Design And Analysis On The Threshold Function Secret Sharing With Information-Theoretic Security

Functional Secret Sharing(FSS)is a cryptographic primitive recently introduced by Boyle et al.,motivated by increasing the efficiency of private information access protocol.Unlike traditional secret sharing,in which the secret shared between participants is a certain value,and the secret shared in FSS is a function.In the distribution phase,the dealer splits the secret function into several sub-functions and sends them to the corresponding participants privately.The set of participants who are authorized in the reconstruction phase work together to reconstruct the function value of function at a certain point in its domain.Existing FSS schemes are constructed based on the pseudo-random generators,so these schemes are computationally secure.That is,the security of the scheme can only resist adversaries with limited computing power.In addition,these schemes require all participants work together during the reconstruction phase,and they cannot be applied to real-world scenarios flexibly.In response to these problems,the work of this article starts from improving the security and practicability of the FSS scheme.Some FSS schemes are then designed that can resist attack by any adversary with unlimited computing power,and they can tolerate the absence of some participants during the reconstruction phase.We called them as the threshold FSS scheme with information-theoretic security.The research work of this article mainly includes the following three parts.(1)A new threshold FSS scheme is designed based on the polynomial interpolation technology.First,a formal definition of the threshold FSS scheme is given,and a strict formal definition of security is also given under the game-based security model.According to the security definition,the security problems of one existing threshold FSS scheme is analyzed,and the reasons why it does not meet the security requirements of FSS schemes are elaborated in detail.We then use the polynomial technology to improve this scheme.A new threshold FSS scheme based on the polynomial interpolation technology was designed and we proved it is information-theoretic security.(2)Based on the Shamir's scheme,three threshold FSS schemes with information-theoretic security are constructed.The initial scheme is designed without considering the transmission cost,and the length of its subfunction is as same as the size of the domain of the secret function.Afterwards,we use the tensor product to transform the longer vector into several shorter vectors.The length of the subfunction in the initial scheme is reduced based on the tensor product.Two improvement schemes with higher transmission efficiency are then constructed in turn.However,none of these schemes can satisfy the simplicity,compression,and function privacy required by FSS to improve the efficiency of private information access protocol.(3)A threshold FSS scheme with information-theoretic security is constructed by using multi-variable polynomial techniques.This scheme can simultaneously meet the three properties of simplicity,compressibility,and function privacy.The core technique is to convert the calculation of the function value of the secret function at the public point in the FSS to the calculation of the function value of the public function at the secret point.In view of the close relationship between FSS and private information retrieval protocol,a robust private information retrieval protocol is designed effectively based on our proposed threshold FSS scheme.