Threshold Cryptography with Chinese Remainder Theorem a Dissertation Submitted to the Department of Computer Engineering and the Institute of Engineering and Science of Bi̇lkent University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

THRESHOLD CRYPTOGRAPHY WITH CHINESE REMAINDER THEOREM Kamer Kaya Ph.D. in Computer Engineering Supervisor: Asst. Prof. Dr. Ali Aydın Selçuk August, 2009 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 iv