When companies collaborate across a supply chain, they must constantly compare and match sensitive numbers: transaction amounts, exchange rates, prices, and quantities that flow between Internet of Things devices scattered across warehouses, factories, and logistics networks. A new study published in Cluster Computing by Xin Liu and colleagues proposes a family of secure computational protocols that allow enterprises to compute the intersection and union of sets containing rational numbers without ever revealing the underlying values to each other. The work targets a subtle but consequential problem in applied cryptography: most existing secure computation techniques handle only integers, and the standard workaround of scaling rational numbers by a fixed multiplier can silently destroy precision, opening the door to privacy leaks and wrong business decisions.
The core insight of the paper is that transaction data in cross-enterprise supply chain collaboration has a fundamentally rational structure. A price of 3.75 units, an exchange rate of 0.917, or a quantity of 12.5 items cannot be faithfully represented as an integer without some loss. Previous approaches to processing rational numbers in secure computing typically multiply every value by the same scaling factor to convert it into an integer, perform the secure computation, and interpret the result afterwards. The authors argue that this practice is risky in two distinct ways. First, any loss of rational number precision can lead to erroneous judgments, for example when two nearly identical prices must be distinguished to detect fraud or verify a contract. Second, simplistic conversion of rational numbers to integers may itself create privacy risks, because the scaling operation can leak structural information about the data if the factor is chosen carelessly.
To avoid these pitfalls, the research team transforms rational numbers into matrices and designs secure protocols for set intersection and set union that operate directly on this matrix representation. The cryptographic engine underneath is the Elgamal homomorphic encryption algorithm, a public key cryptosystem introduced by Taher Elgamal in 1985 and based on the hardness of the discrete logarithm problem. Homomorphic encryption allows mathematical operations to be performed on ciphertexts, so that the encrypted data can be manipulated without decryption. In the setting studied here, the multiplicative homomorphic property of Elgamal enables the participating enterprises to jointly evaluate whether elements appear in both of their sets, which is the intersection, or in at least one of them, which is the union, while each party’s individual inputs remain hidden.
A crucial feature of the new protocols is that they are designed and analyzed under two different adversary models. In the semi-honest model, participants follow the protocol exactly as specified but try to learn extra information from the messages they legitimately observe. This is the classical setting for much of secure multiparty computation, a field that traces back to Andrew Yao’s foundational 1982 work on protocols for secure computation and the famous millionaires’ problem, in which two parties want to know who is richer without disclosing their actual wealth. The malicious model is far more demanding: participants may deviate arbitrarily from the protocol, sending malformed messages, replaying old values, or aborting at strategic moments in order to cheat or extract information. Protocols that are secure against semi-honest adversaries can collapse entirely when faced with active cheating.
The authors prove the security of their protocols under the malicious model using the ideal-practical, or real-ideal, simulation paradigm, which is the standard gold standard for security proofs in modern cryptography. In this framework, one imagines an ideal world in which the parties hand their inputs to a trusted third party who performs the computation and returns only the output. A protocol is deemed secure if whatever an adversary can achieve by attacking the real protocol could also be achieved in that ideal world, meaning the protocol leaks nothing beyond what the ideal functionality itself reveals. By constructing a simulation-based argument for the malicious setting, the researchers demonstrate that even participants who actively misbehave cannot learn the other parties’ rational number inputs or force an incorrect result to be accepted.
The practical motivation comes from the growing dependence of IoT enterprises on automated, cross-organizational data exchange. Modern supply chains are studded with sensors, RFID tags, and connected devices that generate continuous streams of transactional records. Enterprises want to answer questions such as whether a supplier’s price list overlaps with a competitor’s, whether the same batch of goods appears in two different logistics records, or which exchange rates are shared across regional partners. Set intersection and union are the natural mathematical operations for these tasks, but performing them on plaintext data would force companies to expose commercially sensitive information to partners who may be competitors at the same time as they are collaborators. Secure multiparty computation resolves this tension by letting the computation proceed on encrypted inputs, and the paper positions its rational-number-aware protocols as a missing piece for supply chain scenarios where fractional values dominate.
The study situates itself within a rich body of prior work on secure set operations and rational number computation. Earlier research has produced protocols for secure set membership determination, private intersection-sum computation, and the secure computation of the maximum value of the sum of corresponding elements of an intersection set. Other groups have developed secure multiparty computation of rational intervals, secure determination of interval positional relationships, and privacy-preserving two-party rational set computation. Related efforts have extended secure computation to graph intersections and concatenations, and recent surveys have collected efficient protocol families and their applications. Work in adjacent domains, including privacy-preserving authentication for smart cities, secure data analysis in healthcare IoT systems, blockchain-assisted verifiable data computing, and privacy-preserving collaborative routing, illustrates how broadly the demand for secure joint computation has spread across the Internet of Things landscape.
Beyond correctness and security, the paper contributes a comparative efficiency analysis. The authors analyze the efficiency of existing protocols against their proposed construction and argue that the new approach is practical for real deployment. The matrix-based representation of rational numbers plays a double role here: it preserves exact precision, avoiding the rounding errors introduced by uniform scaling, and it provides a uniform structure on which the Elgamal-based operations can be applied systematically. Because Elgamal encryption is well studied and its security rests on a long-standing hardness assumption, the construction inherits a degree of cryptographic maturity that newer, less scrutinized techniques may lack. The authors also report simulation results supporting the practicality claim, and they state that all data generated or analyzed during the study are included in the article itself.
The research was carried out by a team spanning Tianjin Renai College, the Inner Mongolia University of Science and Technology in Baotou, and the North China University of Technology in Beijing, with correspondence to Lanying Liang. The work received support from multiple Chinese funding bodies, including the National Natural Science Foundation of China, the Natural Science Foundation of Inner Mongolia, and several regional talent and technology development programs. The article was received in July 2025, accepted in August 2026, and published on 3 September 2026 in volume 29 of Cluster Computing as article number 722.
For enterprises weighing the trade-off between data utility and data exposure, the message of this research is that precision itself is a security property. A protocol that silently rounds 0.917 to 0.92 does not merely produce a slightly wrong answer; it can produce a confidently wrong answer that propagates through pricing, auditing, and compliance decisions, and it may do so in ways that leak information about the original figures. By keeping rational numbers exact throughout the computation and by hardening the protocols against actively malicious participants, the new work offers IoT-driven supply chains a template for collaboration in which partners can compute exactly the answers they need, and nothing more. As connected devices multiply and supply chains grow more intertwined, techniques of this kind may become a standard layer of the infrastructure that lets rival firms share a market without sharing their secrets.
Subject of Research: Secure multiparty computation protocols for privacy-preserving rational number set operations in IoT supply chain collaboration
Article Title: Secure computational protocols for supply chain security of IoT enterprises based on intersection and union of rational numbers sets under the malicious model
Article References: Liu, X., Wang, Y., Liang, L., Xu, G., Gu, Y., & Zhang, B. (2026). Secure computational protocols for supply chain security of IoT enterprises based on intersection and union of rational numbers sets under the malicious model. Cluster Computing, 29(12), Article 722. https://doi.org/10.1007/s10586-026-06479-x
Image Credits: AI Generated
DOI: 10.1007/s10586-026-06479-x
Keywords: secure multiparty computation, rational numbers, set intersection, set union, Elgamal encryption, homomorphic encryption, malicious model, IoT, supply chain security, privacy-preserving computation, cryptography, Cluster Computing
News Source: Denise Maddox. (October 11, 2026). New Cryptographic Protocols Keep Rational Numbers Intact in IoT Supply Chain Data Sharing. Scienmag.
