Longjiang Qu

On the 2 m -variable symmetric Boolean functions with maximum algebraic immunity

The properties of the 2m-variable symmetric Boolean functions with maximum algebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.

Balanced rotation symmetric boolean functions with maximum algebraic immunity

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, even-variable-balanced RSBFs with maximum algebraic immunity (AI) are investigated. At first, we give an original construction of 2 m -variable-balanced RSBFs with maximum AI. Then we improve the construction to obtain more 2 m -variable-balanced RSBFs with maximum AI, and these new RSBFs have higher non-linearity than all previously obtained RSBFs. Further, we generalise our construction of 2 m -variable RSBFs to a new construction that can generate any even-variable RSBFs.

Construction of Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity

Rotation symmetric Boolean functions (RSBFs) which are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, we study the construction of RSBFs with maximum algebraic immunity. First, a new construction of RSBFs on odd number of variables with maximum possible Algebraic Immunity is given. Then by using the relationship between some flats and support of a n -variables Boolean function f , we prove that a construction of RSBFs on even number of variables has maximum possible Algebraic Immunity. Furthermore, we study the nonlinearity of functions by our construction.

New constructions of semi-bent functions in polynomial forms

Semi-bent functions are a kind of useful functions in cryptography. In this paper we give new constructions of n -variable quadratic semi-bent functions in polynomial forms for both odd and even n , and we also present many n -variable semi-bent functions with few trace terms when n is even.

Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity

Rotation symmetric Boolean functions (RSBFs) have been used as components of different cryptosystems. In this paper, we investigate n-variable (n even and n ⩾ 12) RSBFs to achieve maximum algebraic immunity (AI), and provide a construction of RSBFs with maximum AI and nonlinearity. These functions have higher nonlinearity than the previously known nonlinearity of RSBFs with maximum AI. We also prove that our construction provides high algebraic degree in some case.

Balanced 2p-variable rotation symmetric Boolean functions with maximum algebraic immunity

Abstract In this paper, we study the construction of Rotation Symmetric Boolean Functions (RSBFs) which achieve a maximum algebraic immunity (AI). For the first time, a construction of balanced 2 p -variable ( p is an odd prime) RSBFs with maximum AI was provided, and the nonlinearity of the constructed RSBFs is not less than 2 2 p − 1 − ( 2 p − 1 p ) + ( p − 2 ) ( p − 3 ) + 2 ; this nonlinearity result is significantly higher than the previously best known nonlinearity of RSBFs with maximum AI.

A note on vectorial bent functions

In this paper we give three methods on the constructions of vectorial bent functions from F2^n to F2^n^2, where n is a positive even integer. The first two kinds of functions are based on monomial bent functions. The third kind of functions are based on partial spreads bent functions, and those vectorial bent functions can achieve the largest algebraic degree.

Linear complexity of binary sequences with interleaved structure

In this study, the minimal polynomials and the linear complexity of interleaved binary sequences are investigated. Both the linear complexity and the minimal polynomials of low correlation zone sequences constructed by Zhou et al. are completely determined. Besides, an open problem proposed by Li and Tang is discussed. At last, a sufficient condition and a necessary condition are presented about when the linear complexity of the interleaved sequences constructed by Tang et al. attains the maximum.

Construction of highly nonlinear resilient S-boxes with given degree

We provide two new construction methods for nonlinear resilient S-boxes with given degree. The first method is based on the use of linear error correcting codes together with highly nonlinear S-boxes. Given a [u, m, t + 1] linear code where u = n−d−1, d > m, we show that it is possible to construct (n, m, t, d) resilient S-boxes which have currently best known nonlinearity. Our second construction provides highly nonlinear (n, m, t, d) resilient S-boxes which do not have linear structure, then an improved version of this construction is given.

Enumeration of balanced symmetric functions over GF(p)

Symmetric functions display some interesting properties since this class of functions are invariant under permutation of indices. In this paper, we prove that the construction and enumeration of the number of balanced symmetric functions over GF(p) are equivalent to solving an equation system and enumerating the solutions, as a result we obtain the exact number of n-variable balanced symmetric functions by searching the solutions of the equation system. When n and p become large, we give a lower bound on number of balanced symmetric functions over GF(p), and the lower bound provides best known result.