Secret Sharing Scheme And Signature Schemes Over The Ring Of Eisenstein Integers

With the rapid development of computer technology and network technology, more and more government services, e-commerce, and personal affairs need to implement through the computer network. How to ensure information security has become the concern of the people.In the related fields of secret communication, people mainly use the encryption method for confidential information. The key of confidentiality is the core of the encryption, so the safety of the key affects the safety of communication system directly. In public key cryptosystems,secret sharing ideas are an important part. It is not only very useful in public key cryptographic key management, but also has very extensive applications in the data security and network management. Digital signature is a method of electronically stored information signature. It is one of the core technologies of information security problem. It is widely used in military, communications, electronic commerce, etc. So the study of secret sharing schemes and digital signature schemes are of practical significance.Whether it is a secret sharing scheme or a digital signature scheme, their safety are based on the hard-to-solve problem. It is the factorization problem or a discrete logarithm problem,etc. The more difficult is to solve these problems, the more secure is the corresponding cryptosystems. Since factorization of elements over the ring of Eisenstein integers is much more complex than of elements over the integer, it is of theoretical and practical significance to construct secret sharing scheme, signature scheme and public key cryptosystems over the ring of Eisenstein integers.This thesis first proposes a new verifiable secret sharing scheme over the ring of Eisenstein integers. Verification analyses, security analyses and performance analyses are given for the proposed scheme. The scheme is based on the Shamir threshold scheme, the RSA cryptosystem, the difficulty of factoring elements in the ring of Eisenstein integers and intractability of the Discrete Logarithm. Then, the extensions of the El Gamal public key cryptosystem and digital signature schemes with appendix and message recovery to the ring of Eisenstein integers are given and proved. The security of the proposed extensions is based on the difficulty to solve the Discrete Logarithm problem over the ring of Eisenstein integers.