An Improved Outsourcing Algorithm to Solve Quadratic Congruence Equations in Internet of Things
Xiulan Li, Jingguo Bi, Chengliang Tian, Hanlin Zhang, Jia Yu, Yanbin Pan · IEEE Internet of Things Journal · 2021
Solving quadratic congruence equations is an expensive operation widely employed in cryptographic constructions for secure Internet of Things applications. Recently, two outsourcing algorithms were proposed by Zhanget al.to solve quadratic congruence equations by employing Cippolla’s algorithm. It was claimed that all the inputs and outputs can be obscured in these two algorithms. However, we present two passive attacks in this article to show that all the inputs and outputs can be recovered efficiently by just a curious server, which implies the two outsourcing algorithms are insecure. To fix them, we further propose an improved outsourcing algorithm to solve quadratic congruence equations, which is more efficient and the privacy of actual inputs and outputs can be protected very well.