## On bent and semi-bent quadratic
Boolean functions

Pascale Charpin, Enes Pasalic and Cédric Tavernier

INRIA-projet Codes, Technical University of Denmark, Thales Communication

` Pascale.Charpin@inria.fr`

` E.Pasalic@mat.dtu.dk`

` cedric.tavernier@fr.thalesgroup.com`
Regular paper in IEEE Transactions on Information Theory.

To appear.

### Abstract

The maximum length sequences, also called m-sequences,
have received a lot of attention since the late
sixties.

In terms of LFSR synthesis they are usually generated
by certain power polynomials over finite field and in

addition
characterized by a low cross correlation and high nonlinearity.
We say that such sequence is generated by a

* semi-bent* function.

Some new families of such function, represented by
$f(x)=\sum_{i=1}^{\frac{n-1}{2}}c_iTr(x^{2^i+1})$,

*n* odd and
the *c_i* are binary, have recently been
introduced by Khoo, Gong and Stinson.
We first generalize their results

to even *n*.
We further investigate the conditions on the choice of *c_i*
for explicit definitions
of new infinite families

having three and four trace
terms. Also a class of nonpermutation polynomials whose
composition with a quadratic

function yields again a
quadratic semi-bent function is specified.
The treatment of semi-bent functions is then presented

in a much wider
framework. We show how bent and semi-bent functions are interlinked,
that is, the concatenation

of two suitably chosen semi-bent
functions will yield a bent function and vice versa. Finally this
approach is

generalized so that the construction of both bent and
semi-bent functions of any degree in certain range for any

*n >= 7* is presented, *n* being the number of input variables.
**Keywords : **
Boolean function, m-sequence, quadratic mapping,
semi-bent function, bent function,

nonlinearity, linear permutation.