK) proof system. Moreover, up to now there is no CH scheme with post-quantum f-CR security in the standard model. Therefore, no CH can support redactable blockchains in a post-quantum way without relying on random oracles. In this paper, we introduce a variant of CH, namely tagged chameleon hash (tCH). Tagged chameleon hash takes a tag into hash evaluations and collision finding algorithms. We define two security notions for tCH, restricted collision resistance (r-CR) and full collision resistance (f-CR), and prove the equivalence between r-CR and f-CR when tCH works in the one-time tag mode. We propose a tCH scheme from lattices without using any NIZK proof, and prove that its restricted collision resistance is reduced to the Short Integer Solution (SIS) assumption in the standard model. We also show how to apply tCH to a blockchain in one-time tag mode so that the blockchain can be compiled to a redactable one. Our tCH scheme provides the first post-quantum solution for redactable blockchains, without resorting to random oracles or NIZK proofs. Besides, we also construct a more efficient tCH scheme with r-CR tightly reduced to SIS in the random oracle model, which may be of independent interest.
2024
PKC
Threshold Structure Preserving Signatures: Strong and Adaptive Security under Standard Assumptions
Structure-preserving signatures (SPS) have emerged as an important cryptographic building block, as their compatibility with the Groth-Sahai (GS) NIZK framework allows to construct protocols under standard assumptions with reasonable efficiency. Over the last years there has been a significant interest in the design of threshold signature schemes. However, only very recently Crites et al. (ASIACRYPT 2023) have introduced threshold SPS (TSPS) along with a fully non-interactive construction. While this is an important step, their work comes with several limitations. With respect to the construction, they require the use of random oracles, interactive complexity assumptions and are restricted to so called indexed Diffie-Hellman message spaces. Latter limits the use of their construction as a drop-in replacement for SPS. When it comes to security, they only support static corruptions and do not allow partial signature queries for the forgery. In this paper, we ask whether it is possible to construct TSPS without such restrictions. We start from an SPS from Kiltz, Pan and Wee (CRYPTO 2015) which has an interesting structure, but thresholdizing it requires some modifications. Interestingly, we can prove it secure in the strongest model (TS-UF-1) for fully non-interactive threshold signatures (Bellare et al., CRYPTO 2022) and even under fully adaptive corruptions. Surprisingly, we can show the latter under a standard assumption without requiring any idealized model. All known constructions of efficient threshold signatures in the discrete logarithm setting require interactive assumptions and idealized models. Concretely, our scheme in type III bilinear groups under the SXDH assumption has signatures consisting of 7 group elements. Compared to the TSPS from Crites et al. (2 group elements), this comes at the cost of efficiency. However, our scheme is secure under standard assumptions, achieves strong and adaptive security guarantees and supports general message spaces, i.e., represents a drop-in replacement for many SPS applications. Given these features, the increase in the size of the signature seems acceptable even for practical applications.
2024
PKC
Towards Practical Multi-key TFHE: Parallelizable, Key-Compatible, Quasi-linear Complexity
Multi-key homomorphic encryption is a generalized notion of homomorphic encryption supporting arbitrary computation on ciphertexts, possibly encrypted under different keys. In this paper, we revisit the work of Chen, Chillotti and Song (ASIACRYPT 2019) and present yet another multi-key variant of the TFHE scheme. The previous construction by Chen et al. involves a blind rotation procedure where the complexity of each iteration gradually increases as it continuously operates on ciphertexts under different keys. Hence, the complexity of gate bootstrapping grows quadratically with respect to the number of associated keys. Our scheme is based on a new blind rotation algorithm which consists of two separate phases. We first split a given multi-key ciphertext into several single-key ciphertexts, take each of them as input to the blind rotation procedure, and obtain accumulators corresponding to individual keys. Then, we merge these single-key accumulators into a single multi-key accumulator. In particular, we develop a novel homomorphic operation between single-key and multi-key ciphertexts to instantiate our pipeline. Therefore, our construction achieves an almost linear time complexity since the gate bootstrapping is dominated by the first phase of blind rotation which requires only independent single-key operations. It also enjoys with great advantages of parallelizability and key-compatibility. We implement the proposed scheme and provide its performance benchmark. For example, our 16-key gate bootstrapping takes about 5.65s, which is 4.38x faster compared to the prior work.
2024
PKC
Updatable Policy-Compliant Signatures
Policy-compliant signatures (PCS) are a recently introduced primitive by Badertscher et al. [TCC 2021] in which a central authority distributes secret and public keys associated with sets of attributes (e.g., nationality, affiliation with a specific department, or age) to its users. The authority also enforces a policy determining which senders can sign messages for which receivers based on a joint check of their attributes. For example, senders and receivers must have the same nationality, or only senders that are at least 18 years old can send to members of the computer science department. PCS further requires attribute-privacy -- nothing about the users' attributes is revealed from their public keys and signatures apart from whether the attributes satisfy the policy or not. The policy in a PCS scheme is fixed once and for all during the setup. Therefore, a policy update requires a redistribution of all keys. This severely limits the practicality of PCS. In this work, we introduce the notion of updatable policy-compliant signatures (UPCS) extending PCS with a mechanism to efficiently update the policy without redistributing keys to all participants. We define the notion of UPCS and provide the corresponding security definitions. We then provide a generic construction of UPCS based on digital signatures, a NIZK proof system, and a so-called secret-key two-input partially-hiding predicate encryption (2-PHPE) scheme. Unfortunately, the only known way to build the latter for general two-input predicates is using indistinguishability obfuscation. We show that the reliance on the heavy tool of 2-PHPE is inherent to build UPCS by proving that non-interactive UPCS implies 2-PHPE. To circumvent the reliance on 2-PHPE, we consider interactive UPCS, which allows sender and receiver to interact during the message signing. In this setting, we present two UPCS schemes: the first one requires only a digital signature scheme, a NIZK proof system, and secure two-party computation. This scheme works for arbitrary policies, but requires senders and receivers to engage in the two-party computation for each policy update. Our second scheme additionally requires a (single-input) predicate-encryption scheme and only requires the sender and receiver to interact ones independent of the updates. In contrast to 2-PHPE, single-input predicate encryption supporting certain predicate classes are known to exist (e.g., from pairings) under more concrete and well-understood assumptions.
2024
PKC
Updatable, Aggregatable, Succinct Mercurial Vector Commitment from Lattice
Vector commitments (VC) and their variants attract a lot of attention due to their wide range of usage in applications such as blockchain and accumulator. Mercurial vector commitment (MVC), as one of the important variants of VC, is the core technique for building more complicated cryptographic applications, such as the zero-knowledge set (ZKS) and zero-knowledge elementary database (ZK-EDB). However, to the best of our knowledge, the only post-quantum MVC construction is trivially implied by a generic framework proposed by Catalano and Fiore (PKC '13) with lattice-based components which causes \emph{large} auxiliary information and \emph{cannot satisfy} any additional advanced properties, that is, updatable and aggregatable. A major difficulty in constructing a \emph{non-black-box} lattice-based MVC is that it is not trivial to construct a lattice-based VC that satisfies a critical property called ``mercurial hiding". In this paper, we identify some specific features of a new falsifiable family of basis-augmented SIS assumption ($\mathsf{BASIS}$) proposed by Wee and Wu (EUROCRYPT '23) that can be utilized to construct the mercurial vector commitment from lattice \emph{satisfying} updatability and aggregatability with \emph{smaller} auxiliary information. We \emph{first} extend stateless update and differential update to the mercurial vector commitment and define a \emph{new} property, named updatable mercurial hiding. Then, we show how to modify our constructions to obtain the updatable mercurial vector commitment that satisfies these properties. To aggregate the openings, our constructions perfectly inherit the ability to aggregate in the $\mathsf{BASIS}$ assumption, which can break the limitation of \emph{weak} binding in the current aggregatable MVCs. In the end, we show that our constructions can be used to build the various kinds of lattice-based ZKS and ZK-EDB directly within the existing framework.
2024
PKC
Vector Commitments With Proofs of Smallness: Short Range Proofs and More
Vector commitment schemes are compressing commitments to vectors that make it possible to succinctly open a commitment for individual vector positions without revealing anything about other positions. We describe vector commitments enabling constant-size proofs that the committed vector is small (i.e., binary, ternary, or of small norm). As a special case, we obtain range proofs featuring the shortest proof length in the literature with only $3$ group elements per proof. As another application, we obtain short pairing-based NIZK arguments for lattice-related statements. In particular, we obtain short proofs (comprised of $3$ group elements) showing the validity of ring LWE ciphertexts and public keys. Our constructions are proven simulation-extractable in the algebraic group model and the random oracle model.
2024
PKC
Witness Encryption for Succinct Functional Commitments and Applications
Witness encryption (WE), introduced by Garg, Gentry, Sahai, and Waters (STOC 2013) allows one to encrypt a message to a statement x for some NP language L, such that any user holding a witness for x ∈ L can decrypt the ciphertext. The extreme power of this primitive comes at the cost of its elusiveness: a practical construction from established cryptographic assumptions is currently out of reach. In this work, we investigate a new notion of encryption that has a flavor of WE and that we can build only based on bilinear pairings, for interesting classes of computation. We do this by connecting witness encryption to functional commitments (FC). FCs are an advanced notion of commitments that allows fine-grained openings, that is non-interactive proofs to show that a commitment cm opens to v such that y = G(v), with the crucial feature that both commitments and openings are succinct. Our new WE notion, witness encryption for (succinct) functional commitment (WE-FC), allows one to encrypt a message with respect to a triple (cm, G, y), and decryption is unlocked using an FC opening that cm opens to v such that y = G(v). This mechanism is similar to the notion of witness encryption for NIZK of commitments [Benhamouda and Lin, TCC’20], with the crucial difference that ours supports commitments and decryption time whose size and complexity do not depend on the length of the committed data v. Our main contributions are therefore the formal definition of WE-FC, a generic methodology to compile an FC in bilinear groups into an associated WE-FC scheme (semantically secure in the generic group model), and a new FC construction for NC1 circuits that yields a WE-FC for the same class of functions. Similarly to [Benhamouda and Lin, TCC’20], we show how to apply WE-FC to construct multiparty reusable non-interactive secure computation (mrNISC) protocols. Crucially, the efficiency profile of WE-FC yields mrNISC protocols whose offline stage has shorter communication (only a succinct commitment from each party). As an additional contribution, we discuss further applications of WE-FC and show how to extend this primitive to better suit these settings.
2024
PKC
Zero Knowledge Protocols and Signatures from the Restricted Syndrome Decoding Problem
The Restricted Syndrome Decoding Problem (R-SDP) corresponds to the Syndrome Decoding Problem (SDP) with the additional constraint that all entries of the solution error vector must live in a fixed subset of the finite field. In this paper, we study how this problem can be applied to the construction of signatures derived from Zero-Knowledge (ZK) protocols. First, we show that R-SDP appears to be well-suited for this type of application: ZK protocols relying on SDP can easily be modified to use R-SDP, resulting in significant reductions in the communication cost. We then introduce and analyze a variant of R-SDP, which we call R-SDP(G), with the property that solution vectors can be represented with a number of bits that is slightly larger than the security parameter (which clearly provides an ultimate lower bound). This enables the design of competitive ZK protocols. We show that existing ZK protocols can greatly benefit from the use of R-SDP, achieving signature sizes in the order of 7 kB, which are smaller than those of several other schemes submitted to the additional call of NIST.