Threshold cryptography with Chinese remainder theorem

Abstract

Information security has become much more important since electronic communication is started to be used in our daily life. The content of the term information security varies according to the type and the requirements of the area. However, no matter which algorithms are used, security depends on the secrecy of a key which is supposed to be only known by the agents in the first place. The requirement of the key being secret brings several problems. Storing a secret key on only one person, server or database reduces the security of the system to the security and credibility of that agent. Besides, not having a backup of the key introduces the problem of losing the key if a software/hardware failure occurs. On the other hand, if the key is held by more than one agent an adversary with a desire for the key has more flexibility of choosing the target. Hence the security is reduced to the security of the least secure or least credible of these agents. Secret sharing schemes are introduced to solve the problems above. The main idea of these schemes is to share the secret among the agents such that only predefined coalitions can come together and reveal the secret, while no other coalition can obtain any information about the secret. Thus, the keys used in the areas requiring vital secrecy like large-scale finance applications and commandcontrol mechanisms of nuclear systems, can be stored by using secret sharing schemes. Threshold cryptography deals with a particular type of secret sharing schemes. In threshold cryptography related secret sharing schemes, if the size of a coalition exceeds a bound t, it can reveal the key. And, smaller coalitions can reveal no information about the key. Actually, the first secret sharing scheme in the literature is the threshold scheme of Shamir where he considered the secret as the constant of a polynomial of degree t − 1, and distributed the points on the polynomial to the group of users. Thus, a coalition of size t can recover the polynomial and reveal the key but a smaller coalition can not. This scheme is widely accepted by the researchers and used in several applications. Shamir’s secret sharing scheme is not the only one in the literature. For example, almost concurrently, Blakley proposed another secret sharing scheme depending on planar geometry and Asmuth and Bloom proposed a scheme depending on the Chinese Remainder Theorem. Although these schemes satisfy the necessary and sufficient conditions for the security, they have not been considered for the applications requiring a secret sharing scheme. Secret sharing schemes constituted a building block in several other applications other than the ones mentioned above. These applications simply contain a standard problem in the literature, the function sharing problem. In a function sharing scheme, each user has its own secret as an input to a function and the scheme computes the outcome of the function without revealing the secrets. In the literature, encryption or signature functions of the public key algorithms like RSA, ElGamal and Paillier can be given as an example to the functions shared by using a secret sharing scheme. Even new generation applications like electronic voting require a function sharing scheme. As mentioned before, Shamir’s secret sharing scheme has attracted much of the attention in the literature and other schemes are not considered much. However, as this thesis shows, secret sharing schemes depending on the Chinese Remainder Theorem can be practically used in these applications. Since each application has different needs, Shamir’s secret sharing scheme is used in applications with several extensions. Basically, this thesis investigates how to adapt Chinese Remainder Theorem based secret sharing schemes to the applications in the literature. We first propose some modifications on the Asmuth-Bloom secret sharing scheme and then by using this modified scheme we designed provably secure function sharing schemes and security extensions.