J.-W. Liu’s scientific contributions

The determination of the algebraic representations of coordinates of finite field elements with the elements themselves as the variables, which is a new property of finite fields, is investigated. Based on the partition of equivalent classes, the resolving of a linear system of equations and the calculation of the dual basis of the standard basis, three methodologies are presented. With those results, we have successfully given an essential explanation to the simplicity of the algebraic representation of Rijndael S-box and provided a direct proof to the equivalence between any two coordinate functions of Rijndael S-box.

First, we design an approach to determining the corresponding trace function of a linear transformation over finite fields, on the basis of which we have proposed an effective methodology to compute the algebraic representations of coordinates of finite field elements in the form of trace functions. Second, by calculating the dual basis of the standard basis, we develop another method to find the algebraic representations of coordinates of finite field elements, also in the form of trace functions. The necessary data for both methods are only n elements over the corresponding fields with time complexity each of O(n3). This is a great improvement in contrast to the existing data and time complexity of O(pn-1) and O(np2n-2) respectively. Finally, based on the algebraic representations of coordinates which could be computed easily with our new approaches, we have given a direct proof of the equivalence between any two coordinate functions of Rijndael S-box and we have depicted this equivalence with only one matrix of order eight over GF(28). Compared to the existing ones, in which the equivalence is found by the search of the affine parameters and is illustrated with no less than 56 matrices of order eight over GF(2), our proof seems more straightforward and simpler.

The NESSIE project is a three-year project with the main objective to put forward a portfolio of strong cryptographic primitives of various types. We have focused on the three block ciphers selected most recently. We study their mechanisms, performance and design principles and compare the security against the well-known cryptanalysis. It is pointed out that MISTY1 and Camellia are secure against the differential and linear cryptanalysis and can also withdraw the interpolation attacks, slide attacks and related-key attacks. But they can both be described with a set of multivariate quadratic or linear equations and this may form a potential algebraic weakness. The security analysis of SHACAL-2 will likely lead to a new methodology of cryptanalysis of block ciphers. We believe that the NESSIE project will help to promote the standardization of our own cryptographic primitives.