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16 February 2024
Xiaoyu Ji, Junru Li, Yifan Song
Secure multiparty computation (MPC) allows a set of $n$ parties to jointly compute a function on their private inputs. In this work, we focus on the information-theoretic MPC in the \emph{asynchronous network} setting with optimal resilience ($t
Our work closes this gap by presenting the first ACSS protocol that achieves $O(n\kappa)$ bits per sharing. When combined with the compiler from ACSS to AMPC by Choudhury and Patra [IEEE Trans. Inf. Theory '17], we obtain an AMPC with $O(n^2\kappa)$ bits per multiplication gate, improving the previously best-known result by a factor of $n^2$. Moreover, with a concurrent work that improves the compiler by Choudhury and Patra by a factor of $n$, we obtain the first AMPC with $O(n\kappa)$ bits per multiplication gate.
Our work closes this gap by presenting the first ACSS protocol that achieves $O(n\kappa)$ bits per sharing. When combined with the compiler from ACSS to AMPC by Choudhury and Patra [IEEE Trans. Inf. Theory '17], we obtain an AMPC with $O(n^2\kappa)$ bits per multiplication gate, improving the previously best-known result by a factor of $n^2$. Moreover, with a concurrent work that improves the compiler by Choudhury and Patra by a factor of $n$, we obtain the first AMPC with $O(n\kappa)$ bits per multiplication gate.
Alessandro Budroni, Jesús-Javier Chi-Domínguez, Giuseppe D'Alconzo, Antonio J. Di Scala, Mukul Kulkarni
The Linear Code Equivalence (LCE) Problem has received increased attention in recent years due to its applicability in constructing efficient digital signatures. Notably, the LESS signature scheme based on LCE is under consideration for the NIST post-quantum standardization process, along with the MEDS signature scheme that relies on an extension of LCE to the rank metric, namely Matrix Code Equivalence (MCE) Problem. Building upon these developments, a family of signatures with additional properties, including linkable ring, group, and threshold signatures, has been proposed. These novel constructions introduce relaxed versions of LCE (and MCE), wherein multiple samples share the same secret equivalence. Despite their significance, these variations have often lacked a thorough security analysis, being assumed to be as challenging as their original counterparts. Addressing this gap, our work delves into the sample complexity of LCE and MCE --- precisely, the sufficient number of samples required for efficient recovery of the shared secret equivalence. Our findings reveal, for instance, that one shouldn't use the same secret twice in the LCE setting since this enables a polynomial time (and memory) algorithm to retrieve the secret. Consequently, our results unveil the insecurity of two advanced signatures based on variants of the LCE Problem.
Vipul Goyal, Chen-Da Liu-Zhang, Yifan Song
Secure multi-party computation (MPC) allows a set of $n$ parties to jointly compute a function over their private inputs.
The seminal works of Ben-Or, Canetti and Goldreich [STOC '93] and Ben-Or, Kelmer and Rabin [PODC '94] settled the feasibility of MPC over asynchronous networks. Despite the significant line of work devoted to improving the communication complexity, current protocols with information-theoretic security and optimal resilience $t
In this work we make progress towards closing this gap. We provide a novel MPC protocol that makes black-box use of an asynchronous complete secret-sharing (ACSS) protocol, where the cost per multiplication reduces to the cost of distributing a constant number of sharings via ACSS, improving a linear factor over the state of the art by Choudhury and Patra [IEEE Trans. Inf. Theory '17].
Instantiating ACSS with the protocol by Choudhury and Patra [J. Crypto '23] we achieve an MPC protocol with $\mathcal{O}(n^3C)$ communication. Moreover, with a recent concurrent work achieving ACSS with linear cost per sharing, we achieve an MPC with $\mathcal{O}(nC)$ communication.
In this work we make progress towards closing this gap. We provide a novel MPC protocol that makes black-box use of an asynchronous complete secret-sharing (ACSS) protocol, where the cost per multiplication reduces to the cost of distributing a constant number of sharings via ACSS, improving a linear factor over the state of the art by Choudhury and Patra [IEEE Trans. Inf. Theory '17].
Instantiating ACSS with the protocol by Choudhury and Patra [J. Crypto '23] we achieve an MPC protocol with $\mathcal{O}(n^3C)$ communication. Moreover, with a recent concurrent work achieving ACSS with linear cost per sharing, we achieve an MPC with $\mathcal{O}(nC)$ communication.
Yifan Song, Xiaxi Ye
In this work, we study the communication complexity of perfectly secure MPC protocol with guaranteed output delivery against $t=(n-1)/3$ corruptions. The previously best-known result in this setting is due to Goyal, Liu, and Song (CRYPTO, 2019) which achieves $O(n)$ communication per gate, where $n$ is the number of parties.
On the other hand, in the honest majority setting, a recent trend in designing efficient MPC protocol is to rely on packed Shamir sharings to speed up the online phase. In particular, the work by Escudero et al. (CCS 2022) gives the first semi-honest protocol that achieves a constant communication overhead per gate across all parties in the online phase while maintaining overall $O(n)$ communication per gate. We thus ask the following question: ``Is it possible to construct a perfectly secure MPC protocol with GOD such that the online communication per gate is $O(1)$ while maintaining overall $O(n)$ communication per gate?''
In this work, we give an affirmative answer by providing an MPC protocol with communication complexity $O(|C|+\mathsf{Depth}\cdot n+n^5)$ elements for the online phase, and $O(|C|\cdot n+\mathsf{Depth}\cdot n^2 + n^4)$ elements for the preprocessing phase, where $|C|$ is the circuit size and $\mathsf{Depth}$ is the circuit depth.
On the other hand, in the honest majority setting, a recent trend in designing efficient MPC protocol is to rely on packed Shamir sharings to speed up the online phase. In particular, the work by Escudero et al. (CCS 2022) gives the first semi-honest protocol that achieves a constant communication overhead per gate across all parties in the online phase while maintaining overall $O(n)$ communication per gate. We thus ask the following question: ``Is it possible to construct a perfectly secure MPC protocol with GOD such that the online communication per gate is $O(1)$ while maintaining overall $O(n)$ communication per gate?''
In this work, we give an affirmative answer by providing an MPC protocol with communication complexity $O(|C|+\mathsf{Depth}\cdot n+n^5)$ elements for the online phase, and $O(|C|\cdot n+\mathsf{Depth}\cdot n^2 + n^4)$ elements for the preprocessing phase, where $|C|$ is the circuit size and $\mathsf{Depth}$ is the circuit depth.
Kaisei Kajita, Go Ohtake, Tsuyoshi Takagi
Adaptor signatures have attracted attention as a tool to ad-dress scalability and interoperability issues in blockchain applications, for example, such as atomic swaps for exchanging di˙erent cryptocur-rencies. Adaptor signatures can be constructed by extending of common digital signature schemes that both authenticate a message and disclose a secret witness to a speci˝c party. In Asiacrypt 2021, Aumayr et al. formulated the two-party adaptor signature as an independent crypto-graphic primitive. In this study, we extend the their adaptor signature scheme formulation to N party adaptor signature scheme, present its generic construction, and de˝ne the security to be satis˝ed. Next, we present a concrete construction based on Schnorr signatures and discuss the security properties.
Nobuyuki Sugio
Large language models (LLMs), exemplified by the advanced AI tool ChatGPT in 2023, have demonstrated remarkable capabilities in generating sentences, images, and program codes, driven by their development from extensive datasets. With over 100 million users worldwide, ChatGPT stands out as a leader among LLMs. Previous studies have shown its proficiency in generating program source codes for the symmetric-key block ciphers AES, CHAM, and ASCON. This study ventures into the implementation of cryptanalytic program source codes for a lightweight block cipher using ChatGPT, exploring its potential and limitations in the field of cryptography.
Hiroki Okada, Tsuyoshi Takagi
Threshold public key encryption (ThPKE) is PKE that can be decrypted by collecting "partial decryptions" from t (≤ N) out of N parties. ThPKE based on the learning with errors problem (LWE) is particularly important because it can be extended to threshold fully homomorphic encryption (ThFHE). ThPKE and ThFHE are fundamental tools for constructing multiparty computation (MPC) protocols: In 2023, NIST initiated a project (NIST IR 8214C) to establish guidelines for implementing threshold cryptosystems. Because MPC often requires simulation-security (SS), ThPKE schemes that satisfy SS (SS-ThPKE) are also important. Recently, Micciancio and Suhl (ePrint 2023/1728) presented an efficient SS-ThPKE scheme based on LWE with a polynomial modulus. However, the scheme requires to use a nonstandard problem called “known-norm LWE” for the security proof because the norm ∥e∥ of the error of the public key is leaked from the partial decryptions. This leads to the following two challenges: 1) The construction based on LWE incurs a security loss of approximately 13 bits for 128-bit security. 2) No construction based on (standard) Ring-LWE has been presented. In this paper, we address both of these challenges: we propose an efficient SS-ThPKE scheme whose security is (directly) reduced from standard (Ring-)LWE with a polynomial modulus. The core technique of our construction is what we call "error sharing". We distribute shares of a small error ζ via secret sharing, and use them to prevent leakage of ∥e∥ from partial decryptions.
Sönke Jendral
CRYSTALS-Dilithium is a post-quantum secure digital signature algorithm currently being standardised by NIST. As a result, devices making use of CRYSTALS-Dilithium will soon become generally available and be deployed in various environments. It is thus important to assess the resistance of CRYSTALS-Dilithum implementations to physical attacks.
In this paper, we present an attack on a CRYSTALS-Dilithium implementation in hedged mode in ARM Cortex-M4 using fault injection. Voltage glitching is performed to skip computation of a seed during the generation of the signature. We identified settings that consistently skip the desired function without crashing the device. After the successful fault injection, the resulting signature allows for the extraction of the secret key vector. Our attack succeeds with probability 0.582 in a single trace.
We also propose countermeasures against the presented attack.
Hao Chung, Tim Roughgarden, Elaine Shi
Users bid in a transaction fee mechanism (TFM) to get their transactions included and confirmed by a blockchain protocol. Roughgarden (EC'21) initiated the formal treatment of TFMs and proposed three requirements: user incentive compatibility (UIC), miner incentive compatibility (MIC), and a form of collusion-resilience called OCA-proofness. Ethereum's EIP-1559 mechanism satisfies all three properties simultaneously when there is no contention between transactions, but loses the UIC property when there are too many eligible transactions to fit in a single block. Chung and Shi (SODA'23) considered an alternative notion of collusion-resilience, called c-side-constract-proofness (c-SCP), and showed that, when there is contention between transactions, no TFM can satisfy UIC, MIC, and c-SCP for any c at least 1. OCA-proofness asserts that the users and a miner should not be able to "steal from the protocol" and is intuitively weaker than the c-SCP condition, which stipulates that a coalition of a miner and a subset of users should not be able to profit through strategic deviations (whether at the expense of the protocol or of the users outside the coalition).
Our main result is the first proof that, when there is contention between transactions, no (possibly randomized) direct-revelation TFM satisfies UIC, MIC, and OCA-proofness. This result resolves the main open question in Roughgarden(EC'21). We also suggest several relaxations of the basic model that allow our impossibility result to be circumvented.
Our main result is the first proof that, when there is contention between transactions, no (possibly randomized) direct-revelation TFM satisfies UIC, MIC, and OCA-proofness. This result resolves the main open question in Roughgarden(EC'21). We also suggest several relaxations of the basic model that allow our impossibility result to be circumvented.
Hart Montgomery, Sikhar Patranabis
We show that key exchange and two-party computation are exactly equivalent to monoid actions with certain structural and hardness properties. To the best of our knowledge, this is the first "natural" characterization of the mathematical structure inherent to any key exchange or two-party computation protocol, and the first explicit proof of the necessity of mathematical structure for public-key cryptography. We then utilize these characterizations to show a new black-box separation result, while also achieving a simpler and more general version of an existing black-box separation result. Concretely, we obtain the following results:
- Two-Party Key Exchange. We show that that any two-party noninteractive key exchange protocol is equivalent to the existence of an abelian monoid equipped with a natural hardness property, namely (distributional) unpredictability. More generally, we show that any $k$-round (two-party) key exchange protocol is essentially equivalent to the existence of a (distributional) unpredictable monoid with certain commutator-like properties. We then use a generic version of this primitive to show a simpler and more general version of Rudich's (Crypto '91) black-box separation of $k$-round and $(k+1)$-round key exchange.
- Two-Party Computation. We show that any maliciously secure two-party computation protocol is also equivalent to a monoid action with commutator-like properties and certain hardness guarantees. We then use a generic version of this primitive to show a black-box separation between $k$-round semi-honest secure two-party computation and $(k+1)$-round maliciously secure two-party computation. This yields the first black-box separation (to our knowledge) between $k$-round and $(k+1)$-round maliciously secure two-party computation protocols.
We believe that modeling cryptographic primitives as mathematical objects (and our approach of using such modeling for black-box separations) may have many other potential applications and uses in understanding what sort of assumptions and mathematical structure are necessary for certain cryptoprimitives.
- Two-Party Key Exchange. We show that that any two-party noninteractive key exchange protocol is equivalent to the existence of an abelian monoid equipped with a natural hardness property, namely (distributional) unpredictability. More generally, we show that any $k$-round (two-party) key exchange protocol is essentially equivalent to the existence of a (distributional) unpredictable monoid with certain commutator-like properties. We then use a generic version of this primitive to show a simpler and more general version of Rudich's (Crypto '91) black-box separation of $k$-round and $(k+1)$-round key exchange.
- Two-Party Computation. We show that any maliciously secure two-party computation protocol is also equivalent to a monoid action with commutator-like properties and certain hardness guarantees. We then use a generic version of this primitive to show a black-box separation between $k$-round semi-honest secure two-party computation and $(k+1)$-round maliciously secure two-party computation. This yields the first black-box separation (to our knowledge) between $k$-round and $(k+1)$-round maliciously secure two-party computation protocols.
We believe that modeling cryptographic primitives as mathematical objects (and our approach of using such modeling for black-box separations) may have many other potential applications and uses in understanding what sort of assumptions and mathematical structure are necessary for certain cryptoprimitives.
Miranda Christ, Sam Gunn
We construct pseudorandom error-correcting codes (or simply pseudorandom codes), which are error-correcting codes with the property that any polynomial number of codewords are pseudorandom to any computationally-bounded adversary. Efficient decoding of corrupted codewords is possible with the help of a decoding key.
We build pseudorandom codes that are robust to substitution and deletion errors, where pseudorandomness rests on standard cryptographic assumptions. Specifically, pseudorandomness is based on either $2^{O(\sqrt{n})}$-hardness of LPN, or polynomial hardness of LPN and the planted XOR problem at low density.
As our primary application of pseudorandom codes, we present an undetectable watermarking scheme for outputs of language models that is robust to cropping and a constant rate of random substitutions and deletions. The watermark is undetectable in the sense that any number of samples of watermarked text are computationally indistinguishable from text output by the original model. This is the first undetectable watermarking scheme that can tolerate a constant rate of errors.
Our second application is to steganography, where a secret message is hidden in innocent-looking content. We present a constant-rate stateless steganography scheme with robustness to a constant rate of substitutions. Ours is the first stateless steganography scheme with provable steganographic security and any robustness to errors.
We build pseudorandom codes that are robust to substitution and deletion errors, where pseudorandomness rests on standard cryptographic assumptions. Specifically, pseudorandomness is based on either $2^{O(\sqrt{n})}$-hardness of LPN, or polynomial hardness of LPN and the planted XOR problem at low density.
As our primary application of pseudorandom codes, we present an undetectable watermarking scheme for outputs of language models that is robust to cropping and a constant rate of random substitutions and deletions. The watermark is undetectable in the sense that any number of samples of watermarked text are computationally indistinguishable from text output by the original model. This is the first undetectable watermarking scheme that can tolerate a constant rate of errors.
Our second application is to steganography, where a secret message is hidden in innocent-looking content. We present a constant-rate stateless steganography scheme with robustness to a constant rate of substitutions. Ours is the first stateless steganography scheme with provable steganographic security and any robustness to errors.
Manuel Barbosa, Kai Gellert, Julia Hesse, Stanislaw Jarecki
In the past three decades, an impressive body of knowledge has been built around secure and private password authentication. In particular, secure password-authenticated key exchange (PAKE) protocols require only minimal overhead over a classical Diffie-Hellman key exchange. PAKEs are also known to fulfill strong composable security guarantees that capture many password-specific concerns such as password correlations or password mistyping, to name only a few. However, to enjoy both round-optimality and strong security, applications of PAKE protocols must provide unique session and participant identifiers. If such identifiers are not readily available, they must be agreed upon at the cost of additional communication flows, a fact which has been met with incomprehension among practitioners, and which hindered the adoption of provably secure password authentication in practice.
In this work, we resolve this issue by proposing a new paradigm for truly password-only yet securely composable PAKE, called bare PAKE. We formally prove that two prominent PAKE protocols, namely CPace and EKE, can be cast as bare PAKEs and hence do not require pre-agreement of anything else than a password. Our bare PAKE modeling further allows us to investigate a novel "reusability" property of PAKEs, i.e., whether $n^2$ pairwise keys can be exchanged from only $n$ messages, just as the Diffie-Hellman non-interactive key exchange can do in a public-key setting. As a side contribution, this add-on property of bare PAKEs leads us to observe that some previous PAKE constructions relied on unnecessarily strong, "reusable" building blocks. By showing that ``non-reusable'' tools suffice for standard PAKE, we open a new path towards round-optimal post-quantum secure password-authenticated key exchange.
Vladimir Shpilrain, Bianca Sosnovski
Cayley hash functions are based on a simple idea of using a pair of semigroup elements, $A$ and $B$, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the semigroup. The main advantage of Cayley hash functions compared to, say, hash functions in the SHA family is that when an already hashed document is amended, one does not have to hash the whole amended document all over again, but rather hash just the amended part and then multiply the result by the hash of the original document. Some authors argued that this may be a security hazard, specifically that this property may facilitate finding a second preimage by splitting a long bit string into shorter pieces. In this paper, we offer a way to get rid of this alleged disadvantage and keep the advantages at the same time. We call this method ``Cayley hashing with cookies" using terminology borrowed from the theory of random walks in a random environment. For the platform semigroup, we use $2\times 2$ matrices over $F_p$.
Hyeonbum Lee, Jae Hong Seo
Nova is a new type of recursive proof system that uses folding scheme as its core building block. This brilliant idea of folding relations can significantly reduce recursion overhead. In this paper, we study some issues related to Nova's soundness proof that relies on the soundness of the folding scheme in a recursive manner.
First, its proof strategy, due to its recursive nature, inevitably expands the running time of the recursive extractor polynomially for each additional recursive step. This constrains Nova's soundness model to have only logarithmically bounded recursive steps, and so the soundness proof in this limited model does not guarantee the soundness for linear rounds in the security parameter, e.g., 128 rounds for 128 bit security. On the other hand, there are no known attacks on arbitrary depth recursion of Nova, leaving a gap between theoretical security guarantees and real-world attacks. We aim to bridge this gap in two opposite directions. In a negative direction, we present a recursive proof system that is unforgeable in a log-round model but forgeable if used in linear rounds. This shows that the soundness proof in the log-round model might tell nothing about real-world uses that require linearly long rounds. In a positive direction, we show that when Nova uses a specific group-based folding scheme, its knowledge soundness over polynomial rounds can be proven in the algebraic group model with our refinements. To the best of our knowledge, this is the first result to show Nova's polynomial rounds soundness.
Second, the folding scheme is converted non-interactively via the Fiat-Shamir transformation and then arithmetized into R1CS. Therefore, the soundness of Nova using the non-interactive folding scheme essentially relies on the heuristic random oracle instantiation in the standard model. In our new soundness proof for Nova in the algebraic group model, we replace this heuristics with a cryptographic hash function with special property. We view this hash function as an independent object of interest and expect it to help further understand the soundness of Nova.
First, its proof strategy, due to its recursive nature, inevitably expands the running time of the recursive extractor polynomially for each additional recursive step. This constrains Nova's soundness model to have only logarithmically bounded recursive steps, and so the soundness proof in this limited model does not guarantee the soundness for linear rounds in the security parameter, e.g., 128 rounds for 128 bit security. On the other hand, there are no known attacks on arbitrary depth recursion of Nova, leaving a gap between theoretical security guarantees and real-world attacks. We aim to bridge this gap in two opposite directions. In a negative direction, we present a recursive proof system that is unforgeable in a log-round model but forgeable if used in linear rounds. This shows that the soundness proof in the log-round model might tell nothing about real-world uses that require linearly long rounds. In a positive direction, we show that when Nova uses a specific group-based folding scheme, its knowledge soundness over polynomial rounds can be proven in the algebraic group model with our refinements. To the best of our knowledge, this is the first result to show Nova's polynomial rounds soundness.
Second, the folding scheme is converted non-interactively via the Fiat-Shamir transformation and then arithmetized into R1CS. Therefore, the soundness of Nova using the non-interactive folding scheme essentially relies on the heuristic random oracle instantiation in the standard model. In our new soundness proof for Nova in the algebraic group model, we replace this heuristics with a cryptographic hash function with special property. We view this hash function as an independent object of interest and expect it to help further understand the soundness of Nova.
Eugene Frimpong, Alexandros Bakas, Camille Foucault, Antonis Michalas
Functional Encryption (FE) is a cutting-edge cryptographic technique that enables a user with a specific functional decryption key to determine a certain function of encrypted data without gaining access to the underlying data. Given its potential and the fact that FE is still a relatively new field, we set out to investigate how it could be applied to resource-constrained environments. This work presents what we believe to be the first lightweight FE scheme explicitly designed for resource-constrained devices. We also propose a use case protocol that demonstrates how our scheme can secure an Internet of Things (IoT) architecture where relevant devices collect data and securely deliver them to a storage server, where an analyst can request access to the encrypted data. Finally, we conduct thorough experiments on two commercially available resource-constrained devices to provide compelling evidence of our approach's practicality and efficiency. Although the results of our evaluations show that there is room for improvement in the proposed scheme, this work represents one of the first attempts to apply FE to the IoT setting that can directly impact people's daily lives and the everyday operations of organizations.
Seongtaek Chee, Kyung Chul Jeong, Tanja Lange, Nari Lee, Alex Pellegrini, Hansol Ryu
We analyse Layered ROLLO-I, a code-based cryptosystem submitted to the Korean post quantum cryptography competition, of which four versions have been proposed. We show that the first two versions do not provide the claimed security against rank decoding attacks and give reductions to small instances of ROLLO-I for which such attacks are even more effective. Finally, we provide two efficient message recovery attacks, affecting every security level of the first three versions of Layered ROLLO-I and security levels 128 and 192 of the fourth version.
Changrui Mu, Shafik Nassar, Ron D. Rothblum, Prashant Nalini Vasudevan
A zero-knowledge proof enables a prover to convince a verifier that $x \in S$, without revealing anything beyond this fact. By running a zero-knowledge proof $k$ times, it is possible to prove (still in zero-knowledge) that $k$ separate instances $x_1,\dots,x_k$ are all in $S$. However, this increases the communication by a factor of $k$. Can one do better? In other words, is (non-trivial) zero-knowledge batch verification for $S$ possible?
Recent works by Kaslasi et al. (TCC 2020, Eurocrypt 2021) show that any problem possessing a non-interactive statistical zero-knowledge proof (NISZK) has a non-trivial statistical zero-knowledge batch verification protocol. Their results had two major limitations: (1) to batch verify $k$ inputs of size $n$ each, the communication in their batch protocol is roughly $\textrm{poly}(n,\log{k})+O(k)$, which is better than the naive cost of $k \cdot \textrm{poly}(n)$ but still scales linearly with $k$, and, (2) the batch protocol requires $\Omega(k)$ rounds of interaction.
In this work we remove both of these limitations by showing that any problem in $NISZK$ has a non-interactive statistical zero-knowledge batch verification protocol with communication $\textrm{poly}(n,\log{k})$.
Recent works by Kaslasi et al. (TCC 2020, Eurocrypt 2021) show that any problem possessing a non-interactive statistical zero-knowledge proof (NISZK) has a non-trivial statistical zero-knowledge batch verification protocol. Their results had two major limitations: (1) to batch verify $k$ inputs of size $n$ each, the communication in their batch protocol is roughly $\textrm{poly}(n,\log{k})+O(k)$, which is better than the naive cost of $k \cdot \textrm{poly}(n)$ but still scales linearly with $k$, and, (2) the batch protocol requires $\Omega(k)$ rounds of interaction.
In this work we remove both of these limitations by showing that any problem in $NISZK$ has a non-interactive statistical zero-knowledge batch verification protocol with communication $\textrm{poly}(n,\log{k})$.
Ren Taguchi, Atsushi Takayasu
Thus far, several papers estimated concrete quantum resources of Shor’s algorithm for solving a binary elliptic curve discrete logarithm problem. In particular, the complexity of computing quantum inversions over a binary field F2n is dominant when running the algorithm, where n is a degree of a binary elliptic curve. There are two major methods for quantum inversion, i.e., the quantum GCD-based inversion and the quantum FLT-based inversion. Among them, the latter method is known to require more qubits; however, the latter one is valuable since it requires much fewer Toffoli gates and less depth. When n = 571, Kim-Hong’s quantum GCD-based inversion algorithm (Quantum Information Processing 2023) and Taguchi-Takayasu’s quantum FLT-based inversion algorithm (CT-RSA 2023) require 3, 473 qubits and 8, 566 qubits, respectively. In contrast, for the same n = 571, the latter algorithm requires only 2.3% of Toffoli gates and 84% of depth compared to the former one. In this paper, we modify Taguchi-Takayasu’s quantum FLT-based inversion algorithm to reduce the required qubits. While Taguchi-Takayasu’s FLT-based inversion algorithm takes an addition chain for n−1 as input and computes a sequence whose number is the same as the length of the chain, our proposed algorithm employs an uncomputation step and stores a shorter one. As a result, our proposed algorithm requires only 3, 998 qubits for n = 571, which is only 15% more than Kim-Hong’s GCD-based inversion algorithm. Furthermore, our proposed algorithm preserves the advantage of FLT-based inversion since it requires only 3.7% of Toffoli gates and 77% of depth compared to Kim-Hong’s GCD-based inversion algorithm for n = 571.
Surya Mathialagan, Spencer Peters, Vinod Vaikuntanathan
We study succinct non-interactive arguments (SNARGs) and succinct non-interactive arguments of knowledge (SNARKs) for the class $\mathsf{UP}$ in the reusable designated verifier model. $\mathsf{UP}$ is an expressive subclass of $\mathsf{NP}$ consisting of all $\mathsf{NP}$ languages where each instance has at most one witness; a designated verifier SNARG (dvSNARG) is one where verification of the SNARG proof requires a private verification key; and such a dvSNARG is reusable if soundness holds even against a malicious prover with oracle access to the (private) verification algorithm.
Our main results are as follows.
(1) A reusably and adaptively sound zero-knowledge (zk) dvSNARG for $\mathsf{UP}$, from subexponential LWE and evasive LWE (a relatively new but popular variant of LWE). Our SNARGs achieve very short proofs of length $(1 + o(1)) \cdot \lambda$ bits for $2^{-\lambda}$ soundness error.
(2) A generic transformation that lifts any ``Sahai-Waters-like'' (zk) SNARG to an adaptively sound (zk) SNARG, in the designated-verifier setting. In particular, this shows that the Sahai-Waters SNARG for $\mathsf{NP}$ is adaptively sound in the designated verifier setting, assuming subexponential hardness of the underlying assumptions. The resulting SNARG proofs have length $(1 + o(1)) \cdot \lambda$ bits for $2^{-\lambda}$ soundness error. Our result sidesteps the Gentry-Wichs barrier for adaptive soundness by employing an exponential-time security reduction.
(3) A generic transformation that lifts any adaptively sound (zk) SNARG for $\mathsf{UP}$ to an adaptively sound (zk) SNARK for $\mathsf{UP}$, while preserving zero-knowledge. The resulting SNARK achieves the strong notion of black-box extraction. There are barriers to achieving such SNARKs for all of $\mathsf{NP}$ from falsifiable assumptions, so our restriction to $\mathsf{UP}$ is, in a sense, necessary.
Applying (3) to our SNARG for $\mathsf{UP}$ from evasive LWE (1), we obtain a reusably and adaptively sound designated-verifier zero-knowledge SNARK for $\mathsf{UP}$ from subexponential LWE and evasive LWE. Moreover, applying both (2) and (3) to the Sahai-Waters SNARG, we obtain the same result from LWE, subexponentially secure one-way functions, and subexponentially secure indistinguishability obfuscation. Both constructions have succinct proofs of size $\mathsf{poly}(\lambda)$. These are the first SNARK constructions (even in the designated-verifier setting) for a non-trivial subset of $\mathsf{NP}$ from (sub-exponentially) falsifiable assumptions.
(1) A reusably and adaptively sound zero-knowledge (zk) dvSNARG for $\mathsf{UP}$, from subexponential LWE and evasive LWE (a relatively new but popular variant of LWE). Our SNARGs achieve very short proofs of length $(1 + o(1)) \cdot \lambda$ bits for $2^{-\lambda}$ soundness error.
(2) A generic transformation that lifts any ``Sahai-Waters-like'' (zk) SNARG to an adaptively sound (zk) SNARG, in the designated-verifier setting. In particular, this shows that the Sahai-Waters SNARG for $\mathsf{NP}$ is adaptively sound in the designated verifier setting, assuming subexponential hardness of the underlying assumptions. The resulting SNARG proofs have length $(1 + o(1)) \cdot \lambda$ bits for $2^{-\lambda}$ soundness error. Our result sidesteps the Gentry-Wichs barrier for adaptive soundness by employing an exponential-time security reduction.
(3) A generic transformation that lifts any adaptively sound (zk) SNARG for $\mathsf{UP}$ to an adaptively sound (zk) SNARK for $\mathsf{UP}$, while preserving zero-knowledge. The resulting SNARK achieves the strong notion of black-box extraction. There are barriers to achieving such SNARKs for all of $\mathsf{NP}$ from falsifiable assumptions, so our restriction to $\mathsf{UP}$ is, in a sense, necessary.
Applying (3) to our SNARG for $\mathsf{UP}$ from evasive LWE (1), we obtain a reusably and adaptively sound designated-verifier zero-knowledge SNARK for $\mathsf{UP}$ from subexponential LWE and evasive LWE. Moreover, applying both (2) and (3) to the Sahai-Waters SNARG, we obtain the same result from LWE, subexponentially secure one-way functions, and subexponentially secure indistinguishability obfuscation. Both constructions have succinct proofs of size $\mathsf{poly}(\lambda)$. These are the first SNARK constructions (even in the designated-verifier setting) for a non-trivial subset of $\mathsf{NP}$ from (sub-exponentially) falsifiable assumptions.
Keita Emura, Shingo Sato, Atsushi Takayasu
Keyed homomorphic public key encryption (KHPKE) is a variant of homomorphic public key encryption, where only users who have a homomorphic evaluation key can perform a homomorphic evaluation. Then, KHPKE satisfies the CCA2 security against users who do not have a homomorphic evaluation key, while it satisfies the CCA1 security against users who have the key. Thus far, several KHPKE schemes have been proposed under the standard Diffie-Hellman-type assumptions and keyed fully homomorphic encryption (KFHE) schemes have also been proposed from lattices. As a natural extension, there is an identity-based variant of KHPKE; however, the security is based on a $q$-type assumption and there are no attribute-based variants. Moreover, there are no identity-based variants of KFHE schemes due to the complex design of the known KFHE schemes. In this paper, we obtain two results for constructing the attribute-based variants. First, we propose an attribute-based KFHE (ABKFHE) scheme from lattices. We start by designing the first KFHE scheme secure solely under the LWE assumption in the standard model. Since the design is conceptually much simpler than known KFHE schemes, we just replace their building blocks with attribute-based ones and obtain the proposed ABKFHE schemes. Next, we propose an efficient attribute-based KHPKE (ABKHE) scheme from a pair encoding scheme (PES). Due to the benefit of PES, we obtain various ABKHE schemes that contain the first identity-based KHPKE scheme secure under the standard $k$-linear assumption and the first pairing-based ABKHE schemes supporting more expressive predicates.