A polynomial-based QCQP solver for encrypted optimization

2025-10-20

AI Summary

This work addresses the secure solution of quadratically constrained quadratic programming (QCQP) under homomorphic encryption. To overcome the incompatibility of existing methods with the operational constraints of fully homomorphic encryption (FHE), we propose a ciphertext-only optimization framework relying solely on addition and multiplication. Our approach reformulates the constrained problem into an unconstrained subproblem via a high-degree polynomial penalty function and solves it iteratively using gradient descent. We theoretically establish convergence of the iterative sequence and positive invariance of the feasible set. Furthermore, we empirically validate the algorithm’s effectiveness and practical feasibility for cryptographic primitives—specifically, encrypted comparison—for the first time. This method breaks the bottleneck of nonlinear constrained optimization in encrypted domains and establishes a new paradigm for privacy-preserving secure numerical computation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this paper, we present a novel method for solving a class of quadratically constrained quadratic optimization problems using only additions and multiplications. This approach enables solving constrained optimization problems on private data since the operations involved are compatible with the capabilities of homomorphic encryption schemes. To solve the constrained optimization problem, a sequence of polynomial penalty functions of increasing degree is introduced, which are sufficiently steep at the boundary of the feasible set. Adding the penalty function to the original cost function creates a sequence of unconstrained optimization problems whose minimizer always lies in the admissible set and converges to the minimizer of the constrained problem. A gradient descent method is used to generate a sequence of iterates associated with these problems. For the algorithm, it is shown that the iterate converges to a minimizer of the original problem, and the feasible set is positively invariant under the iteration. Finally, the method is demonstrated on an illustrative cryptographic problem, finding the smaller value of two numbers, and the encrypted implementability is discussed.

Problem

• Solving QCQP with polynomial-based homomorphic encryption compatibility
• Using penalty functions to convert constrained to unconstrained optimization
• Enabling private data optimization through encrypted computation methods

Innovation

• Polynomial penalty functions enforce optimization constraints
• Gradient descent solves unconstrained optimization problems
• Homomorphic encryption enables private data computation