International Association
for Cryptologic Research

In this work, we initiate the formal study of oblivious batch updates over outsourced encrypted Bloom filters, focusing on scenarios where a storage-limited sender must insert or delete batches of elements in a Bloom filter maintained on an untrusted server. Our survey identifies only two prior approaches (CCS 2008 and CCS 2012) that can be adapted to this problem. However, they either fail to provide adequate security in dynamic scenarios or incur prohibitive update costs that scale with the filter’s maximum capacity rather than the actual batch size.

To address these limitations, we introduce a new cryptographic primitive, $\textit{Oblivious Bloom Filter Insertion}$ ($\textsf{OBFI}$), and propose novel constructions. At the core of our design is a novel building block, $\textit{Oblivious Bucket Distribution}$ ($\textsf{OBD}$), which enables a storage-limited sender to distribute a large array of elements, uniformly sampled from a finite domain, into small, fixed-size buckets in a data-oblivious manner determined by element order. The design of $\textsf{OBD}$ is further supported by identifying and proving a new structural property of such arrays, which establishes tight and explicit probabilistic bounds on the number of elements falling within predefined subranges of the domain.

Our $\textsf{OBFI}$ constructions achieve adaptive data-obliviousness and ensure that batch update costs scale primarily with the batch size. Depending on the variant, the sender’s storage requirement ranges from $O(\lambda)$, where $\lambda$ is the security parameter, down to $O(1)$. Finally, we demonstrate the practicality of $\textsf{OBFI}$ by integrating it into representative Bloom-filter-based cryptographic protocols for Searchable Symmetric Encryption, Public-key Encryption with Keyword Search, and Outsourced Private Set Intersection, thereby obtaining batch-updatable counterparts with state-of-the-art security and performance.

A Batched Threshold Encryption (BTE) scheme enables a committee of servers to perform a lightweight (in terms of communication and computation) threshold decryption of an arbitrary batch of ciphertexts from a larger pool, while ensuring the privacy of ciphertexts that are outside the batch. Such a primitive has a direct application in designing encrypted mempools for MEV protection in modern blockchains. Bormet et al. (USENIX 2025) recently proposed a BTE scheme called “BEAT-MEV” which is concretely efficient for small to moderate batch sizes.

In this work, we improve and extend the BEAT-MEV scheme in multiple ways. First, we improve the computational cost from quadratic to quasilinear in the batch size, thus making it practical for large batch sizes. This improvement is achieved by substituting the key-homomorphic punctured PRF used in BEAT-MEV with an FFT-friendly alternative. Second, we extend the ideas in their scheme to the weighted setting, where each server in the committee has an associated 'weight' value (e.g., stake weight of validators in PoS blockchains), while crucially ensuring that the communication cost remains independent of the weights. In contrast, BEAT-MEV with naive virtualization would incur communication cost linear in the total weight. Third, for handling the small failure rate inherent in BEAT-MEV scheme due to index collisions across different clients at the time of encryption, we propose a generalization of their suggested approach which offers an option to trade off between ciphertext size and server communication for a given failure rate.

We implement and evaluate our scheme and compare it with BEAT-MEV to demonstrate our concrete improvement. In the unweighted setting, we improve the computational cost (without increasing the communication cost) by ≈ 6× for a batch size of 512 ciphertexts. In the weighted setting, we improve the communication cost (without compromising computation time), over BEAT-MEV with naive virtualization, by ≈ 50× for 100 validators with total stake weight 5000 distributed as per the latest Solana stake distribution.

Given a circuit $G: \{0, 1\}^n \to \{0, 1\}^m$ with $m > n$, the *range avoidance* problem ($\text{Avoid}$) asks to output a string $y\in \{0, 1\}^m$ that is not in the range of $G$. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of *proof complexity generators* --- circuits $G: \{0, 1\}^n \to \{0, 1\}^m$ where $m > n$ but for every $y\in \{0, 1\}^m$, it is infeasible to prove the statement "$y\not\in\mathrm{Range}(G)$" in a given propositional proof system.

This paper connects these two problems with the existence of *demi-bits generators*, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). $\bullet$ We show that the existence of demi-bits generators implies $\text{Avoid}$ is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich's PRG, we prove the hardness of $\text{Avoid}$ even when the instances are constant-degree polynomials over $\mathbb{F}_2$. $\bullet$ We show that the dual weak pigeonhole principle is unprovable in Cook's theory $\mathsf{PV}_1$ under the existence of demi-bits generators secure against $\mathbf{AM}/_{O(1)}$, thereby separating Jeřábek's theory $\mathsf{APC}_1$ from $\mathsf{PV}_1$. Previously, Ilango, Li, and Williams (STOC '23) obtained the same separation under different (and arguably stronger) cryptographic assumptions. $\bullet$ We transform demi-bits generators to proof complexity generators that are *pseudo-surjective* in certain parameter regime. Pseudo-surjectivity is the strongest form of hardness considered in the literature for proof complexity generators.

Our constructions are inspired by the recent breakthroughs on the hardness of $\text{Avoid}$ by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use *randomness extractors* to significantly simplify the construction and the proof.

Succinct zero-knowledge arguments (zk-SNARKs) enable a prover to convince a verifier of the truth of a statement via a succinct and efficiently verifiable proof without revealing any additional information about the secret witness. A barrier to practical deployment of zk-SNARKs is their high proving cost. With this motivation, we study server-aided zk-SNARKs, where a client/prover outsources most of its work to a single, untrusted server while the server learns nothing about the witness or even the proof. We formalize this notion and show how to realize server-aided proving for widely deployed zk-SNARKs, including Nova, Groth16, and Plonk.

The key building block underlying our designs is a new primitive, encrypted multi-scalar multiplication (EMSM), that enables private delegation of multi-scalar multiplications (MSMs). We construct an EMSM from variants of the learning parity with noise assumption in which the client does $O(1)$ group operations, while the server’s work matches that of the plaintext MSM.

We implement and evaluate our constructions. Compared to local proving, our techniques lower the client's computation by up to $20\times$ and reduce the proving latency by up to $9\times$.

Fully Homomorphic Encryption (FHE) schemes typically experience significant data expansion during encryption, leading to increased computational costs and memory demands during homomorphic evaluations compared to their plaintext counterparts. This work builds upon prior methods aimed at reducing ciphertext expansion by leveraging matrix secrets under the Matrix-LWE assumption. In particular, we consider a ciphertext format referred to in this work as common mask (CM) ciphertexts, which comprises a shared mask and multiple message bodies. Each body encrypts a distinct message while reusing the common random mask. We demonstrate that all known FHEW/TFHE style ciphertext variants and operations can be naturally extended to this CM format. Our benchmarks highlight the potential for amortizing operations using the CM structure, significantly reducing overhead. For instance, in the boolean setting, we have up to a 51% improvement when packing 8 messages. Beyond ciphertext compression and amortized evaluations, the CM format also enables the generalization of several core-TFHE operations. Specifically, we support applying distinct lookup tables on different encrypted messages within a single CM ciphertext and private linear operations on messages encrypted within the same CM ciphertext.

Secure multiparty computation (MPC) enables mutually distrustful parties to jointly compute functions over private data without revealing their inputs. A central paradigm in MPC is the secret-sharing-based model, where secret sharing underpins the efficient realization of arithmetic, comparison, numerical, and Boolean operations on shares of private inputs. In this paper, we systematize protocols for these operations, with particular attention to two foundational contributions \cite{ChidaGHIKLN18,NO07} that devised secure multiplication and comparison. Our survey provides a unified, self-contained exposition that highlights the composability, performance trade-offs, and implementation choices of these protocols. We further demonstrate how they support practical privacy-preserving systems, including recommender systems, distributed optimization platforms, and e-voting infrastructures. By clarifying the protocol landscape and connecting it to deployed and emerging applications, we identify concrete avenues for improving efficiency, scalability, and integration into real-world MPC frameworks. Our goal is to bridge theory and practice, equipping both researchers and practitioners with a deeper understanding of secret-sharing-based MPC as a foundation for privacy technologies.

We outline how to generalize the Guruswami-Sudan list decoder anal- ysis from Ben–Sasson, Carmon, Ishai, Kopparty and Saraf [BCI+20] in order to obtain a “global” proximity gap, called mutual correlated agree- ment in Arnon, Chiesa, Fenzi and Yogev [WHIR 2024], or strong correlated agreement in Zeilberger [Khatam 24].

Silence Laboratories is looking for cryptography research interns for the summer of 2026. Candidates must be students enrolled in a PhD program, or otherwise be able to demonstrate a comparable level of proficiency in cryptography. The internship will be centered on a privacy preserving computation product being built at Silence, and will involve investigating the underlying theory (likely a subfield of MPC) and ideally result in a prototype implementation.

At Silence, you can expect unique and impactful opportunities to put theory to practice, freedom in how you approach your work, competitive pay, and a chance to work with some of the top research and engineering talent.

The internship will be fully remote and for a duration of 3 months, although exact dates can be flexible. Please send your CV and a link to your homepage (if available) to internships@silencelaboratories.com with the title “Summer 2026 Application [Your Name]” by Dec 20th 2025 for full consideration.

Closing date for applications:

Contact: internships@silencelaboratories.com

Secure lookup table (LUT) protocols allow retrieving values from a table at secret indices, and have become a promising approach for the secure evaluation of non-linear functions. Most existing LUT protocols target the two-party setting, where the best protocols achieve a communication cost of $O(N)$ for a table of size $N$. MAESTRO (Morita et al., USENIX Security 2025) represents the state-of-the-art LUT protocol for AES in the three-party honest-majority setting, with a communication cost of $O(N^{1/2})$; malicious security is achieved with distributed zero-knowledge proofs. However, it only supports single-input tables over characteristic-2 fields $\mathbb{F}_{2^k}$ and lacks support for multi-input tables over rings $\mathbb{Z}_{2^k}$, which are more widely used in modern computation. Moreover, the $O(N^{1/2})$ cost remains expensive for large-scale applications; their efficient distributed zero-knowledge proofs are specialized for AES and cannot be easily applied to $\mathbb{Z}_{2^k}$.

In this work, we present MARLUT, a new generalized and optimized LUT construction supporting multi-input tables over both rings $\mathbb{Z}_{2^k}$ and fields $\mathbb{F}_{2^k}$ with malicious security. We achieve this by (1) extending the semi-honest LUT protocol from MAESTRO, utilizing high-dimensional tensors to reduce its communication cost to $O(N^{1/3})$, and (2) designing a new distributed zero-knowledge proof for inner-product relations over $\mathbb{Z}_{2^k}$. Our distributed zero-knowledge proof is more efficient than the state-of-the-art work (Li et al., CCS 2024) and may be of independent interest. Experiments show that on a table of size $2^{16}$, our semi-honest LUT protocol reduces the offline computational and communication cost by a factor of $5.95$ and $3.23$, respectively. Our distributed zero-knowledge proofs show up to $7.07\times$ and $4.97\times$ speedups over the state-of-the-art protocol on ring $\mathbb{Z}_{2^8}$ and $\mathbb{Z}_{2^{16}}$, respectively.

The first contribution of the paper is to put forward an abstract definition of the Grain family of stream ciphers which formalises the different components that are required to specify a particular member of the family. Our second contribution is to provide new and strengthened definitions of the components. These include definining new classes of nonlinear Boolean functions, improved definition of the state update function during initialisation, choice of the tap positions, and the possibility of the linear feedback shift register being smaller than the nonlinear feedback shift register. The third contribution of the paper is to put forward seven concrete proposals of stream ciphers by suitably instantiating the abstract family, one at the 80-bit security level, and two each at the 128-bit, 192-bit, and the 256-bit security levels. At the 80-bit security level, compared to the well known Grain~v1, the new proposal uses Boolean functions with improved cryptographic properties \textit{and} an overall lower gate count. At the 128-bit level, compared to ISO/IEC standard Grain-128a, the new proposals use Boolean functions with improved cryptographic properties; one of the proposals require a few extra gates, while the other has an overall lower gate count. At the 192-bit, and the 256-bit security levels, there are no proposals in the literature with smaller gate counts.

Password-based Authenticated Key Exchange (${\sf PAKE}$) is a widely acknowledged, promising security mechanism for establishing secure communication between devices. It enables two parties to mutually authenticate each other over insecure networks and generate a session key using a low-entropy password. However, the existing $\mathsf{PAKE}$ protocols encounter significant challenges concerning both security and efficiency in the context of the \textit{Internet of Things} (IoT). In response to these challenges, we contribute to the advancement of post-quantum secure $\mathsf{PAKE}$ protocols tailored for IoT applications, enriching the existing landscape. In this study, we introduce two novel protocols, $\mathsf{PAKE}$-\textup{I} and $\mathsf{PAKE}$-\textup{II}, designed to address these concerns and enhance the security standards of $\mathsf{PAKE}$ protocol. While $\mathsf{PAKE}$-\textup{I} is secure under lattice-based hardness assumptions, $\mathsf{PAKE}$-\textup{II} derives its security from isogeny-based hard problems. Our lattice-based protocol $\mathsf{PAKE}$-\textup{I} is secure based on the \textit{Pairing with Errors} ($\mathsf{PWE}$) assumption and the \textit{Decision Ring Learning with Errors} ($\mathsf{DRLWE}$) assumption and our isogeny-based protocol $\mathsf{PAKE}$-\textup{II} is secure based on the hardness of the \textit{Group Action Inverse Problem} ($\mathsf{GAIP}$) and the \textit{Commutative SuperSingular Diffie-Hellman} ($\mathsf{CSSDH}$) problem in the Random Oracle Model $(\mathsf{ROM})$. We present a comprehensive security proof in a conventional game-based indistinguishability security model that addresses offline dictionary attacks, replay attacks, compromise attacks for both parties (client and server) and perfect forward secrecy. Additionally, our proposed $\mathsf{PAKE}$ protocols are the first post-quantum secure $\mathsf{PAKE}$s that achieve identity privacy and resistance to pre-computation attacks. Through rigorous performance evaluations, the paper demonstrates that the proposed $\mathsf{PAKE}$ schemes are ultralight and exhibit notable advantages in terms of total computation cost and enhanced security properties when compared to the existing protocols. More positively, both the proposed $\mathsf{PAKE}$ are optimal in the sense that they achieve mutual authentication explicitly in only three rounds which is the least number of rounds required for acquiring mutual authentication between two parties.

Smart contract-based decentralized applications (dApps) have become an ever-growing way to facilitate complex on-chain operations. Oracle services strengthened this trend by enabling dApps to access real-world data and respond to events happening outside the blockchain ecosystem. A large number of academic and industrial oracle solutions have emerged, capturing various designs, capabilities, and security assumptions/guarantees. This rapid development makes it challenging to comprehend the landscape of oracles, understand their trade-offs, and build on them.

To address these challenges, we develop a systematization of knowledge for blockchain oracle services. To the best of our knowledge, our work is the first to provide extensive study of oracles while empirically investigating their capabilities in practice. After examining the general design framework of oracles, we develop a multi-dimensional systematization framework assessing existing solutions based on their capabilities, trust and security assumption/guarantees, and their underlying design architecture. To further aid in this assessment, we conduct a number of empirical experiments to examine oracle deployed in practice, thus offering additional insights about their deployment maturity, usage popularity, performance, and ease-of-use. We go on to distill a number of insights and gaps, thus providing a guide for practitioners (on the use of these oracles) and researchers (by highlighting gaps and open problems).

With the rapid advancement of cloud computing technology, outsourcing massive datasets to cloud servers has become a prominent trend, making secure and efficient data sharing mechanisms a critical requirement. Attribute-based proxy re-encryption (ABPRE) has emerged as an ideal solution due to its support for fine-grained, one-to-many access control and robust ciphertext transformation capabilities. However, existing ABPRE schemes still exhibit shortcomings in addressing forward security issues caused by long-term private key leakage, threats from quantum computer attacks, and vulnerabilities to honest re-encryption attacks (HRA). To simultaneously resolve these challenges, this paper introduces a novel cryptographic primitive termed puncturable attribute-based proxy re-encryption with switchable tags (PABPRE-ST), constructing a secure cloud data sharing scheme that supports fine-grained revocation. By integrating puncturable encryption (PE) mechanisms into the ABPRE framework, the scheme achieves fine-grained ciphertext revocation based on tags. In PABPRE-ST, data owners embed tags into ciphertexts, enabling data users to puncture specific tags and thereby revoke access to corresponding ciphertexts at a granular level. Furthermore, the scheme allows delegators to switch ciphertext tags, enhancing sharing flexibility. We formalize the security definitions for the proposed puncturable attribute-based proxy re-encryption scheme and prove its security under the learning with errors (LWE) assumption, which is widely believed to be resistant to quantum computer attacks. Security analysis demonstrates that the proposed scheme achieves HRA security in the standard model.

Quantum voting allows us to design voting scheme by quantum mechanics. The existing quantum voting protocols mainly use quantum entangled states. However, the existing protocols rarely consider the problem of repeated voting and tampered voting by malicious voters, and hybrid quantum voting protocol has not been discussed. In this paper, we use EFI pairs (Entity-Friendly Integer pairs) instead quantum entangled states to address the shortage of existing protocols, and propose a new quantum voting protocol. Our protocol is structured to avoid repeated voting by any voter, and can prevent the leakage of voters' voting information. The security of our protocol can be finally reduced to a classical assumption i.e. BQP = QMA. Combined with quantum key distribution (QKD), we further optimize the protocol to prevent malicious adversaries from interfering with the final voting results. Moreover, we use extended noisy trapdoor claw-free function (ENTCF) to construct the first hybrid quantum voting protocol, which allows a classical voter to interact with a quantum center through a classical channel to complete the voting process.

In a batched identity-based encryption (IBE) scheme, ciphertexts are associated with a batch label $\mathsf{tag}^*$ and an identity $\mathsf{id}^*$ while secret keys are associated with a batch label $\mathsf{tag}$ and a set of identities $S$. Decryption is possible whenever $\mathsf{tag} = \mathsf{tag}^*$ and $\mathsf{id}^* \in S$. The primary efficiency property in a batched IBE scheme is that the size of the decryption key for a set $S$ should be independent of the size of $S$. Batched IBE schemes provide an elegant cryptographic mechanism to support encrypted memory pools in blockchain applications.

In this work, we introduce a new algebraic framework for building pairing-based batched IBE. Our framework gives the following:

First, we obtain a selectively-secure batched IBE scheme under a $q$-type assumption in the plain model. Both the ciphertext and the secret key consist of a constant number of group elements. This is the first pairing-based batched IBE scheme in the plain model. Previous pairing-based schemes relied on the generic group model and the random oracle model.

Next, we show how to extend our base scheme to a threshold batched IBE scheme with silent setup. In this setting, users independently choose their own public and private keys, and there is a non-interactive procedure to derive the master public key (for a threshold batched IBE scheme) for a group of users from their individual public keys. We obtain a statically-secure threshold batched IBE scheme with silent setup from a $q$-type assumption in the plain model. As before, ciphertexts and secret keys in this scheme contain a constant number of group elements. Previous pairing-based constructions of threshold batched IBE with silent setup relied on the generic group model, could only support a polynomial number of identities (where the size of the public parameters scaled linearly with this bound), and ciphertexts contained $O(\lambda / \log \lambda)$ group elements, where $\lambda$ is the security parameter.

Finally, we show that if we work in the generic group model, then we obtain a (threshold) batched IBE scheme with shorter ciphertexts (by 1 group element) than all previous pairing-based constructions (and without impacting the size of the secret key).

Our constructions rely on classic algebraic techniques underlying pairing-based IBE and do not rely on the signature-based witness encryption viewpoint taken in previous works.

We present a generic, automatable framework to reduce the demand for fresh randomness in first-order masked circuits while preserving security in the glitch-extended probing model. The method analyzes the flow of randomness through a circuit to establish security rules based on the glitch-extended probing model. These rules are then encoded as an interference graph, transforming the optimization challenge into a graph coloring problem, which is solved efficiently with a DSATUR heuristic. Crucially, the optimization only rewires randomness inputs without altering core logic, ensuring seamless integration into standard EDA flows and applicability to various gadgets like DOM-indep (Domain-Oriented Masking) and HPC (Hardware Private Circuits). On 32-bit adder architectures, the framework substantially reduces randomness requirements by 79–90%; for instance, the Kogge–Stone adder's requirement of 259 unique random inputs is reduced to 27. All optimized designs were evaluated using PROLEAD, with the leakage results indicating compliance with first-order glitch-extended probing security.

The Unbalanced Oil and Vinegar (UOV) construction is the foundation of several post-quantum digital signature algorithms currently under consideration in NIST's standardization process for additional post-quantum digital signature schemes. This paper introduces new single fault injection attacks against the signing procedure of deterministic variants of signature schemes based on the UOV construction. We show how these attacks can be applied to attack MAYO and PROV, two signature schemes submitted to the NIST call for additional post-quantum signature schemes. The attacks are demonstrated with reference implementations that run on an ARM Cortex-M4 processor. Our attacks do not require precise triggering or precise fault injection capabilities. Any type of fault in large portions of the code has the potential to result in successful key recovery. We demonstrate our attacks with very cheap equipment and simple clock glitching techniques, enabling the recovery of the secret key with either two faulty signatures or one correct signature and one faulty signature in the case of MAYO and one correct signature and two faulty signatures in case of PROV. The fact that our attacks do not require precise fault injection capabilities and can be successful with only a few signatures makes them particularly powerful, hence harmful for the implementation security of post-quantum digital signature schemes.

The existing lattice-based signature and IBE schemes suffer from the non-compactness of public keys or larger reduction loss in the security analysis. Thus we solve and improve those deficiencies as follows: – First, we construct a lattice-based short signature scheme with a compact verification key in the standard model based on the ring short integer solution (RSIS) assumption. Under the same com- pactness, the ring modulus of our signature scheme is significantly smaller than the compact sig- nature scheme of Alperin-Sheriff (PKC 2015). More importantly, our signature scheme achieves better reduction loss than all the previous confined guessing-based signatures. In other words, our signature scheme achieves better security and efficiency simultaneously. – Secondly, we further design a short signature scheme with a nearly compact public key size and an even smaller reduction loss. Our second signature scheme achieves even better reduction loss than our first signature scheme yet at the cost of increasing the public key to a super-constant number of ring vectors. – Last but not least, we construct an adaptively secure compact IBE scheme from the lattice as- sumptions and the truncation collision-resistant hash functions (TCRHF) introduced by Jager and Kurek (ASIACRYPT 2018). Note that the previous TCRHF-based IBE schemes are not even close to compactness. The above improvements mainly benefited from our compact design of the tag functions and their more compact homomorphic evaluations. We also believe that our newly designed tag function may find new applications in designing other cryptographic schemes, like ABE and others.

Designated Verifier zero-knowledge Succinct Non-Interactive Arguments of Knowledge (DV-zkSNARKs) are cryptographic argument systems in which the ability to verify proofs is restricted to a designated verifier. Unlike publicly verifiable zkSNARKs, these constructions ensure that only an authorized party can validate the correctness of the proof. Existing lattice-based DV-zkSNARK constructions typically rely on either linear-only encryption (LOE) or linear targeted malleability (LTM). The former does not guarantee security against quantum adversaries, while the latter restricts knowledge soundness to the non-adaptive setting. To address these limitations, we propose an inner product argument system that relies solely on the hardness of the Module Short Integer Solution (MSIS) assumption and achieves knowledge soundness in the random oracle model. This construction enables a designated verifier, holding a secret key, to succinctly verify inner product of a committed witness with an arbitrary vector. By combining our argument system with a linear probabilistic checkable proof (LPCP) compiler, to the best of our knowledge, we obtain the first DV-zkSNARK construction based on standard assumptions. Our implementation achieves prover and verification times comparable to the state of the art, while reducing public parameter size by a factor of 10, at the cost of a 2.5× increase in proof size.

Optical computing has garnered significant attention in recent years due to its high-speed parallel processing and low power consumption capabilities. It has the potential to replace traditional electronic components and systems for various computation tasks. Among these applications, leveraging optical techniques to address information security issues has emerged as a critical research topic. However, current attempts are predominantly focused on areas such as image encryption and information hiding, with limited exploration of other modern information security concepts, including zero-knowledge proof (ZKP). In this paper, we propose an optical ZKP method based on single-pixel imaging (SPI). By utilizing the flexibility of SPI, our proposed approach can directly acquire randomly permuted results of the source problem's solution in the form of encoded images, thereby encrypting and verifying the original solution. ZKP for the source problem can be realized with optical computing based on a proving protocol without disclosing additional information. Simulated and experimental results show that our proposed method can be effectively applied to two typical ZKP problems: Sudoku and Hamiltonian cycle problem.