The Studies On Bent Function And Its Sequence

Boolean function is an important component of cryptographic algorithm. The relevant safety indicators which it has are the principal foundation on researching and analyzing the performance of cryptographic algorithm against cryptographic attacking. Bent function has maximum nonlinearity, so it has highest efficiency on againsting linear cryptanalysis and differential cryptanalysis. Riera and Parker puts forward the concept of Negabent-Hadamard transformation firstly when they studied more criterias for Bent function, then the concept of Negabent function was generated, not like Bent function, the number of its variable could not only be even. So the Negabent function will be the key and hot point after Bent function.This paper studies the following aspects from the properties and constructions of Bent function, Bent sequence, Negabent function and Bent-Negabent function which focus on the safety indicator of Boolean function.Firstly, the advantages and disadvantages of bent function’s properties is analyzed comprehensively, Then the main construction at present of Bent function was analyzed and summarized. A new construction of Bent sequences is given based on the permutation matrix, the linearity between the new Bent sequences and the original Bent sequences isn’t connected by this construction.Secondly, some properties of Nega-Hadamard transformation about Boolean function and some safety indicators of Nega-Hadamard function are studied, given and proved, a problem is pointed out when use a decision of Negabent function to construct Negabent function, at the same time a direct sum construction and dual construction is given.Finally, this paper improving the construction of Maiorana-McFarland Bent function, the construction of Maiorana-McFarland Bent-Negabent function is put forward based on the property of orthogonal matrix which keeps the weight of vector. And find the Negabent functions which is not belong to Maiorana-McFarland Bent functions. Furthermore, the disadvantages of construction for Bent-Negabent which was given by Parker is pointed out, it is been improved and extended in this paper, the condition which keeps the Bent-Negabent function invariance via the linear transformation constructed by this method is found at the moment.