Secret sharing of $(k,n)$-threshold is naturally used to assist in security-assurance applications in smart systems. It can ensure a high level of data security and reliability. Most of existing secret sharing schemes are employing interpolating polynomial or Chinese remainder theorem, while the hyperplane geometry of Blakley’ s scheme has not been widely used. One of the reasons is that the hyperplane geometry-based scheme leaks information about the secret even via less than the threshold number of shadows. In this paper, we propose a novel secret sharing scheme based on the high-dimensional rotation paraboloid instead of the hyperplanes to overcome this weakness. The secret is corresponding to the unique focal point, and the shadows are corresponding to points on it. We prove that linearly independent $k$ points can determine a unique $(k-1)$-dimensional rotation paraboloid, which is the theoretical support of secret distribution and reconstruction. Moreover, we realize verification via two well-known terms: cheater identification and cheating detection. We identify the validity of shadows submitted by participants for cheater identification and verify whether the reconstructed secret is the original one or not for cheating detection, so that dishonest behavior in the secret sharing scheme can be detected. Security analysis shows our scheme is secure against different types of attacks.

秘密分享 $（ķ，ñ）$-threshold自然用于辅助智能系统中的安全性保证应用程序。它可以确保高水平的数据安全性和可靠性。现有的大多数秘密共享方案都采用插值多项式或中文余数定理，而Blakley方案的超平面几何并未得到广泛使用。原因之一是基于超平面几何的方案甚至通过少于阈值数量的阴影也会泄漏有关机密的信息。在本文中，我们提出了一种基于高维旋转抛物面而不是超平面的新型秘密共享方案，以克服这一弱点。秘密对应于唯一焦点，阴影对应于其上的点。我们证明线性独立$ķ$ 点可以确定唯一 $（ķ-\mathrm{1个}）$维旋转抛物面，这是秘密分布和重建的理论支持。此外，我们通过两个众所周知的术语来实现验证：作弊者识别和作弊检测。我们确定参与者提交的用于作弊者识别的阴影的有效性，并验证重构的机密是否是用于作弊检测的原始机密，以便可以检测到机密共享方案中的不诚实行为。安全分析表明，我们的方案可以抵抗各种类型的攻击。