Yupu Hu | Searcharxiv

arXiv2012

Secondary Constructions of Bent Functions and Highly Nonlinear Resilient Functions

In this paper, we first present a new secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, we apply the construction using as initial functions some specific bent functions and then provide several concrete constructions of bent functions. The second part of the paper is devoted to the constructions of resilient functions. We give a generalization of the indirect sum construction for constructing resilient functions with high nonlinearity. In addition, we modify the generalized construction to ensure a high nonlinearity of the constructed function.

cs.CR↗

arXiv2012

The Termination of Algorithms for Computing Gröbner Bases

The F5 algorithm is generally believed as one of the fastest algorithms for computing Gröbner bases. However, its termination problem is still unclear. Recently, an algorithm GVW and its variant GVWHS have been proposed, and their efficiency are comparable to the F5 algorithm. In the paper, we clarify the concept of an admissible module order. For the first time, the connection between the reducible and rewritable check is discussed here. We show that the top-reduced S-Gröbner basis must be finite if the admissible monomial order and the admissible module order are compatible. This paper presents a complete proof of the termination and correctness of the GVWHS algorithm. What is more, it can be seen that the GVWHS is in fact an F5-like algorithm. Different from the GVWHS algorithm, the F5B algorithm may generate redundant sig-polynomials. Taking into account this situation, we prove the termination and correctness of the F5B algorithm. And we notice that the original F5 algorithm slightly differs from the F5B algorithm in the insertion strategy on which the F5-rewritten criterion is based. Exploring the potential ordering of sig-polynomials computed by the original F5 algorithm, we propose an F5GEN algorithm with a generalized insertion strategy, and prove the termination and correctness of it. Therefore, we have a positive answer to the long standing problem of proving the termination of the original F5 algorithm.

math.AC↗