International Association for Cryptologic Research

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18 February 2026

Prabhanjan Ananth, Divyanshu Bhardwaj, Aparna Gupte
ePrint Report ePrint Report
Classical non-interactive secure computation, despite being extensively studied, suffers from an inherent barrier: adversaries can learn the entire residual function via resetting attacks. We investigate whether quantum resources can circumvent this barrier and restrict adversarial leakage. Our results are as follows:

1. $\textbf{Definitions}$: We introduce new security definitions for the one-message MPC and 2PC settings that restrict the amount of adversarial leakage compared to prior classical definitions. 2. $\textbf{MPC}$: There exist information-theoretically secure one-message multi-party computation protocols in the oracle model in both the quantum pre-processing and classical pre-processing settings. 3. $\textbf{2PC}$: There exist semi-honest secure one-message two-party computation for (randomized) pseudorandom functionalities in the plain model based on LWE and maliciously secure one-message two-party computation for (randomized) constrained functionalities in the CRS model based on iO. Prior work by [Gupte, Liu, Raizes, Roberts and, Vaikuntanathan STOC 2025] achieved semi-honest security based on iO.

Our results demonstrate the power of quantum information to circumvent barriers in classical secure computation.
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Jintong Yu
ePrint Report ePrint Report
Deep Learning-based Non-profiled Side-Channel Analysis (DL-NSCA) enables automated feature extraction and obviates the need for a profiling device. However, existing methods mostly rely on leakage from non-linear operations and require prior knowledge of the target algorithm and its plaintexts/ciphertexts, which restricts their applicability to black-box scenarios and proprietary implementations.

Motivated by the ubiquity of plaintext-key XOR operations in cryptographic algorithms, this paper presents a new SCA perspective based on leakage from linear operations to enable cross-algorithm attacks. We first theoretically demonstrate that leakage from invertible linear operations, referred to as blind leakage, introduces two symmetric maxima of the correlations between key guesses and leakage (one for non-blind leakage), rendering existing distinguishers ineffective. To address this issue, we then propose a distinguisher, VS-GBA, applicable to both blind and non-blind leakage, and improve its robustness using a Gaussian Borda voting scheme. Experimental results on a high-noise ARM Cortex-M4 device demonstrate that our method enables effective cross-algorithm attacks on masked AES, ASCON, and PRESENT, and outperforms non-blind leakage attacks. Furthermore, attacking XTS-AES demonstrates that blind leakage exploitation extends the boundaries of DL-NSCA to scenarios where plaintexts/ciphertexts are XORed with a tweak.
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Prabhanjan Ananth, Divyanshu Bhardwaj
ePrint Report ePrint Report
Quantum one-time programs, introduced by Broadbent, Gutowski, and Stebila [CRYPTO 2013], is a compiler that takes as input a program $f$ and converts it into a quantum state $\sigma_f$ such that given $\sigma_f$, it is possible to only obtain the output of $f$ on a single input. Our goal is to investigate the class of functionalities for which we can design one-time programs in the plain model. We show the following:

1. We show that, assuming learning with errors, one-time programs for (randomized) pseudorandom functionalities exist in the plain model. In contrast, the result by Gupte, Liu, Raizes, Roberts, and Vaikuntanathan [STOC 2025] relied upon indistinguishability obfuscation.

2. We show that, assuming indistinguishability obfuscation, one-time programs for any class of randomized functionalities exist in the plain model, as long as the class of functionalities satisfy a natural security notion called constrained security.

Additionally, we give constructions of one-time encryption and one-time message authentication codes based on learning with errors. Finally, we show that a novel variant of one-time programs imply one-message secure two-party computation protocols for non-trivial functionalities in the CRS model.
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Rohit Chatterjee, Yunqi Li, Prashant Nalini Vasudevan
ePrint Report ePrint Report
We study the implications of the existence of weak Zero-Knowledge (ZK) protocols for worst-case hard languages. These are protocols that have completeness, soundness, and zero-knowledge errors (denoted $\epsilon_c$, $\epsilon_s$, and $\epsilon_z$, respectively) that might not be negligible. Under the assumption that there are worst-case hard languages in NP, we show the following:

1. If all languages in NP have NIZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+\epsilon_z < 1 $, then One-Way Functions (OWFs) exist.

This covers all possible non-trivial values for these error rates. It additionally implies that if all languages in NP have such NIZK proofs and ec is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition $ \epsilon_c+\sqrt{\epsilon_s}+\epsilon_z < 1 $ [Chakraborty et al., CRYPTO 2025].

2. If all languages in NP have k-round public-coin ZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+(2k-1)\epsilon_z < 1 $, then OWFs exist.

3. If, for some constant k, all languages in NP have k-round public-coin ZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+k\epsilon_z < 1 $, then infinitely-often OWFs exist.
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Hiroki Furue, Yasuhiko Ikematsu
ePrint Report ePrint Report
The unbalanced oil and vinegar signature scheme (UOV) was proposed by Kipnis et al. in 1999 as a multivariate-based scheme. UOV is regarded as one of the most promising candidates for post-quantum cryptography owing to its short signatures and fast performance. Recently, Ran proposed a new key recovery attack on UOV over a field of even characteristic, reducing the security of its proposed parameters. Furthermore, Jin et al. generalized Ran’s attack to schemes over a field of arbitrary characteristic by exploiting the structure of the symmetric algebra. In this work, we propose a new framework for recovering the secret subspace of UOV over a finite field $\mathbb{F}_{p^e}$ by generalizing these preceding results. First, we show that a key recovery against UOV can be successfully performed using the XL algorithm by exploiting the structure of the $p$-truncated polynomial ring $R^{(p)}=\mathbb{F}_{p^e}[x_1,\dots,x_n]/ \langle x_1^p,\dots,x_n^p\rangle$. This result simplifies the description of the attacks proposed by Jin et al.\ by formulating them in terms of the polynomial ring, independent of the structure of the symmetric algebra. Second, we generalize this result to the polynomial rings of more general forms, namely, the $p^\ell$-truncated polynomial rings $R^{(p^\ell)}$ for any $1 \le \ell \le e$. This result is due to our description in terms of the polynomial ring and can relax the constraints on the solving degree of the XL algorithm using $R^{(p^\ell)}$ by taking a larger $\ell$. Finally, we consider performing the reconciliation and intersection attacks using the $p^\ell$-truncated polynomial rings against UOV. In particular, we newly take into account the intersection attack using this framework, which has not been considered in previous analyses. Based on our complexity estimation, we confirm that the optimal complexity of the reconciliation attack using the proposed framework is consistent with that of the symmetric-algebra attack by Jin et al. We further show that the intersection attack using the proposed framework outperforms the reconciliation attack against the proposed parameters of UOV and reduces the security of multiple parameters compared to their claimed security levels. In addition, we show that our complexity estimation of the reconciliation attack using the proposed framework reduces the security of multiple parameters of SNOVA compared to previously known attacks.
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Wujie Xiong, Hao Zhou, Yutong Ye, Ruoming Jin, Lei Xu
ePrint Report ePrint Report
Sparse General Matrix-Matrix Multiplication (SpGEMM) is a fundamental but computationally intensive operation that underpins many scientific workloads, including numerous AI applications. With the increasing demands for data security, privacy-preserving computation techniques such as Fully Homomorphic Encryption (FHE) have gained significant attention for their ability to process sensitive data without decryption. Nonetheless, executing SpGEMM within the framework of FHE presents significant challenges. The most effective SpGEMM algorithms exploit matrix sparsity to minimize computational costs; however, FHE obscures both the data values and the sparsity structures. Prior FHE-based privacy-preserving computation frameworks either ignore the inherent sparsity of matrices and rely on dense General Matrix-Matrix Multiplication (GEMM), incurring substantial overhead from redundant homomorphic multiplications, or they attempt to exploit sparsity by encrypting only the non-zero values, which inadvertently exposes sensitive positional information. To address this gap and achieve a better balance between efficiency and privacy, we propose an efficient FHE-compatible SpGEMM framework that enables oblivious data and position processing under a Single Instruction Multiple Data (SIMD) scheme. Moreover, we extend our method to support an arbitrary number of sparse matrices (FHE-SpGEMCM). The efficiency analysis shows that our method achieves an average homomorphic computation cost of $(n_A n_B)^2 / (n^2 N)$, where $n_A$ and $n_B$ represent the number of nonzero elements in $A$ and $B$, respectively, $n$ is the shared inner dimension of the multiplication, and $N$ denotes the batch size used in FHE. Experimental results demonstrate that for square matrices of scale $2^9$, our scheme achieves an average speedup of $439.25\times$ and a $10.68\times$ reduction in memory consumption compared to state-of-the-art baselines that ignore sparsity. Furthermore, when the scale increases to $2^{13}$, our method yields up to a $1201.77\times$ speedup over baselines that only exploit the sparsity of a single matrix.
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Haolin Liu, Adrien Siproudhis, Samuel Experton, Peter Lorenz, Christina Boura, Thomas Peyrin
ePrint Report ePrint Report
Neural network model extraction has recently emerged as an important security concern, as adversaries attempt to recover a network’s parameters via black-box queries. Carlini et al. proposed in CRYPTO’20 a model extraction approach inspired by differential cryptanalysis, consisting of two steps: signature extraction, which extracts the absolute values of network weights layer by layer, and sign extraction, which determines the signs of these signatures. However, in practice this signature-extraction method is limited to very shallow networks only, and the proposed sign-extraction method is exponential in time. Recently, Canales-Martínez et al. (Eurocrypt’24) proposed a polynomial-time sign-extraction method, but it assumes the corresponding signatures have already been successfully extracted and can fail on so-called low-confidence neurons.

In this work, we first revisit and refine the signature extraction process by systematically identifying and addressing for the first time critical limitations of Carlini et al.'s signature-extraction method. These limitations include rank deficiency and noise propagation from deeper layers. To overcome these challenges, we propose efficient algorithmic solutions for each of the identified issues, greatly improving the capabilities of signature extraction. Our approach permits the extraction of much deeper networks than previously possible.

In addition, we propose new methods to improve numerical precision in signature extraction, and enhance the sign extraction part by combining two polynomial methods to avoid exponential exhaustive search in the case of low-confidence neurons. This leads to the very first end-to-end model extraction method that runs in polynomial time.

We validate our attack through extensive experiments on ReLU-based neural networks, demonstrating significant improvements in extraction depth. For instance, our attack extracts consistently at least eight layers of neural networks trained on either the MNIST or CIFAR-10 datasets, while previous works could barely extract the first three layers of networks of similar width. Our results represent a crucial step toward practical attacks on larger and more complex neural network architectures.
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Geoffroy Couteau, Naman Kumar
ePrint Report ePrint Report
Following the recent work of (Doerner et al., CCS 2025), we introduce an improved reduction of OLE to OT. We prove the following: there is a perfectly-secure OLE-to-OT reduction over any $n$-bit field $\mathbb{F}$ using almost-linear communication $n\cdot 2^{O(\log^{*} n)}$ and almost-constant rounds $O((\log^{*} n)^2)$. In the course of proving our result, we also introduce new perfectly-secure protocols for computing shares of the equality and greater-than predicate on $n$-bit strings in the OT-hybrid model using $O(n)$ communication and $\log^{*} n$ rounds. These results are of independent interest.
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Ryann Cartor, Nathan Daly, Giulia Gaggero, Jason T. LeGrow, Andrea Sanguineti, Silvia Sconza
ePrint Report ePrint Report
We present One Round "Cheating" Adaptor Signatures (OR- CAS): a novel and efficient construction of adaptor signature schemes from CSI-FiSh. Our protocol improves substantially on existing group action-based schemes: Unlike IAS (Tairi et al., FC 2021), our scheme does not require expensive non-interactive zero-knowledge proofs, and unlike adaptor MCSI-FiSh (Jana et al., CANS 2024) our construction does not require any modification to the underlying digital signature scheme. We prove the protocol’s security under the strong security no- tions of Dai et al. (Indocrypt 2022) and Gerhart et al. (Eurocrypt 2024).
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Marvin Beckmann, Christian Majenz
ePrint Report ePrint Report
Ring signatures are a powerful primitive that allows a member to sign on behalf of a group, without revealing their identity. Recently, ring signatures have received additional attention as an ingredient for post-quantum deniable authenticated key exchange, e.g., for a post-quantum version of the Signal protocol, employed by virtually all end-to-end-encrypted messenger services. While several ring signature constructions from post-quantum assumptions offer suitable security and efficiency for use in deniable key exchange, they are currently proven secure in the random oracle model (ROM) only, which is insufficient for post-quantum security.

In this work, we provide four security reductions in the quantum-accessible random oracle model (QROM) for two generic ring signature constructions: two for the AOS framework and two for a construction paradigm based on ring trapdoors, whose generic backbone we formalize. The two security proofs for AOS ring signatures differ in their requirements on the underlying sigma protocol and their tightness. The two reductions for the ring-trapdoor-based ring signatures exhibit various differences in requirements and the security they provide. We employ the measure-and-reprogram technique, QROM straightline extraction tools based on the compressed oracle, history-free reductions and QROM reprogramming tools. To make use of Rényi divergence properties in the QROM, we study the behavior of quantum algorithms that interact with an oracle whose distribution is based on one of two different distributions over the set of outputs. We provide tight bounds for the statistical distance, show that the Rényi divergence can not be used to replace the entire oracle and provide a workaround.
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Pyrros Chaidos, Pooya Farshim, Denis Firsov, Dimitar Jetchev, Aggelos Kiayias, Markulf Kohlweiss, Anca Nitulescu
ePrint Report ePrint Report
We develop formal code-based security definitions for blockchain bridges and apply them to several bridge architectures deployed in practice. We derive both traditional pen-and-paper proofs and on the other, automated guarantees against bounded counterexamples. The latter is achieved via bounded model checking of our formally specified properties, implemented in Quint, a specification language and model checker closely related to TLA+.

Our definitions are expressed in a precise, code-based variant of the Universal Composition (UC) framework. This enables the modular use of hybrid functionalities—even for property-based definitions—and is essential for bounded model checking, where underlying primitives must be idealized.

Accordingly, we idealize and model-check all building blocks used in our protocols. Notably, we formulate a novel UC ideal functionality for Advanced Threshold Signatures (ATS) and modelcheck it for attacks to ensure its robustness
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Ngoc Khanh Nguyen, Jan Niklas Siemer
ePrint Report ePrint Report
In this work, we investigate the landscape of emerging lattice-based assumptions tailored for anonymous credentials, focusing on variants of the Short Integer Solution (SIS) problem augmented with auxiliary hints. We provide a tight reduction from the Generalised ISISf (GenISISf) (Dubois et al., PKC 2025) assumption to its interactive variant IntGenISISf, enabling the construction of proof-friendly signature schemes without incurring the significant efficiency loss observed in prior works. In particular, our results directly apply to the anonymous credential scheme proposed by Bootle et al. (CRYPTO 2023), and circumvent the 4X blow-up in the credential size due to their security loss. We also identify families of functions f for which GenISISf is as hard as SIS, leading to the first (strongly) unforgeable standard-model signature scheme from SIS without relying on chameleon hash functions.

Moreover, we analyse the “one-more”-type lattice assumptions, showing in particular that Randomised One-More-ISIS (Baldimtsi et al., ASIACRYPT 2024) is at least as hard as standard One-More-ISIS (Agrawal et al., ACM CCS 2022). Further, we inspect different, yet equivalent, variations of Randomised One-More-ISIS which could be of independent interest. Finally, we compare the structural properties of GenISISf and One-More-ISIS, highlighting both shared techniques and fundamental differences. We believe our results contribute to a clearer understanding of the assumptions underpinning efficient, lattice-based anonymous credential systems.
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Max Cartor, Ryann Cartor, Hiroki Furue, Daniel Smith-Tone
ePrint Report ePrint Report
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 addi- tional 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 in- sights 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.
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Tianyu Zheng, Shang Gao, Xun Liu
ePrint Report ePrint Report
Proof-carrying data (PCD) is a powerful cryptographic primitive for ensuring computational integrity in distributed settings. State-of-the-art PCD constructions based on accumulation schemes achieve practical prover efficiency and support a wide range of applications. However, realizing zero-knowledge for accumulation-based PCD remains challenging, particularly for high-degree relations. Existing solutions often incur substantial overhead due to the need for zero-knowledge in both the underlying non-interactive arguments of knowledge (NARKs) and accumulation schemes. In this work, we present new theoretical and practical improvements for zero-knowledge PCD. First, we propose a novel construction that eliminates the need for zero-knowledge NARKs by separating the compliance predicate and accumulation verification, thereby reducing proof size and improving efficiency. Second, we design an efficient zero-knowledge accumulation scheme for special sound protocols, introducing techniques such as masking vectors and zero-knowledge sum-check protocols to ensure privacy with minimal overhead. The theoretical analysis demonstrates that our construction achieves logarithmic proof size and verification time for d-degree NP relations, and outperforms existing solutions in both asymptotic and concrete complexity.
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Julien Béguinot, Gianluca Brian, Loïc Masure
ePrint Report ePrint Report
Masking, i.e., computing over secret-shared data, is one of the main counter-measures against side-channel analysis, provably secure in the standard noisy-leakage model. However, all the state-of-the-art security proofs rely on a reduction to a more abstract model called random probing (Eurocrypt’14). As a result, the noise requirements of such proofs must scale with the field size of the circuit which, beyond not reflecting real-world physics of target devices, is often prohibitive, especially in the post-quantum era. That is why it is critical to find alternative strategies to the reduction to the random-probing model. In this paper, we establish for the first time a masking security proof bypassing this reduction, answering positively to the above question. Contrary to the common belief that directly working in the noisy-leakage model is not convenient, we show how to reach this goal by leveraging an extension of the Xor lemma. We also show how to leverage the IOS framework (CHES’21) in order to derive composable security directly in the noisy-leakage model. As a result, our bound relies on relaxed noise requirements characterized in a weaker metric than the so far state of the art (Crypto’24, ’19). Moreover, our new proof strategy allows us to derive a security bound for circuits masked with the first-order ISW compiler, for which the optimal noise requirement scales as \(\Theta\left(\left|\mathbb{F}\right|^{1/2}\right)\), whereas proofs using the random-probing model work in a regime of \(\mathcal{O}\left(\left|\mathbb{F}\right|\right)\). The latter contribution illustrates how some current design choices for masked implementations could be concretely affected: we exhibit two exemplary masked implementations such that the former one is more secure in the (more abstract) random-probing model, whereas the latter one is more secure in the (more realistic) noisy-leakage model.
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Diana Ghinea, Darya Melnyk, Tijana Milentijević
ePrint Report ePrint Report
The Multidimensional Approximate Agreement problem ($D$-AA) considers a setting with $n$ parties with inputs in $\mathbb{R}^D$. Out of the $n$ parties, up to $t$ may be byzantine (malicious). The goal is for the honest parties to obtain $\varepsilon$-close outputs that lie in the honest inputs' convex hull. While tight bounds on the resilience of $D$-AA have been found for the purely synchronous and asynchronous models, this is still an open question for the network-agnostic model. Here, the type of network is not known a priori: it may be synchronous, and then the number of byzantine parties is up to $t_s$, or asynchronous, and then the number of byzantine parties is up to $t_a \leq t_s$. In this model, it is known that $n > (D + 1) \cdot t_s + t_a$ is sufficient for deterministic protocols [GLW, SPAA'23], tight for $D = 1$ [GLW, PODC'22], while $n > \max\{(D + 1) \cdot t_s, t_s + (D + 1) \cdot t_a\}$ is tight for randomized protocols concerned with exact agreement [CGWW, DISC'24].

In this work, we establish that, for $D > 1$ the condition $n > \max\{(D + 1) \cdot t_s, t_s + (D + 1) \cdot t_a\}$ is tight for deterministic protocols as well. We identify that the gap in prior deterministic protocols is not geometric, but stems from an asymmetry in the communication primitive that produces parties' "views". The core technical contribution of our work is hence a strengthened network-agnostic Gather primitive that enforces a global structural property on the number of values received by honest parties, eliminating the problematic asymmetry - so that standard safe-area geometric convergence arguments apply under the optimal thresholds.
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Chun Guo, Jian Guo, Xinnian Li, Wenjie Nan
ePrint Report ePrint Report
We present the first general upper bound on permutation-based pseudorandom functions in the information-theoretic setting. We show that any non-compressing PRF, with input and output domain at least \([N]\), making \(t\) black-box calls to any \(t\) public permutations on \([N]\), can be distinguished from a random function over the output domain with at most \(\widetilde{O}\big(N^{t/(t+1)}\big)\) total queries to the PRF and the permutations. Our results suggest that the designs of Chen et al. (Crypto~2019) are optimal, among all possible constructions, in terms of information-theoretic security.

In particular, we propose the generalized key alternating construction, which captures permutation-based PRFs. We then prove that, for any such construction, there exists an explicit distinguisher achieving the tradeoff $Q_fQ_p^{t}=\widetilde{O}\big((2t^2)^{t+1}N^{t}\big) $ with constant advantage, where \(Q_f\) counts PRF queries and \(Q_p\) counts queries to each public permutation. We further extend our bound to blockcipher-based PRFs and to an adaptive setting in which each round may adaptively choose a permutation from a public family of permutations \(\mathcal P\). In this case, the general upper bound becomes \(\widetilde{O}\big(|\mathcal P|\,N^{t/(t+1)}\big)\).
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Giacomo Santato, Riccardo Zanotto
ePrint Report ePrint Report
In this paper, we study the IND-CPA-D security of FHE schemes when combined through scheme switching. We introduce a formal framework that captures security in this setting and provide a rigorous analysis of different constructions. Within this framework, we identify sufficient conditions for the switching algorithm under which the combined schemes achieve IND-CPA-D security, assuming the individual schemes are IND-CPA-D-secure.

We then focus on the specific case of scheme switching from CKKS to TFHE. We analyze how existing switching algorithms designed for vanilla CKKS can be modified to preserve IND-CPA-D security. We demonstrate an attack against the IND-CPA-D security of the PEGASUS construction [LHHMQ21] and we implement a proof-of-concept of this attack against the CKKS-to-FHEW switching mechanism in the OpenFHE library. Finally, we present a generic transformation that converts a vanilla CKKS-based switch into an IND-CPA-D-secure one.
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Petr Chmel, Pavel Hubáček, Dominik Stejskal
ePrint Report ePrint Report
The Algebraic Group Model (AGM) has become a standard framework for analyzing the knowledge soundness of group-based polynomial commitment schemes. In this work, we formally establish inherent limitations of this methodology. We isolate a structural property satisfied by essentially all practical group-based polynomial commitments, which we term AGM-clarity. We prove that for AGM-clear schemes, evaluation binding implies knowledge soundness in the AGM. This collapse reveals that the AGM definition of knowledge soundness does not capture a distinct security property, but is merely a structural consequence of evaluation binding.

To precisely characterize the guarantees on extractability provided by the AGM, we introduce Weak Interpolation Knowledge Soundness (WIKS) in the standard model, which is an extreme relaxation of extractability. We show that WIKS is implied by standard evaluation binding and prove that, for AGM-clear schemes, knowledge soundness in the AGM is equivalent to WIKS. Ultimately, our results demonstrate that proofs in the AGM for practical polynomial commitment schemes do not certify ``knowledge'' in the sense of immediate extractability, as they yield no guarantees beyond those already implied by standard model evaluation binding.
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Junru Li, Yifan Song
ePrint Report ePrint Report
In this work, we study the concrete efficiency of SPDZ-type MPC protocols in the dishonest majority setting with maximum corruption, where $t=n-1$ out of $n$ parties can be corrupted. In the semi-honest setting, the state-of-the-art SPDZ protocol achieves an amortized communication cost of $4n$ field elements per multiplication gate, assuming a pseudorandom correlation generator (PCG) that prepares random Beaver triples silently in the offline phase. However, achieving security against malicious adversaries typically incurs a substantial overhead. Existing works either require communication of at least $10n$ field elements per gate, or additionally require an additively homomorphic encryption scheme. In this work, we construct a maliciously secure SPDZ-type protocol with the same amortized communication complexity as the semi-honest variant, i.e., $4n$ field elements per multiplication gate, assuming a PCG for tensor-product correlations. We note that the tensor-product correlations come free as by-products from almost all existing constructions of PCGs for random Beaver triples. These by-products allow us to construct an efficient preprocessing phase and a recursive check protocol, which enables an online evaluation of the circuit without authenticating all the wire values. The correctness of the online computation can also be checked efficiently with a sublinear communication cost in the circuit size.
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