**ON DILLON'S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS**

*CLAUDE CARLET AND SIHEM MESNAGER*

**Abstract: **One of the classes of bent Boolean functions introduced by John Dillon in his thesis
is family H. While this class corresponds to a nice original construction of bent functions in
bivariate form, Dillon could exhibit in it only functions which already belonged to the well-
known Maiorana-McFarland class. We rst notice that H can be extended to a slightly larger
class that we denote by H. We observe that the bent functions constructed via Niho power
functions, which four examples are known, due to Dobbertin et al. and to Leander-Kholosha,
are the univariate form of the functions of class H. Their restrictions to the vector spaces
uF2n=2 , u 2 F?
2n, are linear. We also characterize the bent functions whose restrictions to the
uF2n=2 's are ane. We answer to the open question raised by Dobbertin et al. in JCT A 2006
on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well,
by explicitely calculating the dual of one of these functions. We observe that this Niho function
also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing
whether H (or H) is a subclass of the Maiorana-McFarland completed class. We then show that
the condition for a function in bivariate form to belong to class H is equivalent to the fact that
a polynomial directly related to its denition is an o-polynomial and we deduce eight new cases
of bent functions in H which are potentially new bent functions and most probably not ane
equivalent to Maiorana-McFarland functions.

**Category / Keywords: **Boolean function, Bent function, Maximum nonlinearity, Walsh-Hadamard trans- form, Partial Spread class, Niho function, O-polynomial.

**Date: **received 7 Nov 2010, last revised 26 Nov 2010

**Contact author: **mesnager at math jussieu fr

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**Version: **20101126:202021 (All versions of this report)

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