Secure planar convex hull protocol for large-scaled point sets in semi-honest model

Zhao Wenjie -, Mingxing He

Cheng Wang, Ri‐Gui Zhou

Wen-Jie Liu, Solomon Danquah Danso

Convex hull computation is a fundamental problem in secure multi-party computational geometry (SMCG), classified under secure multi-party computation (SMC) for finding the convex hull of a set of points. Existing quantum solutions largely depend on quantum homomorphic encryption (QHE), which introduces significant computational overhead due to frequent key updates by a trusted third party (TTP). Furthermore, most current protocols lack a mechanism for input commitment, making them vulnerable to post-computation input tampering or denial by the TTP. To overcome these limitations, we propose an efficient convex hull protocol that utilizes quantum secret commitment (QSC) as a more secure alternative to QHE. Our protocol enables a designated party (the committer) to securely commit to input values in a manner that guarantees both binding (no post-hoc alteration) and hiding (input secrecy). We introduce a novel value comparison protocol within an Ideal Quantum K-Party model, ensuring privacy-preserving convex hull computation without reliance on QHE. Rigorous security analysis demonstrates significant improvements in computational efficiency and resilience against quantum adversaries. Our protocol represents a pivotal advancement for quantum-secure multi-party computations and lays the groundwork for scalable, future-proof privacy-preserving geometry in quantum computing contexts.

Bai Liu, Xinguo Wang, Run-Hua SHI +1 more

Abstract With the growing demand for location privacy protection and collaborative computing in spatial crowdsourcing (SC) tasks, traditional encryption methods based on computational complexity face threats from quantum computing and efficiency bottlenecks. To address this, this paper proposes a secure multi-party quantum comparison (SMQC) protocol based on quantum homomorphic encryption. By utilizing a quantum comparator and a key update mechanism, this protocol enables secure comparison of numerical values among multiple parties while maintaining encryption. Building on this foundation, the SMQC protocol is further applied to the SC environment to construct a single-point collaborative convex hull computation protocol. Through extreme point determination and convex hull boundary verification, this protocol efficiently performs multi-party collaborative convex hull computation while protecting the location privacy of participants. Analysis indicates that the convex hull computation framework built on this protocol is reliable and secure, further expanding the application of quantum-secure solutions in SC scenarios such as single-point collaborative obstacle avoidance and collaborative spatial protection.

Hamidreza Amini Khorasgani

Secure multi-party computation (MPC) enables mutually distrusting parties to compute functions over private inputs without revealing additional information. A general MPC protocol is the one that can be used to securely compute any function. General MPC requires either an honest majority or oblivious transfer, which can be provided via computational assumptions, a black-box hybrid, or trusted hardware. However, in the setting of dishonest majority, there are still many tasks that can be computed securely in the information-theoretic plain model without relying on oblivious transfer or any other computational hardness assumptions. Such protocols are perfectly secure, fast, and do not require any setup or preprocessing.This thesis investigates the feasibility and complexity of secure two-party computation, which inherently falls under the dishonest majority setting, with randomized output in the information-theoretic plain model against honest-but-curious adversaries. For deterministic output functions, seminal work by Chor, Kushilevitz, and Beaver provided a complete characterization of which computations admit secure protocols. In contrast, randomized functionalities give rise to a substantially richer landscape, where questions of round complexity, communication complexity, and even decidability have remained unresolved for decades.The thesis develops a geometric framework for analyzing secure computation by encoding candidate private-coin secure protocols for a given functionality as points in high-dimensional spaces. Within this representation, the space of all secure protocols forms recursively defined geometric sets whose structure can be studied using tools from real algebraic geometry and convex analysis. This perspective enables feasibility, search, and optimization problems in secure computation to be reduced to structural properties of these sets.Using this framework, the thesis establishes that determining whether a given two-party functionality with randomized output admits a secure protocol is decidable. A finite procedure is provided that either constructs a secure protocol with optimal round and communication complexity or identifies a geometric obstruction to security, resolving a foundational open problem in secure computation and information complexity. These results rely on showing that certain lamination hulls arising in the analysis are semi-algebraic, which is achieved by reducing the problem to solving systems of linear inequalities over convex sets. To this end, the thesis introduces a Gaussian-elimination-inspired algorithm and a new set operator, the positive geometric join, leading to a class of algebraically well-behaved geometric objects called hemihedra.The thesis also studies the round and communication complexity of secure computation with randomized output and disproves a longstanding folklore conjecture asserting that round complexity depends only on the cardinalities of the input and output domains. Explicit constructions demonstrate the existence of securely realizable functions with arbitrarily high round complexity, even for fixed size inputs and outputs, while matching upper bounds are proved for small output alphabets. These results reveal a close connection between secure computation and lamination hulls originally studied in hydrodynamics, highlighting unexpected geometric structure underlying information-theoretic security.

Jonathan I. Katz, Chiu‐Yuen Koo