A unified construction of weightwise perfectly balanced Boolean functions

At Eurocrypt 2016, Méaux et al. presented FLIP, a new family of stream ciphers {that aimed to enhance the efficiency of homomorphic encryption frameworks. Motivated by FLIP, recent research has focused on the study of Boolean functions with good cryptographic properties when restricted to subsets of the space $\mathbb{F}_2^n$. If an $n$-variable Boolean function has the property of balancedness when restricted to each set of vectors with fixed Hamming weight between $1$ and $n-1$, it is a weightwise perfectly balanced (WPB) Boolean function. In the literature, a few algebraic constructions of WPB functions are known, in which there are some constructions that use iterative method based on functions with low degrees of 1, 2, or 4. In this paper, we generalize the iterative method and contribute a unified construction of WPB functions based on functions with algebraic degrees that can} be any power of 2. For any given positive integer $d$ not larger than $m$, we first provide a class of $2^m$-variable Boolean functions with a degree of $2^{d-1}$. Utilizing these functions, we then present a construction of $2^m$-variable WPB functions $g_{m;d}$. In particular, $g_{m;d}$ includes four former classes of WPB functions as special cases when $d=1,2,3,m$. When $d$ takes other integer values, $g_{m;d}$ has never appeared before. In addition, we prove the algebraic degree of the constructed WPB functions and compare the weightwise nonlinearity of WPB functions known so far in 8 and 16 variables

Engineering Dynamics and Life Sciences

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”