Jiacheng Yang, Yongxin Zhang, Hong Lei, Zijian Bao · 6 authors
In the process of integrating the digital economy with the real economy, a vast and diverse supply of data has emerged. Among these, the exponential growth of data in vehicular ad-hoc networks (VANETs) hold immense commercial value. This further drives the demand for building large-scale data marketing platforms to support trading between vehicles and businesses in order to reduce the cost of local management. However, this must address several challenges related to security and performance, such as fairness, privacy protection, and data delivery efficiency. Therefore, this paper proposes a privacy-preserving large-scale data marketing system (PLDM), aiming to address these challenges. Specifically, this solution is based on blockchain to build a decentralized trusted third party to ensure the fairness of the trading process. In addition, we combine the$\Sigma$-protocol and Merkle tree to prove the validity of both the encryption of data to be traded and the identities of the trading participants. This not only achieves privacy protection for data and identities but also reduces the computational costs for vehicles. We provide the security analysis and experimental evaluation ofPLDM. And the results show thatPLDMperforms well in fairness and privacy protection, supporting efficient delivery of large-scale data and low on-chain computational costs.
Digital banking fraud has grown into a multifaceted threat requiring both strong cryptographic protections and advanced machine learning defenses. This paper proposes a unified framework integrating lattice-based cryptography (for post-quantum resilience and privacy) with federated graph neural networks (for collaborative fraud detection) to address the gap in current financial security architectures. We simulate real-world fraud scenarios –including synthetic identity schemes and transaction laundering – using a mix of publicly reported incidents (e.g., the 2023 Log4Shell exploitation, 2022 SolarWinds-style supply chain compromise, and the Mirai botnet’s IoT spread) as motivating cases. Our methodology leverages threshold homomorphic encryption for privacy-preserving analytics, along with adversarial training and diffusion purification to harden models against poisoning and evasion attacks. Experiments using a mixed dataset of EU banking transactions and simulated breach logs show that our approach achieves over 99% accuracy in detecting novel fraud patterns, while reducing false positives by 35% compared to conventional classifiers. We validate these improvements with statistical significance (p<0.001) and illustrate them via ROC curves and network topology maps. Key findings include the identification of specific trade-offs between cryptographic overhead and detection latency, and the observation that cross-institutional intelligence sharing (using zero-knowledge proofs) can halve the response time to coordinated attacks. These results suggest that combining cryptography with machine learning can close existing vulnerabilities in digital banking and guide future work in robust, privacy-aware fraud prevention.
This review article examines the state of blockchain-enabled identity management in Internet of Things (IoT) networks, focusing on decentralized and secure mechanisms for device identification, authentication, and access control. Traditional centralized identity systems face limitations such as single points of failure, scalability bottlenecks, and vulnerability to breaches. We systematically survey recent literature on blockchain-based frameworks applied to IoT, categorizing approaches by blockchain platform, identity credential models, consensus mechanisms, and smart contract implementations. The analysis highlights key performance metrics such as system latency, throughput, resource overhead, and energy consumption, and compares existing prototypes deployed across diverse IoT scenarios. We assess the security and privacy implications, including resistance to spoofing, Sybil attacks, unauthorized access, data tampering, and insider threats. Additionally, the review identifies open research challenges such as managing identity lifecycle in constrained devices, achieving interoperability across heterogeneous networks, balancing decentralization with scalability, and integrating with emerging technologies like edge computing and zero-knowledge proofs. Finally, we offer recommendations for future research directions and practical deployment strategies to advance blockchain-based identity solutions in IoT ecosystems. Our comprehensive synthesis aims to guide researchers and practitioners in developing robust, scalable, and trustworthy identity frameworks using blockchain for the evolving IoT landscape.
This study introduces a framework that integrates blockchain, decentralized identity, and zero-knowledge proofs to enhance the trustworthiness and confidentiality of disaster information sharing. A sustainable model is proposed for real-world applications, supported by a prototype developed on the Decentralized Solutions for Humanity (DS4H) blockchain research network.
Emanuela Podda, Pol Hölzmer, Alexandre Amard, Johannes Sedlmeir · 5 authors
Zero-knowledge proofs allow the implementation of the data minimisation principle imposed by the GDPR in digital identity wallets and the related personal data transactions, therefore representing a reasonable option to be enforced by lawmakers.
Efficient cryptographic verification methods are critical for secure communication in intelligent transportation systems (ITS), especially with the proliferation of Internet of Vehicles (IoV) devices powered by consumer technology. However, traditional zero-knowledge proofs often face efficiency limitations. This paper proposes a novel verifiable decryption scheme for IoV applications within ITS, leveraging the Module Learning with Errors (MLWE) and Module Small Integer Solution (MSIS) problems to address these challenges using widely available consumer technology platforms. The scheme integrates a compression function tailored for IoV devices and consumer technology in intelligent transportation systems, coupled with error estimation techniques, effectively mitigating disparities in equality relationships between the private data of the prover and verifier during the encryption and decryption process. This enables the prover to selectively share partial information, reducing discrepancies and transforming the verifiable decryption problem into a proof that a vector in the ring satisfies a linear relationship. A rigorous theoretical analysis demonstrates the schemes correctness, security, communication overhead, and computational complexity, reducing its soundness and zero-knowledge properties to the hardness assumption of the MSIS problem. The paper also recommends two parameter sets for different security levels that are feasible for deployment on consumer technology devices. To evaluate the schemes practicality, a C-language implementation was developed and tested on typical consumer technology hardware. Experimental results show significant advantages in proof size and computation time for a single ciphertext compared to existing schemes, making the proposed method highly efficient for ITS scenarios. This verifiable decryption scheme offers a post-quantum cryptographic solution that ensures secure, efficient, and scalable data exchange, aligning with the stringent demands of intelligent transportation systems and leveraging the accessibility of consumer technology.
We introduce the first complete formal solution to corrigibility in the off-switch game, with provable guarantees in multi-step, partially observed environments. Our framework consists of five *structurally separate* utility heads -- deference, switch-access preservation, truthfulness, low-impact behavior via a belief-based extension of Attainable Utility Preservation, and bounded task reward -- combined lexicographically by strict weight gaps. Theorem 1 proves exact single-round corrigibility in the partially observable off-switch game; Theorem 3 extends the guarantee to multi-step, self-spawning agents, showing that even if each head is *learned* to mean-squared error $\varepsilon$ and the planner is $\varepsilon$-sub-optimal, the probability of violating *any* safety property is bounded while still ensuring net human benefit. In contrast to Constitutional AI or RLHF/RLAIF, which merge all norms into one learned scalar, our separation makes obedience and impact-limits provably dominate even when incentives conflict. For settings where adversaries can modify the agent, we prove that deciding whether an arbitrary post-hack agent will ever violate corrigibility is undecidable by reduction to the halting problem, then carve out a finite-horizon "decidable island" where safety can be certified in randomized polynomial time and verified with privacy-preserving, constant-round zero-knowledge proofs.
In our research, we propose the first practically deployable construction of a multi-prover zero-knowledge succinct non-interactive argument of knowledge (zkSNARK) protocol specifically tailored for restricted multiplication straight-line (RMS) programs, a computation model widely applicable in evaluating polynomials. Our protocol ensures input privacy, zero-knowledge, and security against fully malicious provers, all while eliminating the need for any inter-prover communication, making it highly suitable for distributed cloud environments. At the core of our approach is the introduction of the Restricted Quadratic Arithmetic Program model, an algebraic structure aligned with RMS semantics that enables provers to independently generate local proofs. We instantiate our framework using the Pinocchio protocol, resulting in a system that requires only 9 group elements per proof and 10 pairings for verification, nearly matching the efficiency of its single-prover counterpart. By leveraging our multi-prover zkSNARK protocol within a multi-server verification computation framework, we enable secure outsourcing of computations to the cloud of fully untrusted cloud servers. Compared to existing works, our protocol uniquely eliminates the need for any inter-server communication while achieving security even against adversaries controlling all servers.
This chapter explores blockchain's potential to address cloud computing security challenges. Despite cloud computing's scalability and cost efficiency, it faces risks like data breaches and regulatory non-compliance, as seen in the 2019 Capital One AWS breach. Blockchain's decentralized ledger, cryptographic hashing, smart contracts, and consensus mechanisms (e.g., PoW, PoS) enhance security through decentralized access control, secure storage, and intrusion detection. Privacy techniques like homomorphic encryption and zero-knowledge proofs protect data. Case studies, including IBM Food Trust and MedRec, show practical applications. However, scalability, interoperability, regulatory conflicts (e.g., GDPR), and high costs pose barriers. Solutions like sharding and layer-2 protocols aim to overcome these. Future research focuses on scalability, privacy, hybrid cloud integration, and AI-driven security. Blockchain strengthens cloud security but requires innovation to achieve widespread adoption.
Rudraksh Joshi, Amrit K. Goel, Prashant Singh, Jitendra Goyal
Blockchain is a distributed ledger technology designed to ensure transparent and tamper-proof transaction recording without centralized control. Despite its benefits, this transparency can compromise the confidentiality of the data. This paper investigates eight privacy-preserving methods such as encryption, steganography, off-chain hashing, and zero-knowledge proofs. Each technique is explored through working implementations and real-world constraints, including gas cost and legal compliance. A hybrid architecture is proposed that blends on-chain verification with IPFS-based off-chain storage, enabling developers to build scalable and privacy-conscious decentralized applications.
Blockchain provides the opportunity for organizations to execute trustable collaborations through smart contract automations. However, linkability problems exist in blockchain-based collaboration platforms due to privacy leakages, which, when exploited, will result in tracing transaction patterns to users and exposing collaborating organizations and parties. Some privacy-preserving mechanisms have been adopted to reduce linkability problems through the integration of access control systems to smart contracts, off-chain data storage, usage of permissioned blockchain, etc. Still, linkability problems persist in applications deployed in both private and public blockchain networks. Zero-knowledge proof (ZKP) systems provide mechanisms for verifying the correctness of transactions and actions executed on the blockchain without revealing complete information about the transaction. Hence, ZKP systems provide a potential solution to eliminating linkability problems in blockchain-based collaboration systems. The objective of this paper is to identify various linkability problems that exist in blockchain-enabled collaboration systems and understand how ZKP algorithms and smart contract frameworks can be used in addressing the linkability problems. Furthermore, a proof of concept (PoC) is implemented and simulated to demonstrate a ZKP system for a privacy-preserving feedback mechanism that mitigates linkability problems in collaboration systems. The scenario-based results from the PoC evaluation show that a feedback system that includes project participants’ verification through membership proofs, verification of on-time submission of feedback through range proofs, and encrypted calculation of feedback scores through homomorphic arithmetic provides a privacy-aware system for executing collaborations on the blockchain without linking project participants.
Zero-Knowledge Proof (ZKP) has gained widespread application across various domains, demonstrating remarkable success. Among ZKP algorithms, Zero-Knowledge Succinct Non-Interactive Argument of Knowledge (zk-SNARK) is the most widely used. However, despite its advantages of small proof size and succinct verification, zk-SNARK proof generation faces significant challenges due to high computational demands, limiting its practical application. This paper addresses these challenges by accelerating two computationally intensive operations in zk-SNARK proof generation, Number Theory Transformation (NTT) and Multi-Scalar Multiplication (MSM), using FPGAs. In the implementation of NTT hardware accelerators for zk-SNARK applications, the traditional 4-step algorithm often encounters conflicts between off-chip bandwidth and on-chip memory. To resolve this issue, we propose an innovative approach that enhances accelerator performance by recursively applying the 4-step algorithm to create a more efficient 6-step algorithm. For MSM hardware acceleration on FPGAs, existing works are often constrained by limited on-chip memory, restricting the use of longer slice lengths, which are crucial for higher performance when using the commenly used Pippenger algorithm. To overcome this limitation, we introduce the Batch Method, optimizing off-chip memory consumption, enabling the accelerator to use longer slice lengths and achieve superior performance. Experimental results demonstrate that the proposed NTT design achieves 1.76× higher DSP efficiency than the SAM. Meanwhile, the proposed MSM design demonstrates 1.24× higher performance than the MSMAC with aligned frequency and number of PEs. When benchmarked against the GPU implementation GZKP, our MSM design exhibits 1.16× and 1.46× higher performance than GZKP for BLS12-381 and BN-254, respectively. However, the NTT design remains at a disadvantage due to the bandwidth limitation between our platform, Xilinx Alveo U250, and GZKP’s platforms, Nvidia GTX 1080 Ti and Nvidia Tesla V100.
Card-based cryptography enables players to compute logical and arithmetic operations securely, such as bitwise AND and addition of integers. Several multiparty computation protocols and zero-knowledge proof protocols utilizing these secure computations have been developed as its applications. However, the realization of an efficient protocol for an arithmetic operation other than addition and subtraction remains an open problem. This paper proposes card-based protocols, based on integer commitment, for multiplication, division, and square root. Compared to general constructions for protocols for these operations based on binary integer commitment, the proposed protocols exhibit superior simplicity and efficiency. Furthermore, these protocols introduce novel applications for card-based cryptography to secure statistical data aggregation.
Assume that, given a sequence of n integers from 1 to n arranged in random order, we want to sort them, provided that the only acceptable operation is a prefix reversal, which means to take any number of integers (sub-sequence) from the left of the sequence, reverse the order of the sub-sequence, and return them to the original sequence. This problem is called “pancake sorting,” and sorting an arbitrary sequence with the minimum number of operations restricted in this way is known to be NP-hard. In this paper, we consider applying the concept of zero-knowledge proofs to the pancake sorting problem. That is, we design card-based zero-knowledge proof protocols in which a user (the prover) who knows how to sort a given sequence with ℓ operations can convince another user (the verifier) that the prover knows this information without divulging it.
IntroductionIn mathematics and theoretical computer science, a fixed point of an operator $F$ is an entity $x$ such that $F(x) = x$. Fixed-point results appear across many domains: for example, Banach’s Fixed-Point Theorem guarantees a unique fixed point for any contraction mapping on a complete metric space, and the Knaster–Tarski Theorem ensures that every monotone function on a complete lattice has a fixed point. These classical theorems establish existence (and sometimes uniqueness) of solutions to $x = F(x)$ under various conditions. In recursive and computational settings, fixed points enable self-referential definitions – a recursive function can be seen as a fixed point of a functional that “unwinds” one step of the recursion. For instance, the Y-combinator in lambda calculus provides a fixed-point combinator $Y$ such that for any function $W$, $Y(W)$ is a term satisfying $Y(W) = W(Y(W))$. This yields recursive definitions (like the factorial function) as solutions to self-referential equations.This article develops a fully formal framework for recursive fixed points – fixed points obtained via an iterative or recursive process. We focus on the convergence of a sequence (possibly transfinite) of transformations to a self-consistent state. Intuitively, we start with an initial approximation and repeatedly apply a transformation $\phi$; if this process approaches a stable state that no longer changes under $\phi$, we have reached a fixed point. Formally, one may consider an iterative sequence $x_0, x_1 = \phi(x_0), x_2 = \phi(x_1), \dots$ and seek a limit $x_{\infty}$ such that $\phi(x_{\infty}) = x_{\infty}$. Such a limit, if it exists, is a recursive fixed point – the result of infinitely (or transfinitely) many applications of $\phi$. This idea can be generalized beyond simple sequences, using the machinery of ordinal-indexed recursion and category theory to rigorously construct $\phi^\infty$, the outcome of transfinitely many iterations of $\phi$.Recent research by Alpay (2025) introduced Alpay Algebra, a category-theoretic framework where transfinite recursive fixed points play a central role. In this framework, an endofunctor $\phi$ (an operator on objects in a category) can be iterated through ordinal numbers to yield a stable initial fixed point denoted $\phi^{\infty}$. The existence of $\phi^{\infty}$ under broad conditions and its universal properties have been proven with mathematical rigor. The fixed point $\phi^{\infty}$ is recursive in that it is obtained as the limit of an ordinal-indexed chain of iterative approximants (often called the initial chain). Notably, this $\phi^{\infty}$ is not just any solution to $X \cong \phi(X)$; it is the smallest or initial solution, meaning it is generated by the recursive process itself and any other fixed point admits a unique morphism from $\phi^{\infty}$. In other words, $\phi^{\infty}$ encapsulates the “ultimate outcome” of the transformation $\phi$ applied repeatedly without end – a self-consistent structure that remains invariant under $\phi$.This manuscript provides a formal exposition of recursive fixed points. We begin by establishing the mathematical preliminaries (category-theoretic foundations and definitions of transfinite iteration). We then prove the existence and uniqueness of the transfinite fixed point $\phi^{\infty}$ under appropriate conditions, drawing on recent developments in Alpay Algebra. We illustrate these concepts with examples ranging from classical structures (natural numbers, infinite streams) to logic (fixed-point semantics of recursive theories) and AI systems (iterative embedding alignment) to demonstrate the ubiquity of recursive fixed points in theory and practice. Throughout, we use a formal style with symbolic notation – emphasizing symbols over prose – to maximize precision and semantic weight. By the end, we will see that recursive fixed points not only exist, but in fact serve as universal invariants in many self-referential systems, providing a rigorous backbone for understanding phenomena like emergent consistency, identity of processes, and convergence of iterative algorithms.Preliminaries: Category-Theoretic FrameworkOur development uses the language of category theory to formalize recursive processes. We briefly summarize the needed notions (for a comprehensive background, see Mac Lane, 1971). We assume a category $\mathcal{A}$ with an initial object $\varnothing$ (an object with a unique morphism from it to any other object). An endofunctor $\phi: \mathcal{A} \to \mathcal{A}$ is an operator that maps objects to objects and morphisms to morphisms within $\mathcal{A}$. Intuitively, $\phi$ represents one step of a generative or transformative process on the structures in $\mathcal{A}$. We are interested in objects $X$ that satisfy an isomorphism $X \cong \phi(X)$ – such objects are fixed points of the functor $\phi$ (also called $\phi$-algebras that are self-consistent).Transfinite Ordinals and Chains: To capture recursive (potentially infinite) iteration, we consider ordinal numbers $0, 1, 2, \dots, \omega, \omega+1, \dots$ which extend the natural numbers into the transfinite. An ordinal-indexed chain(or transfinite sequence) in $\mathcal{A}$ is a family of objects ${X_{\alpha}}{\alpha < \lambda}$ for some ordinal $\lambda$, together with morphisms connecting them, such that $X{0} = \varnothing$ (the initial object), and for each ordinal $\beta < \lambda$:Successor step: If $\beta = \alpha+1$ is a successor, then $X_{\beta} = \phi(X_{\alpha})$. In other words, each step applies the functor $\phi$ to the previous object.Limit step: If $\beta$ is a limit ordinal (zero is the minimal ordinal, any non-zero ordinal with no immediate predecessor is a limit), then $X_{\beta}$ is defined as the colimit (categorical limit of the diagram) of all earlier $X_{\alpha}$ for $\alpha < \beta$. Intuitively, at a limit stage, $X_{\beta}$ is the “union” or limit of the prior approximations $X_{0}, X_{1}, ..., X_{\alpha}, (\alpha<\beta)$.This process yields an initial chain:X0→ X1=ϕ(X0)→ X2=ϕ2(X0)→ ⋯→ Xω=colim{Xn:n<ω}→ Xω+1=ϕ(Xω)→ ⋯Each stage $X_{\alpha}$ is built “recursively” from the previous ones. We say $\phi$ is continuous (or $\kappa$-accessible) if it preserves colimits of chains of length $<\kappa$ for some regular cardinal $\kappa$ (for example, $\omega$-continuous means it preserves countable colimits). Under such conditions, one can show that the initial chain eventually reaches a stage where applying $\phi$ does not produce a new object. Formally, there exists some ordinal $\mu$ (often $\mu = \kappa$ or earlier) such that $X_{\mu} \cong X_{\mu+1} = \phi(X_{\mu})$. When this first occurs, $X_{\mu}$ is a fixed point of $\phi$. By construction, it is the minimal or initial fixed point, since it arose from the smallest starting object by iterative application of $\phi$. We denote this object as $\mu \phi$ or $\phi^{\infty}$ (Alpay’s notation). It is also called the initial $\phi$-algebra in category-theoretic terms.Definition: The recursive fixed point of $\phi$, denoted $\phi^{\infty}$, is the object (if it exists) at which the transfinite iterative chain stabilizes. Concretely, $\phi^{\infty}$ is an object such that $\phi(\phi^{\infty}) \cong \phi^{\infty}$ and for some ordinal $\mu$, $\phi^{\infty} = X_{\mu}$ with $X_{\mu} \cong \phi(X_{\mu})$, where ${X_{\alpha}}$ is the initial chain defined above.Because $\phi^{\infty}$ arises by iterating $\phi$ starting from the smallest object, it intuitively represents the “limit of applying $\phi$ forever.” This aligns with the idea of a recursive process converging to a fixed point. The existence of $\phi^{\infty}$ is not automatic in every category or for every functor $\phi$; it typically requires conditions like completeness of the category or continuity of $\phi$ as mentioned. The Alpay Algebra framework explicitly assumes such conditions (e.g., working in well-behaved categories with transfinite colimits and $\phi$ preserving those colimits). In fact, Alpay Algebra I establishes that $\phi^{\infty}$ exists for every initial object under ZFC set theory assumptions without additional axioms. All proofs are carried out within standard category-theoretic foundations (following Mac Lane’s paradigm).Existence of the Recursive Fixed Point ($\phi^{\infty}$)We now state and prove (in outline) the Existence Theorem for recursive fixed points. This corresponds to the fixed-point existence results found in Alpay’s work and is analogous to the classical results (Knaster-Tarski, etc.) but in a transfinite categorical setting.Theorem 1 (Existence of $\phi^{\infty}$): Let $\phi: \mathcal{A} \to \mathcal{A}$ be an endofunctor on a category $\mathcal{A}$ that admits all colimits of chains of length up to some regular ordinal $\Lambda$, and assume $\phi$ preserves these colimits (i.e. $\phi$ is $\Lambda$-continuous). If $\mathcal{A}$ has an initial object $X_0$, then the initial chain $(X_{\alpha}){\alpha < \Lambda}$ defined by $X{0} = \varnothing$ and $X_{\alpha+1} = \phi(X_{\alpha})$ (with $X_{\lambda} = \mathrm{colim}{\alpha<\lambda} X{\alpha}$ for limit ordinals $\lambda < \Lambda$) will converge to a fixed point. In other words, there exists some ordinal $\mu < \Lambda$ such that $X_{\mu} \cong \phi(X_{\mu})$. This object $X_{\mu}$ is the recursive fixed point $\phi^{\infty}$. Moreover, $\phi^{\infty}$ is an initial algebra for $\phi$: the unique $\phi$-algebra generated by this transfinite iteration.Proof Sketch: Because $\Lambda$ is regular and $\phi$ preserves colimits of chains of length $<\Lambda$, the chain cannot continue to produce strictly larger (non-isomorphic) objects at every stage without end; if it did, one could take the colimit at stage $\Lambda$ (contradicting regularity or continuity). Thus there must be some stage where stabilization occurs. Formally, consider the sequence of inclusions (morphisms) $X_0 \to X_1 \to X_2 \to \cdots$. Either this sequence continues strictly (no stabilization) for all ordinals $<\Lambda$, or else there is a stage $\mu$ where $X_{\mu} \to X_{\mu+1}$ is an isomorphism. The former scenario is ruled out by a cardinality/cumulativity argument: if no stabilization occurs before $\Lambda$, then $X_{\Lambda} = \mathrm{colim}{\alpha<\Lambda} X{\alpha}$ is a fixed point of $\phi$ at stage $\Lambda$, because $\phi(X_{\Lambda}) = \phi(\mathrm{colim}{\alpha<\Lambda} X{\alpha}) \cong \mathrm{colim}{\alpha<\Lambda} \phi(X{\alpha}) = \mathrm{colim}{\alpha<\Lambda} X{\alpha+1} = \mathrm{colim}{\alpha<\Lambda} X{\alpha} = X_{\Lambda}$ (using continuity of $\phi$). Thus $X_{\Lambda}$ itself would be a fixed point, effectively $X_{\Lambda} \cong \phi(X_{\Lambda})$, achieving stabilization at $\Lambda`. In either case, we obtain some least ordinal $\mu$ (possibly $\mu=\Lambda$ if no earlier stage) such that $X_{\mu} \cong \phi(X_{\mu})$. Define $\phi^{\infty} := X_{\mu}$. By construction, $\phi^{\infty}$ satisfies $\phi(\phi^{\infty}) \cong \phi^{\infty}$. Furthermore, for any $\phi$-algebra $(X,\alpha: \phi(X)\to X)$ (any other fixed structure), we have by initiality of the chain that there is a unique homomorphism from each $X_{\alpha}$ into $X$ commuting with the $\phi$-action; at the limit, this yields a homomorphism $h: \phi^{\infty} \to X$. Thus $\phi^{\infty}$ is the initial object among all solutions of $X \cong \phi(X)$. ∎This theorem formalizes the existence of a transfinitely attained fixed point. In plainer terms, if one keeps applying $\phi$ starting from the simplest object, eventually (perhaps after an infinite number of steps) one stops getting new structures and hits a self-consistent one. That endpoint is $\phi^{\infty}$. All the intermediate steps $X_0 \to X_1 \to \cdots \to X_{\mu}=\phi^{\infty}$ are essentially building up a solution to the equation $X = \phi(X)$ piece by piece (like successive approximations). The condition of $\phi$ preserving colimits ensures that no information is lost in the limit process and that $\phi^{\infty}$ truly is a fixed point.In the context of Alpay Algebra (a formal system introduced by Faruk Alpay), Theorem 1 is a central pillar: it guarantees that for the self-referential processes defined in that framework, a stable identity emerges as a fixed point. Specifically, Alpay Algebra treats $\phi^\infty$ as representing the intrinsic identity of a generative process, since it is the unique invariant state that the process converges to. The existence theorem above matches statements in Alpay’s work such as: “We prove that the fixed point $\phi^\infty$ exists for every initial object ... and [that] $\phi$-iterates converge under regular cardinals”. The iterative construction of $\phi^\infty$ is sometimes called an ordinal-indexed fold or transfinite induction on the functor.It is worth noting that this categorical result generalizes classical fixed-point constructions. For example, in domain theory (a branch of theoretical computer science and math), a special case of this theorem states that any continuous endofunction on a complete partial order has a least fixed point (by taking the $\omega$-chain of iterates starting from the bottom element). That is essentially the $\omega$-continuous case of the above, corresponding to Kleene’s Fixed-Point Theorem for monotone operators on CPOs (which is itself an instance of Tarski’s theorem for lattices). Our transfinite approach extends this idea: even if $\omega$ steps are not enough, one allows transfinite steps until convergence. The result is a powerful guarantee: if the process can keep going without inconsistency, it will find a fixed point. In practical terms, this means any self-referential or recursive definition, under broad conditions, “bottoms out” at a well-defined semantics or structure that does not further change under the defining transformation.Uniqueness and Universal PropertyHaving established existence, we examine the uniqueness and universal property of the recursive fixed point $\phi^{\infty}$. Uniqueness here means $\phi^{\infty}$ is essentially the only fixed point that can be obtained through the recursive construction – if there were another built in the it would be to $\phi^{\infty}$. $\phi^{\infty}$ is the smallest fixed point, and every other fixed point of $\phi$ out” of $\phi^{\infty}$ in a unique and In the of Theorem 1, the recursive fixed point $\phi^{\infty}$ is unique up to isomorphism. Moreover, $\phi^{\infty}$ has the initiality for any object $X$ with an isomorphism \cong \phi(X)$ (i.e. any other fixed point of there exists a unique morphism \phi^{\infty} \to (a the appropriate In any other fixed-point object $X$ admits a unique from $\phi^{\infty}$, which is an isomorphism if $X$ is also obtained via a iterative Sketch: Uniqueness up to isomorphism is a standard category-theoretic argument: if and are initial $\phi$-algebras fixed then by initiality of there is a unique homomorphism I \to and by initiality of a unique homomorphism \to The I \to must the identity on (by uniqueness of the on and = Thus and are \cong This any solutions of the transfinite recursion are we can of the $\phi^{\infty}$ up to the universal any other fixed point we can $X$ with a $\phi$-algebra structure and to Specifically, \to can serve as the structure on \cong is there is a unique $\phi$-algebra homomorphism \phi^{\infty} \to such that = Intuitively, maps the fixed into any fixed structure $X$ in the only with $\phi$. If $X$ itself built by a iterative limit process (e.g., another chain that also a fixed then will be an isomorphism $\phi^{\infty}$, cannot into a end up the and formal terms, there is only one smallest self-consistent structure generated by $\phi$, and any other self-consistent structure a from this smallest one. This $\phi^{\infty}$ the universal fixed point or the identity of the process $\phi$. other solution of $X \cong \phi(X)$ through $\phi^{\infty}$ in a unique The morphism \phi^{\infty} \to can be seen as the or the fixed point $\phi^{\infty}$ into the fixed point If $X$ is not not be or but it exists and is universal property that $\phi^{\infty}$ the of all of the recursive process. In Alpay’s terms, $\phi^{\infty}$ as the of the a fixed point that every other fixed point in a This is analogous to the least fixed point in a lattice is in (or maps every other fixed point in that of Recursive Fixed illustrate the theory with recursive fixed points in various the functor = 1 on the category is a $\phi$-algebra is \to which provides a point (the of the and a function \to (the on the $X$ $X \cong 1 in yields the result that $X$ must be a infinite set to the natural numbers (with the point corresponding to and the function corresponding to the starting from the initial object and iterating $\phi$: $X_0 = = 1 \cong (a = 1 1 \cong = 1 \cong ... one \cong and the colimit as \to is \cong $\omega$ we have \cong 1 has a point and the is to is a fixed point of $\phi$. 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The iterative in (the etc.) is building the chain = = and which satisfies = This is the recursive fixed point. by theory one is the one to – the of a fixed point of the This semantic convergence to a fixed point the of recursive the theory is a point where the process of stabilizes. 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As Electronic Health Records (EHRs) become integral to modern healthcare, concerns regarding security, privacy, and scalability have emerged. Conventional centralized systems face vulnerabilities and lack patient data control, necessitating innovative decentralized solutions. This research introduces a novel AI-blockchain-based architecture for permissioned health networks, leveraging zero-knowledge succinct, non-interactive arguments of knowledge (zkSNARKs). The framework integrates AI-blockchain, InterPlanetary File System (IPFS), and zkSNARKs to enhance EHR security, privacy, and scalability. IPFS is employed for off-chain storage, improving scalability alongside blockchain's tamper-proof logging. Encryption and smart contract-based access controls preserve privacy.
Zeyad Ghaleb Al-Mekhlaf, Murtaja Ali Saare, Jalal Mohammed Hachim Altmemi, Mahmood A. Al-Shareeda · 9 authors
The rapid adoption of Internet of Medical Things (IoMT) devices enables real-time patient monitoring and remote diagnostics and has revolutionized healthcare delivery. Traditional cryptographic schemes like RSA and ECC, which rely on meaningful mathematical challenges, are under great threat from quantum computing, threatening sensitive medical data confidentiality and integrity. This paper proposes a quantum-resistant healthcare security framework based on lattice-based cryptographic primitives such as Learning With Errors (LWE), Ring-LWE (RLWE), and Short Integer Solution (SIS). To this end, we design a five-phase IoMT-friendly framework—Initialization, Registration, Authentication, Data Exchange, and Treatment—where each phase is backed up by lightweight cryptography primitives that can be easily implemented on the low-resource IoMT devices. Relative to the state-of-the-art lattice- and hash-based constructions, our framework involves 50-75% smaller ciphertext sizes, up to a 50% reduction of the communication overhead, and nearly 60% less in computational cost. Furthermore, the solution relies on zero-knowledge proofs, homomorphic encryption as well and attribute-based access control to guarantee strong security and privacy. Using the AVISPA tool, the framework is formally verified, showing its resistance against classical and quantum adversaries. Focusing on tangible healthcare threats, including data tampering and unlicensed access to patient diagnostics, this research paves the way for scalable, efficient, and quantum-resistant medical data protection. Our results pave the way for future investigations into secure post-quantum healthcare and IoT applications.
The rapid digitization of e-government systems has introduced significant privacy challenges, including unauthorized data access and identity theft, which threaten the integrity and trustworthiness of public services. This study proposes a privacy protection framework based on Zero-Knowledge Proofs (ZKP), a cryptographic technique enabling secure verification without revealing sensitive information. The framework addresses critical privacy concerns such as secure identity verification, data confidentiality, and compliance with regulatory standards. By integrating advanced ZKP schemes, including Succinct Non-Interactive Arguments of Knowledge (zk-SNARKs) and Bulletproofs, the framework ensures efficient proof generation and verification while minimizing computational overhead. A performance evaluation demonstrated that the proposed framework reduces privacy risks by 78% and achieves a threefold increase in transaction throughput compared to traditional cryptographic methods, such as Rivest–Shamir–Adleman (RSA) and Public Key Infrastructure (PKI). The scalability and efficiency of the framework were validated through extensive computational overhead analysis and comparative benchmarking. Additionally, trusted setup optimizations and constraint system modeling were employed to enhance the framework’s robustness and adaptability for large-scale e-government applications.
Zero-Knowledge Proofs (ZKPs) are critical for privacy-preserving techniques and verifiable computation. Many ZKP protocols rely on key kernels such as the SumCheck protocol and Merkle Tree commitments to enable their key security properties. These kernels exhibit balanced binary tree computational patterns, which enable efficient hardware acceleration. Although prior work has investigated accelerating these kernels as part of an overarching ZKP protocol, exploiting this common tree pattern remains relatively underexplored. We conduct a systematic evaluation of these tree-based workloads under different traversal strategies, analyzing performance on multi-threaded CPUs and the Multifunction Tree Unit (MTU) hardware accelerator. We introduce a hardware-friendly Hybrid Traversal for binary tree that improves parallelism and scalability while significantly reducing memory traffic on hardware. Our results show that MTU achieves up to $1478\times$ speedup over CPU at DDR-level bandwidth and that our hybrid traversal outperforms breadth-first search by up to $3\times$. These findings offer practical guidance for designing efficient hardware accelerators for ZKP workloads with binary tree structures.
D. Xiao, Shaobo Xu, Chuan Zhang, Licheng Wang · 6 authors
Cross-chain technology facilitates the interoperability among isolated blockchains, where users can transfer and exchange coins. While the heterogeneity between Turing-complete (TC) blockchains like Ethereum and non-Turing-complete (NTC) blockchains like Bitcoin presents a significant challenge for cross-chain transactions. Payment Channel Hubs (PCHs) offer a promising solution for enabling TC-NTC cross-chain payments with high throughput and low confirmation delays. However, existing schemes still face two key challenges: (i) significant computation and communication overhead for variable-amount payment, and (ii) limited unlinkability, i.e., vulnerable to the abort attack. This paper proposes EC2P, the first TC-NTC cross-chain PCH that achieves variable-amount payment unlinkability while resisting the abort attack and minimizing reliance on non-interactive zero-knowledge (NIZK) proofs. EC2P introduces two protocols: the NTC-to-TC and TC-to-NTC payment protocols. The NTC-to-TC payment protocol replaces the traditional puzzle-promise and puzzle-solve paradigm with a semi-blind approach, where only one side is blinded and the blinded side’s interactions are eliminated. This achieves unlinkability and resists the abort attack without NIZK. The TC-to-NTC payment protocol enhances the paradigm by utilizing Turing-complete functionality to constrain the inability to carry out an abort attack. Through rigorous security analysis, we show that EC2P is secure and variable-amount payment unlinkable while resisting the abort attack. We implement EC2P on Ethereum and Bitcoin test networks. Our evaluation demonstrates that EC2P outperforms both in terms of communication and computation overhead and reduces communication costs by 3 orders of magnitude compared to existing variable-amount methods.
Hasib Ahmed Md Khyrul Islam, Huy T. Vo, Aditya Rane
In the era of synthetic media, deepfake manipulations pose a significant threat to information integrity. To address this challenge, we propose TrustDefender, a two-stage framework comprising (i) a lightweight convolutional neural network (CNN) that detects deepfake imagery in real-time extended reality (XR) streams, and (ii) an integrated succinct zero-knowledge proof (ZKP) protocol that validates detection results without disclosing raw user data. Our design addresses both the computational constraints of XR platforms while adhering to the stringent privacy requirements in sensitive settings. Experimental evaluations on multiple benchmark deepfake datasets demonstrate that TrustDefender achieves 95.3% detection accuracy, coupled with efficient proof generation underpinned by rigorous cryptography, ensuring seamless integration with high-performance artificial intelligence (AI) systems. By fusing advanced computer vision models with provable security mechanisms, our work establishes a foundation for reliable AI in immersive and privacy-sensitive applications.
Open access
2 source records
Adversarial Robustness in Machine Learning
Digital Media Forensic Detection
Generative Adversarial Networks and Image Synthesis
Reisha Ali, Akshat Gupta, Maria Francis, Kotaro Kataoka
Decentralized applications (DApps) over blockchains often require the user’s personal information for authentication. However, the public and transparent nature of blockchains can compromise user privacy. Threshold-based anonymous credentials (TAC) provide anonymous and unlinkable authentication, which helps preserve user privacy. Additionally, the design of TAC aligns well with blockchain’s decentralized nature because TAC offers decentralized trust distribution to prevent a single point of failure. However, only a few of them have implementations over blockchains because TAC requires computationally expensive cryptographic operations such as pairings and verification of zero-knowledge proofs (ZKPs) to be done on-chain. Thus, existing TAC systems have not been evaluated in a public permission-less blockchain environment. This evaluation is crucial to assess the efficiency and practicality of deploying TAC in real-world blockchain use cases to make TAC based DApps development feasible. This work presents the design and evaluation of three state-of-the-art TAC systems, RP-Coconut, threshold BBS+ (T-BBS+), and BBS (T-BBS), over the Ethereum blockchain. The evaluation compares the performance of these TAC systems on the Sepolia testnet in terms of execution time and gas usage. Additionally, this work also proposes partial credential verification mechanisms for T-BBS+ and T-BBS that significantly reduce the complexity of identifying valid credentials, thereby lowering the execution time at the user’s end. Furthermore, the implementation for blind issuance and associated ZKPs for T-BBS is provided, which was not previously detailed in the literature and is critical for its correct implementation.