A Study On The Central Map Based On The Boolean Algebraic In Multivariate Public-key Cryptosystems

The Public Key Cryptosystems are pivotal technology in network security fields, they have played a critical role in key management, authentication, encryption, and digital signature. The traditional public key ciphers suffer from a drawback that their speed is relatively low, especially, the presented of quantum algorithms would be a real-world threat to systems based on factoring or the discrete logarithm problem. However there has been fast and intensive development in Multivariate Public Key Cryptosystems(MPKCs for short)in the last two decades due to its high efficiency and have great potential for application in wireless networks.Both affine map and central map are the two components of the MPKCs, and central map is the crucial part that directly influence its security and performance. Therefore, studying the central map in MPKCs has certain theoretical and practical application value. This paper aims to conduct an in-depth study on a new central mapping.In this article, we present an overview of research actuality and developtment foreground of MPKCs nowadays in detail, and its basic composition and construction methods, particularly with the construction of central map. To select a central map is just to select the trapdoor fuction, UOV, STS, MIA and HFE are four basic functions of the trapdoor in MPKCs. The current central map has limitation in security owing to the algebra degree of polynomials is just two. However, we introduce a set of nonlinear boolean functions which algebra degree is at least three, and we suggest a new central map based on the trapdoor of Step-wise Triangular Systems, then the two components of the central map including large-scale boolean matrix and a set of the nonlinear boolean functions are designed respectively.The key to design the large-scale boolean matrix is to construct a "trap door" of this central map, the "trap door" is used to hide its internal structure.Using the thought of linear transformation, large-scale boolean matrix can be translate into small block matrix, so the generation of high level reversible matrix comes down to low level matrix, which could generate the matrix meet the requirements of cryptography. And to design the non-linear boolean functions must to meet conditions that there is only one solution for the central map. In this article, we construct a bijective structures in each layer of this central map, using the thought of bool-replacement to generate a set of eight variables balanced boolean function, combined with the boolean matrix, can esay to find the selution.Usually the MPKCs has a drawback of having a long key length, In this context, we introduce compound balanced boolean functions and a diffusion-confusion function inorder to compress the key length. Futhermore, we analyze the anti-attack capability of this new central map, the analysis results shows that it can resist all known attacks, which has better performance and enable its application in constructing a high power of MPKCs.Finally, this paper implements some algorithms in the process of the central map's generation and solution. Including fast inversion algorithm of boolean matrix, fast solution algorithm of the central map, as well as the algorithm of solving the boolean functions'algebra degree.The research result of this paper is valuable for the design of the central map as well as further research of MPKCs.