## CryptoDB

### Yuriy Tarannikov

#### Publications

Year
Venue
Title
2014
EPRINT
2005
EPRINT
The plateaued functions have a big interest for the studying of bent functions and by the reason that many cryptographically important functions are plateaued. In this paper we study the possible values of the affine rank of spectrum support for plateaued functions. We consider for any positive integer $h$ plateaued functions with a spectrum support of cardinality $4^h$ (the cardinality must have such form), give the bounds on the affine rank for such functions and construct functions where the affine rank takes all integer values from $2h$ till $2^{h+1}-2$. We solve completely the problem for $h=2$, namely, we prove that the affine rank of any plateaued function with a spectrum support of cardinality $16$ is $4$, $5$ or $6$.
2001
ASIACRYPT
2001
FSE
2001
EPRINT
In this paper we consider matrices of special form introduced in [11] and used for the constructing of resilient functions with cryptographically optimal parameters. For such matrices we establish lower bound ${1\over\log_2(\sqrt{5}+1)}=0.5902...$ for the important ratio ${t\over t+k}$ of its parameters and point out that there exists a sequence of matrices for which the limit of ratio of its parameters is equal to lower bound. By means of these matrices we construct $m$-resilient $n$-variable functions with maximum possible nonlinearity $2^{n-1}-2^{m+1}$ for $m=0.5902...n+O(\log_2 n)$. This result supersedes the previous record.
2000
EPRINT
It is proved that the maximal possible nonlinearity of $n$-variable $m$-resilient Boolean function is $2^{n-1}-2^{m+1}$ for ${2n-7\over 3}\le m\le n-2$. This value can be achieved only for optimized functions (i.~e. functions with an algebraic degree $n-m-1$). For ${2n-7\over 3}\le m\le n-\log_2{n-2\over 3}-2$ it is suggested a method to construct an $n$-variable $m$-resilient function with maximal possible nonlinearity $2^{n-1}-2^{m+1}$ such that each variable presents in ANF of this function in some term of maximal possible length $n-m-1$. For $n\equiv 2\pmod 3$, $m={2n-7\over 3}$, it is given a scheme of hardware implementation for such function that demands approximately $2n$ gates EXOR and $(2/3)n$ gates AND.
2000
EPRINT
We use the recent results on the spectral structure of correlation immune and resilient Boolean functions for the investigations of high order correlation immune functions. At first, we give simple proofs of some theorems where only long proofs were known. Next, we introduce the matrix of nonzero Walsh coefficients and establish important properties of this matrix. We use these properties to prove the nonexistence of some high order correlation immune functions. Finally, we establish the order of magnitude for the number of (n-4)th order correlation immune functions of n variables.
2000
EPRINT
In this paper we develop a technique that allows to obtain new effective constructions of highly resilient Boolean functions with high nonlinearity. In particular, we prove that the upper bound $2^{n-1}-2^{m+1}$ on nonlinearity of m-resilient n-variable Boolean functions is achieved for $0.6n-1\le m\le n-2$.

#### Coauthors

Anton Botev (1)
Maria Fedorova (1)
Denis Kirienko (1)
Peter Korolev (1)