The Research Of Quaternary Codes And Their Applications In Cryptography

Error correcting code can be used to construct block cipher and stream cipher components, and to design provable public key cryptosystem depending on its computationally difficult problems. However, previous error correcting codes in the cryptographic applications are mainly the error correction codes over finite fields. In recent years, the error correcting code over rings becomes a hot spot. Especially in 1994, A. Hammons et.al found Kerdock code and Preparata code as binary nonlinear codes are some Z4 linear codes' binary images after Gray mapping, which shows that there is an important corresponding relation between binary nonlinear codes and Z4 linear codes. So how to use quaternary codes design and analysis cryptography schemes is a worthful subject.The following aspects are studied in this paper, and obtained results are listed:Firstly, using quaternary self-dual codes some unimodular lattices are constructed. The research of hard problems on lattices is a focus in the current cryptography schemes' design and analysis. In this thesis, using the quaternary linear codes to construct lattices, we analyze the condition when the quaternary linear codes meet and can construct corresponding lattices. We emphasized the study of all the inseparable self-dual codes' types for the length from 1 to 9 on the Z4 ring and verified construction the process that using quaternary self-dual codes constructed unimodular lattices. Finally, according to the process that using quaternary self-dual codes can construct unimodular lattices, we consider to turn the two lattices' isomorphism problem to the quaternary linear codes which constructed these lattices.Secondly, this thesis demonstrates that there is no nontrivial linear MDS over Z4 ring, and constructs some near-MDS codes over Z4. MDS codes have the best diffusion characteristics and are an important tool in designing block cipher diffusion layers. How to quickly find performance MDS codes in cryptography is very meaningful. By analyzing the existing conclusions about the MDS codes over general rings, we get that: if there is a linear MDS codes over Z4, this code must be free and then prove that there is no nontrivial linear MDS codes over Z4. We explore the construction of near-MDS codes over Z4. According to the construction of near-MDR codes over Z4 ring, we extend the concept of near-MDS codes from over filed to over Z4, and then summary the conditions which near-MDS codes' generator matrix should satisfy, and construct concrete examples of near-MDS codes which can be applied in cryptography.Thirdly, this thesis puts forward a method of constructing quaternary bent function. Bent function is an important tool when designing sequence cipher and block cipher and its implementation has great value in cryptography. We study the definitions of the Boolean bent function, the generalized Boolean bent function and quaternary bent function. According to the classic structure of the Boolean bent function, we construct two Boolean bent functions which have four variables. And then, based on the analysis of the bent characteristics among Boolean functions, generalized Boolean function and quaternary functions, this paper proposed a method of constructing quaternary bent function. Lastly, a concrete example is constructed and their balance property is analyzed.