Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.Comment: 14 pages, 2 table

On q-ary Bent and Plateaued Functions

We obtain the following results. For any prime $q$ the minimal Hamming distance between distinct regular $q$-ary bent functions of $2n$ variables is equal to $q^n$. The number of $q$-ary regular bent functions at the distance $q^n$ from the quadratic bent function $Q_n=x_1x_2+\dots+x_{2n-1}x_{2n}$ is equal to $q^n(q^{n-1}+1)\cdots(q+1)(q-1)$ for $q>2$. The Hamming distance between distinct binary $s$-plateaued functions of $n$ variables is not less than $2^{\frac{s+n-2}{2}}$ and the Hamming distance between distinctternary $s$-plateaued functions of $n$ variables is not less than $3^{\frac{s+n-1}{2}}$. These bounds are tight. For $q=3$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $q=2$ analogous result are well known but for large $q$ it seems impossible. Constructions and some properties of $q$-ary plateaued functions are discussed.Comment: 14 pages, the results are partialy reported on XV and XVI International Symposia "Problems of Redundancy in Information and Control Systems

Implementing Symmetric Cryptography Using Sequence of Semi-Bent Functions

Symmetric cryptography is a cornerstone of everyday digital security, where two parties must share a common key to communicate. The most common primitives in symmetric cryptography are stream ciphers and block ciphers that guarantee confidentiality of communications and hash functions for integrity. Thus, for securing our everyday life communication, it is necessary to be convinced by the security level provided by all the symmetric-key cryptographic primitives. The most important part of a stream cipher is the key stream generator, which provides the overall security for stream ciphers. Nonlinear Boolean functions were preferred for a long time to construct the key stream generator. In order to resist several known attacks, many requirements have been proposed on the Boolean functions. Attacks against the cryptosystems have forced deep research on Boolean function to allow us a more secure encryption. In this work we describe all main requirements for constructing of cryptographically significant Boolean functions. Moreover, we provide a construction of Boolean functions (semi-bent Boolean functions) which can be used in the construction of orthogonal variable spreading factor codes used in code division multiple access (CDMA) systems as well as in certain cryptographic applications