Primal-dual distance bounds of linear codes with application to cryptography

Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower bound and an upper bound on $N(d,d^\perp)$. Further, for small values of $d$ and $d^\perp$, we determine $N(d,d^\perp)$ and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.Comment: 6 pages, using IEEEtran.cls. To appear in IEEE Trans. Inform. Theory, Sept. 2006. Two authors were added in the revised versio

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

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