## CryptoDB

### Sugata Gangopadhyay

#### Publications

**Year**

**Venue**

**Title**

2014

EPRINT

2010

EPRINT

On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions
Abstract

In this paper we consider cubic bent functions obtained by Leander and McGuire
(J. Comb. Th. Series A, 116 (2009) 960-970) which are
concatenations of quadratic Gold functions.
A lower bound of second-order nonlinearities of these
functions is obtained. This bound is compared with the lower
bounds of second-order nonlinearities obtained for functions
belonging to some other classes of functions which are recently
studied.

2010

EPRINT

On second-order nonlinearities of some $\mathcal{D}_0$ type bent functions
Abstract

In this paper we study the lower bounds of second-order nonlinearities
of bent functions constructed by modifying certain cubic Maiorana-McFarland type bent functions.

2009

EPRINT

On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions
Abstract

The $r$-th order nonlinearity of a Boolean function is an important
cryptographic criterion in analyzing the security of stream as well
as block ciphers. It is also important in coding theory as it is
related to the covering radius of the Reed-Muller code $\mathcal{R}(r, n)$.
In this paper we deduce the lower bounds of the second order nonlinearity
of the two classes of Boolean functions of the form
\begin{enumerate}
\item
$f_{\lambda}(x) = Tr_1^n(\lambda x^{d})$ with $d=2^{2r}+2^{r}+1$
and $\lambda \in \mathbb{F}_{2^{n}}$ where $n = 6r$.
\item
$f(x,y)=Tr_1^t(xy^{2^{i}+1})$
where $x,y \in \mathbb{F}_{2^{t}}, n = 2t, n \ge 6$ and
$i$ is an integer such that $1\le i < t$, $\gcd(2^t-1, 2^i+1) = 1$.
\end{enumerate}
For some $\lambda$, the first class gives bent functions whereas
Boolean functions of the second class are all bent, i.e., they achieve
optimum first order nonlinearity.

2008

EPRINT

On Kasami Bent Functions
Abstract

It is proved that no non-quadratic Kasami bent is affine equivalent to Maiorana-MacFarland type bent functions.

2007

EPRINT

On bent functions with zero second derivatives
Abstract

It is proved that a bent function has zero second derivative with respect to $a$, $b$, $a \ne b$, if and only if it is affine on all the flats parallel to the two dimensional subspace $V = \langle a, b \rangle$.

2006

EPRINT

On construction of non-normal Boolean functions
Abstract

Given two non-weakly $k$-normal Boolean functions on $n$ variables a method is proposed to construct a non-weakly $(k+1)$-normal Boolean function on $(n+2)$ variables.

2006

EPRINT

On a new invariant of Boolean functions
Abstract

A new invariant of the set of $n$-variable Boolean functions with respect to the action of $AGL(n,2)$ is studied. Application of this invariant to prove affine nonequivalence of two Boolean functions is outlined. The value of this invariant is computed for $PS_{ap}$ type bent functions.

2004

EPRINT

Crosscorrelation Spectra of Dillon and Patterson-Wiedemann type Boolean Functions
Abstract

In this paper we study the additive crosscorrelation spectra between
two Boolean functions whose supports are union of certain cosets.
These functions on even number of input variables have been introduced by Dillon and we refer to them as Dillon type functions. Our general result shows that the crosscorrelation spectra between any two Dillon type functions are at most $5$-valued. As a consequence we find that the crosscorrelation spectra between two Dillon type bent functions on $n$-variables are at most $3$-valued with maximum possible absolute value at the nonzero points being $\leq 2^{\frac{n}{2}+1}$. Moreover, in the same line, the autocorrelation spectra of Dillon type bent functions at different decimations is studied. Further we demonstrate that these results can be used to
show the existence of a class of polynomials for which the absolute
value of the Weil sum has a sharper upper bound than the Weil bound.
Patterson and Wiedemann extended the idea of Dillon for
functions on odd number of variables. We study the crosscorrelation
spectra between two such functions and then use the results for the
calculating the autocorrelation spectra too.

2003

EPRINT

Direct Sum of Non Normal and Normal Bent Functions Always Produces Non Normal Bent Functions
Abstract

Prof. Claude Carlet has pointed out an error in Theorem 1 of the
paper. The error could not be recovered for the time being.
Thus the statement presented in the title of the paper
is not proved.

2003

EPRINT

Patterson-Wiedemann Construction Revisited
Abstract

In 1983, Patterson and Wiedemann constructed Boolean functions on
$n = 15$ input variables having nonlinearity strictly greater than
$2^{n-1} - 2^{\frac{n-1}{2}}$. Construction of Boolean functions on
odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for $n = 15, 21$. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide
proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all $GF(2)$-linear transformations of $GF(2^{ab})$ which acts on $PG(2,2^a)$.

#### Coauthors

- Ruchi Gode (1)
- P. H. Keskar (1)
- Subhamoy Maitra (3)
- Bimal Mandal (1)
- Enes Pasalic (1)
- Sumanta Sarkar (1)
- Deepmala Sharma (3)
- Brajesh Kumar Singh (1)
- Pantelimon Stanica (2)
- Ruchi Telang (1)