hyperrefSuppressing link with empty target

11institutetext:  The Oskar Klein Centre, Department of Astronomy, Stockholm
University, AlbaNova, SE-10691 Stockholm, Sweden 22institutetext: Department
of Particle Physics and Astrophysics, Weizmann Institute of Science, 234 Herzl
St, 7610001 Rehovot, Israel 33institutetext: Center for Interdisciplinary
Exploration and Research in Astrophysics (CIERA), Northwestern University,
1800 Sherman Ave., Evanston, IL 60201, USA 44institutetext: The Oskar Klein
Centre, Department of Physics, Stockholm University, Albanova University
Center, 106 91 Stockholm, Sweden 55institutetext: Division of Physics,
Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA 66institutetext: MIT-Kavli Institute for Astrophysics and Space
Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA 77institutetext:
Astrophysics Research Institute, Liverpool John Moores University, IC2,
Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
88institutetext: Caltech Optical Observatories, California Institute of
Technology, Pasadena, CA 91125, USA 99institutetext: School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
1010institutetext: IPAC, California Institute of Technology, 1200 E.
California Blvd, Pasadena, CA 91125, USA 1111institutetext: Post Observatory,
Lexington, MA 02421, USA

# Spectroscopic observations of progenitor activity 100 days before a Type Ibn
supernova

S. J. Brennan  11 J. Sollerman  11 I. Irani  22 S. Schulze  3344 P. Chen  22
K. K. Das  55 K. De  66 C. Fransson  11 A. Gal-Yam  22 A. Gkini  11 K. R.
Hinds  77 R. Lunnan  11 D. Perley  77 YJ. Qin  55 R. Stein  55 J. Wise  77 L.
Yan  88 E. A. Zimmerman  22 S. Anand  55 R. J. Bruch  22 R. Dekany  88 A. J.
Drake  55 C. Fremling  88 Brian Healy  99 V. Karambelkar  55 M. M. Kasliwal
55 M. Kong  55 F. J. Masci  1010 R. S. Post  1111 J. Purdum  88 A. Wold  1010

Obtaining spectroscopic observations of the progenitors of core-collapse
supernovae is often unfeasible, due to an inherent lack of knowledge as to
what stars will go supernova and when they will explode. In this Letter, we
present photometric and spectroscopic observations of the progenitor activity
of SN 2023fyq in the preceding 150 days before the He-rich progenitor exploded
as a Type Ibn supernova. The progenitor of SN 2023fyq shows an exponential
rise in flux prior to core-collapse. Complex He I emission line features are
observed, with a P-Cygni like profile, as well as an evolving broad base with
velocities on the order of 10,000 km s-1, possibly due to electron scattering.
The luminosity and evolution of SN 2023fyq is consistent with a faint Type
Ibn, reaching a peak $r$-band magnitude of $-18.1$ mag, although there is some
uncertainty in the distance to the host, NGC 4388, located in the Virgo
cluster. We present additional evidence of asymmetric He-rich material being
present prior to the explosion of SN 2023fyq, as well as after, suggesting
this material has survived the ejecta-CSM interaction. Broad [O I] and the Ca
II triplet lines are observed at late phases, confirming that SN 2023fyq was a
genuine supernova, rather than a non-terminal interacting transient. SN
2023fyq provides insight into the final moments of a massive star’s life,
highlighting that the progenitor is likely highly unstable before core-
collapse.

###### Key Words.:

Supernovae: individual: SN 2023fyq – Supernovae: individual: ZTF22abzzvln –
Stars: massive – Stars: winds, outflows

##  1 Introduction

Massive stars ($\gtrsim$ 8 – 10 M⊙) will eventually end their lives as core-
collapse supernovae (CCSNe; Woosley & Weaver, 1995; Heger et al., 2003; Janka,
2012; Crowther, 2012). With the increasing capabilities of photometric
transient surveys (Bellm, 2014; Chambers et al., 2016; Tonry et al., 2018) a
growing sample of supernova (SN) progenitors have been observed experiencing
outbursts in the weeks – years prior to their ultimate core-collapse (Foley et
al., 2007; Pastorello et al., 2007; Mattila et al., 2008; Ofek et al., 2013;
Fraser et al., 2013; Margutti et al., 2014; Ofek et al., 2014; Strotjohann et
al., 2021; Smith et al., 2022; Hiramatsu et al., 2023). These precursor111In
the literature, precursor and outbursts are used interchangeably in the
context of pre-SN activity, for this Letter we denote these events as the
former. events are difficult to explain, and are a relatively new area of
stellar astrophysics (Quataert & Shiode, 2012; Leung et al., 2021; Strotjohann
et al., 2021), however are often invoked to explain complex circumstellar
material (CSM) shortly before collapse (Matsumoto & Metzger, 2022; Hiramatsu
et al., 2023). Precursor activity is likely accompanied by violent mass
ejections by a currently unclear mechanism. Potential scenarios include binary
interaction (Yoon et al., 2010; Smith, 2011; Kashi et al., 2013; Yoon et al.,
2017; Zapartas et al., 2021; Tsuna et al., 2024), outbursts from Luminous Blue
Variables (LBVs; Humphreys & Davidson, 1994; Smith et al., 2010; Kilpatrick et
al., 2018; Smith et al., 2022), and instabilities towards the end of a massive
stars life (Woosley et al., 2007; Smith & Arnett, 2014; Müller et al., 2016;
Fuller & Ro, 2018; Leung et al., 2021; Wu & Fuller, 2022).

Figure 1: Composite $giy$ color image of the host galaxy of SN 2023fyq from
the Pan-STARRS survey. The transient environment is given in the top right
inset. An $r$-band image of SN 2023fyq from the NOT/ALFOSC from 2023-08-16 is
given in the top left inset. SN 2023fyq exploded $\sim$10′′ from the nucleus
of its host, NGC 4388, in a star-forming region.

This Letter presents spectral and photometric observations of the progenitor
of a Type Ibn SN several months before core-collapse, as well as SN 2023fyq
itself. The precursor activity of SN 2023fyq 222Also known as ZTF22abzzvln,
ATLAS23rwh, and PS23fnw. was discovered by the Zwicky Transient Facility (ZTF;
Graham et al., 2019; Bellm et al., 2019; Masci et al., 2019; Dekany et al.,
2020) at R.A. = 12:25:45.874, Dec. = $+$12:39:48.87 (J2000) on 2023-04-17 at
$g=19.51$ mag (AB) (De, 2023). Spectroscopic observations of SN 2023fyq were
obtained as part of the Census of the Local Universe (CLU) survey (De et al.,
2020) as it was identified as a low-luminosity transient in a nearby galaxy,
consistent with the CLU criteria. SN 2023fyq was later classified by the
Nordic optical telescope Unbiased Transient Survey 2 (NUTS2) as a Type Ib-pec
SN (Valerin et al., 2023) on 2023-07-25, around 5 days pre-peak. SN 2023fyq
occurred in the active spiral galaxy NGC 4388 (Damas-Segovia et al., 2016) in
the Virgo Cluster, as presented in Fig. 1. For consistency with the
literature, we assume a distance of 17.2 Mpc to NGC 4388 measured from Cepheid
Variables (Lianou et al., 2019; Böhringer et al., 1997), giving a distance
modulus ($\mu$) of 31.2 mag, giving SN 2023fyq a peak $r$-band magnitude of
$-18.1$ mag. However the distance is uncertain and may be as high as 25 Mpc
(e.g. Ekholm et al., 2000), meaning SN 2023fyq may be $\sim 1.5$ mag more
luminous. We correct for foreground Milky Way (MW) extinction using ${\rm
R_{V}}=3.1$, ${\rm E(B-V)_{MW}}$ = 0.029 mag (Schlafly & Finkbeiner, 2011),
and the extinction law given by Cardelli et al. (1989). We adopt a redshift of
$z=0.0084$ and the rest-frame phase is reported with respect to the $r$-band
peak magnitude on 2023-07-30 (MJD = 60155.12).

##  2 Observations

###  2.1 Photometric observations

Science-ready images taken as part of the ZTF survey were obtained in $gri$
from the NASA/IPAC Infrared Science Archive
(IPAC333https://irsa.ipac.caltech.edu/applications/ztf/) service. Template
subtracted photometry was performed using the AutoPhOT pipeline (Brennan &
Fraser, 2022). Photometry from the ATLAS forced-photometry
server444https://fallingstar-data.com/forcedphot/ (Tonry et al., 2018; Smith
et al., 2020; Shingles et al., 2021) were obtained in $c$ and $o$ bands.
Several epochs of optical photometry were obtained with the Alhambra Faint
Object Spectrograph and Camera (ALFOSC) on the 2.56 m Nordic Optical Telescope
(NOT) in $gri$, from the Liverpool Telescope (LT) with the optical imaging
component of Infrared-Optical imager: Optical (IO:O) in $griz$, and the
Spectral Energy Distribution Machine (SEDM; Blagorodnova et al., 2018; Rigault
et al., 2019) on the Palomar 60-inch (P60) telescope in $gri$. We also
obtained $gri$ images from a 32-inch Ritchey-Chretien telescope (RC32) at Post
Observatory in Mayhill, New Mexico. UV photometry were acquired using the
Ultra-violet Optical Telescope (UVOT) onboard the Neil Gehrels Swift
Observatory (Gehrels et al., 2004; Roming et al., 2005). We reduced the images
using the Swift HEAsoft555https://heasarc.gsfc.nasa.gov/docs/software/heasoft/
V. 6.26.1. toolset, as detailed in Irani et al. (2023). A UV template image
was constructed using archival observations prior to MJD 59100 and after the
SN declined to the host level. We then removed the local host-galaxy
contribution by subtracting the SN site flux from the fluxes of the individual
epochs.

###  2.2 Spectroscopic observations

Follow-up spectra were obtained using NOT/ALFOSC, the Low Resolution Imaging
Spectrometer (LRIS; Oke et al., 1995) on the 10 m Keck telescope, the Double
Spectrograph (DBSP) on the Palomar 200-inch telescope (P200), the Kast
spectrograph at the Lick Observatory, the Gemini Multi-Object Spectrographs
(GMOS) mounted on the Gemini North 8 m telescope (Hook et al., 2004), and
P60/SEDM (Blagorodnova et al., 2018). Spectra were reduced in a standard
manner, using LPipe (Perley, 2019), DBSP_DRP (Mandigo-Stoba et al., 2022) and
PypeIt (Prochaska et al., 2020a, b, 2020), for Keck/LRIS, P200/DBSP, and
NOT/ALFOSC, pysedm (Rigault et al., 2019; Kim et al., 2022) for SEDM spectra,
the UCSC spectral
pipeline666https://github.com/msiebert1/UCSC_spectral_pipeline for our
Kast/Lick spectrum, and the DRAGONS (Labrie et al., 2019) for our GEMINI/GMOS
spectrum. Observations using the P60, P200,and Keck telescopes were
coordinated using the FRITZ data platform (van der Walt et al., 2019; Coughlin
et al., 2023).

NGC 4388 was observed with the Multi Unit Spectroscopic Explorer (MUSE) at the
8.2 m ESO Very Large Telescope on 12 March 2022 (ID. 108.229J, PI. Venturi).
Observations were reduced using the MUSE pipeline v2.8.7 and the ESO workflow
engine ESOReflex (Freudling et al., 2013; Weilbacher et al., 2020). We
extracted all spectra with the MUSE Python Data Analysis Framework (MPDAF)
version 3.6 (Bacon et al., 2016). The spectrum of the SN site was extracted
with a circular aperture. The aperture radius of $0.5^{\prime\prime}$
translates to a physical scale of 89.5 pc at the projected distance of SN
2023fyq, comparable to the largest giant molecular clouds in the Milky Way.

##  3 Results and Discussion

###  3.1 Photometric evolution of SN 2023fyq and its progenitor activity

Figure 2: Multi band lightcurve of SN 2023fyq. Each band has been offset for
clarity. Vertical grey bars indicate the epochs where spectral observations
were obtained. Absolute magnitudes are given on the right hand Y-axis using
the adopted distance of 17.2 Mpc and are not corrected for any MW or host
extinction.

Figure 2 gives the photometric evolution of SN 2023fyq, highlighting the
peculiar pre-SN evolution, as well as the SN explosion itself. Figure 3
provides the bolometric evolution constructed from the host-subtracted
photometry presented in Fig. 2, and calibrated using the extinction and
distance given in Sect. 1. Due to poor multi-band coverage, we perform
extrapolation using Superbol777https://github.com/mnicholl/superbol (Nicholl,
2018) during the precursor rise.

Figure 3: The blackbody luminosity of SN 2023fyq and its progenitor. Hollow
circles represent blackbody fits using interpolation by Superbol with host-
subtracted $gcroi$ bands. Filled markers are blackbody fits using epochs that
contain $ugri$ bands. The star markers represent blackbody fits to the spectra
given in Fig. 4. Similarly the blackbody radius and temperature are given in
the middle and lower panels respectively.

The blackbody luminosity (${\rm L_{BB}}$) of the progenitor is well fitted by
an accelerating rise (assumed to be exponential), followed by a power-law rise
after the progenitor has likely exploded, with the explosion time (${\rm
t_{exp}}$) taken to be $\sim$13 days before maximum brightness (Fig. 3, upper
panel). Assuming a distance of 17.2 Mpc (see discussion in Sect. 1), we
measure the radiated energy of the pre-SN evolution as ${\rm 1.5\times
10^{47}\leavevmode\nobreak\ erg}$. The radiated energy from the precursor is
consistent with predicted precursor activity from binary interaction (Tsuna et
al., 2024), and similar to precursor activity in other SNe (Strotjohann et
al., 2021). From SN 2023fyq, we measure a peak luminosity of ${\rm 5.9\times
10^{42}\leavevmode\nobreak\ erg\leavevmode\nobreak\ s^{-1}}$ and a total
radiated energy of ${\rm 1.2\times 10^{49}\leavevmode\nobreak\ erg}$. Due to
our sparse photometry around the explosion epoch, we are uncertain whether SN
2023fyq follows the evolution depicted by the analytical fits in Fig. 3, or
rather a multi-component rise (e.g. SN 2021qqp; Hiramatsu et al., 2023). We
adopt a rise time for SN 2023fyq as $\sim$13 days, which is consistent with
what is seen in other Type Ibn SNe (Hosseinzadeh et al., 2017) as shown in
Fig. 3.

The blackbody radius (${\rm R_{BB}}$) of the progenitor remains roughly
constant at $\sim 1\times 10^{14}$ cm up until shortly before the explosion of
the progenitor, although due to the poor photometric coverage around this
time, the errors on these values may be underestimated. This radius
($\sim$1500 R⊙) is unlikely to represent a photosphere of the progenitor star,
but perhaps signifies an extended/inflated envelope, due to the final stages
of a binary merger event (Ivanova et al., 2013; Irani et al., 2023), or a
confined CSM, as this value is consistent with the shock breakout radius
reported for SN 2023ixf (Zimmerman et al., 2023). After a significant amount
of radiation passes through this material, the blackbody radius starts
increasing with $\sim$2000 km s-1 seen in the $-37$d and $-17$d P200/DBSP
spectra in Fig. 3.

Assuming an efficient conversion of kinetic energy to radiation
($\epsilon\approx 1$), we get ${\rm M_{CSM}\approx\frac{2E_{rad}}{\epsilon
v^{2}}}$, giving $0.02\epsilon^{-1}$ M⊙ for the precursor, assuming a velocity
of 1700 km s-1 (taken from the observed P-Cygni minimum). This material would
be very optically thick at $10^{14}$ cm, which is consistent with the
thermalised spectra. For SN 2023fyq, a CSM breakout with $\sim$0.01 M⊙ would
take about 1 – 2 days, with a peak luminosity of a few $10^{42}$ erg s-1,
(e.g. SN 2023ixf; Zimmerman et al., 2023), followed by a cooling phase,
consistent with Fig. 3. The diffusion time (taken to be ${\rm
t_{d}\approx\sqrt{\frac{\kappa M_{CSM}}{vc}}}$; Khatami & Kasen, 2019) gives
$\sim$13 days consistent with the adopted rise time.

We measure the progenitor to have ${\rm log(T_{eff}/K)}\approx 3.8\pm 0.1$ and
${\rm log(L/L_{\odot})}\approx 6.8\pm 0.2$ from host-subtracted photometry.
These values are reminiscent of the Giant Eruption for Eta Carina, which
ejected $\sim$10 M⊙ of material, but did not unbind the star (Smith, 2011;
Prieto et al., 2014). Without detailed late-time stellar evolutionary
modelling, the mechanism powering the pre-SN evolution is unclear.

###  3.2 Spectroscopic evolution

Figure 4: Spectral observations of SN 2023fyq and its progenitor. Each
spectrum has been normalised and offset for clarity, as well as corrected for
redshift and MW extinction We note that the $+15$ d appearance of SN 2023fyq
matches well to the Type Ibn SN 2010al (Pastorello et al., 2013). We do not
correct the above spectra for host contamination or extinction, although the
Balmer (H$\alpha$/H$\beta$) decrement points to substantial host attenuation
of ${\rm E(B-V)_{host}\approx 0.4\pm 0.1}$ (Osterbrock, 1989). We mark regions
of telluric contamination using the $\oplus$ symbol and grey vertical bands.

Spectroscopic observations for the progenitor activity and SN 2023fyq are
presented in Fig. 4. Observations of CCSN progenitors are rare, with the only
previous examples, to our knowledge, being SN 2009ip (Smith et al., 2010;
Foley et al., 2011; Pastorello et al., 2013) and SN 2015bh (Elias-Rosa et al.,
2016; Thöne et al., 2017). In both cases, the spectra are reminiscent of
interacting transients, such as Type IIn SNe (albeit less luminous) or LBVs,
with strong emission lines originating from photo-ionized hydrogen in the
circumstellar environment and an overall blue continuum. In this scenario, any
information about the progenitor was likely obscured by the strong CSM
interaction.

Figure 5: Evolution of the He I $\lambda$5876 emission profile. Pre-SN spectra
show complex P-Cygni like profiles. Soon after the SN has occurred, an
additional absorption component is seen in the $-$6d Lick/KAST spectrum at
around $-$5000 km s-1. At late times, the P-Cygni absorption is no longer
observed, but rather a double-peaked emission profile consisting of host
emission and redshifted He I emission at similar velocities as seen in the
$-$93d Keck/LRIS spectrum. Spectra marked with an asterisk have been re-binned
for clarity, with the raw spectrum given in grey. We manually mask out
telluric lines and artefacts from the sky subtractions.

In agreement with stellar evolutionary theory, the progenitor of the Type Ibn
SN 2023fyq display features expected for He-rich stars (Crowther, 2007; Yoon
et al., 2010). Figure 5 shows the evolution of He I $\lambda$5876, during the
pre- and post- SN epochs, as well as the post-peak appearance. Similar to the
H$\alpha$ profile seen for SN 2009ip 888Although SN 2009ip is a H-rich Type
IIn SN and SN 2023fyq a H-poor Type Ibn SN, we make qualitative comparisons
between the appearance of their progenitor star/environments. in 2009
(Pastorello et al., 2013), SN 2023fyq shows a complex P-Cygni like profile for
He I $\lambda$5876 (and similarly for other He I lines) with an absorption
centered at $-1700$ km s-1 and extending out to $-2500$ km s-1. Perhaps
similarly to H$\alpha$ in late time spectra of SN 2009ip (Fraser et al.,
2015), the He I in SN 2023fyq shows a redshifted emission peak, centered at
+1400 km s-1, which is present during the slow pre-SN rise, as well as post
explosion.

A spectrum of the explosion site from VLT/MUSE taken $\sim$2 years prior to SN
2023fyq is given in Fig. 4. We investigate if any potential flux from the
progenitor activity is present in this spectrum by comparing it to a similar
spectrum extracted 2.4′′ from the explosion site. No differences are observed
at wavelengths corresponding to He I emission lines meaning any progenitor
flux is negligible, Due to the sodium laser guide star, the wavelength range
covering He I $\lambda$5876 is masked, preventing any investigation of
potential host contamination of He I $\lambda$5876 for SN 2023fyq. We suspect
this redshifted component is originating from He I $\lambda$5876 as it is seen
in several He I lines in the pre-SN spectra, as well as a similar emission
component at similar velocities seen in the post-SN spectra. This would mean
that the asymmetric material responsible for this emission was not destroyed
in the SN explosion. SN ejecta interacting with asymmetric CSM has been used
to explain irregular emission line profiles (e.g. Leloudas et al., 2015;
Andrews & Smith, 2018; Pursiainen et al., 2022; Hosseinzadeh et al., 2022;
Smith et al., 2023) and bumpy light curves (Nyholm et al., 2017; Woosley,
2018; Wang et al., 2022), and SN 2023fyq provides the first clear
spectroscopic evidence of asymmetric structure prior to core-collapse.

A broad component is observed for He I $\lambda$5876 profile in the $-37$d and
$-17$d spectra, with wings extending to $\pm 13,000$ km s-1 and a full width
half maximum (FWHM) around 14,000 km s-1. Such velocities are normally
associated with SN ejecta, although have also been observed in non-terminal
eruptions (Pastorello et al., 2013; Smith et al., 2018). To investigate if
this broad component evolves, we measure the flux in a small wavelength bin
around 5800 Å in the continuum-subtracted flux-calibrated spectra. An order of
magnitude increase in flux from $-103$ to $-10$ days is observed. This
confirms that the broad component is rapidly evolving in strength prior to the
explosion of SN 2023fyq rather than not being detectable in earlier spectra
due to low signal-to-noise.

While these features may represent fast moving material, an alternative
explanation could be electron scattering. Some mechanisms cause the progenitor
to be surrounded by a dense CSM (such as acoustic waves or a merger; Smith,
2011; Yoon et al., 2017; Fuller & Ro, 2018; Chevalier, 2012; Fransson et al.,
2022; Tsuna et al., 2024), and may lead to shock dissipation and emission of
radiation in the optically thick CSM. This may explain the general rise of the
light curve by diffusion. As photons reach optically thin material, the broad
component is observed in the Pre-SN spectra. This also explains the roughly
constant radius and the slowly rising effective temperature.

The post-SN He I $\lambda$5876 appears double peaked with similar shaped
profiles also seen in the ${{Ca\textsc{II}}\leavevmode\nobreak\ \lambda\lambda
8498,8542,8662}$ near-infrared triplet. Double peaked emission lines have been
observed in Type Ibn SNe (e.g. SN 2018bcc, see Fig. 5; Karamehmetoglu et al.,
2021), as well as Type IIn SNe (Brennan et al., 2022a; Hiramatsu et al.,
2023), and are attributed to asymmetric SN ejecta or CSM or both (Andrews et
al., 2019; Pursiainen et al., 2022). While SN 2023fyq does not show a
symmetric double peaked profile, the offset He I emission does suggest an
asymmetric ejecta/CSM configuration. Given the similarity between the pre- and
post- SN appearance, these asymmetries are likely caused by pre-existing CSM
structure.

SN 2023fyq resembles a Type Ibn SN as shown in Fig. 3 and 4. Given the complex
pre-SN appearance of the He I $\lambda$5876, we expect the SN ejecta to
collide with the pre-existing CSM. It is possible that the narrow He I
component arises mainly from the underlying H II region rather than from the
progenitor activity, although such strong emission is mainly expected for
highly star-forming regions (Thone et al., 2015), whereas the environment of
SN 2023fyq seems more mundane.

A long standing question for SN 2009ip-like transients is whether the
transient is indeed a SN or rather a “SN impostor” (Van Dyk et al., 2000;
Mauerhan et al., 2013; Fraser et al., 2013; Brennan et al., 2022b; Smith et
al., 2022). As shown in Figs. 4 and 5, SN 2023fyq shows broad emission from
${\rm[{O\textsc{I}}]\leavevmode\nobreak\ \lambda\lambda 6300,6364}$ at nebular
phases, showing that SN 2023fyq is a genuine CCSN (Jerkstrand et al., 2014;
Kuncarayakti et al., 2015), whose progenitor underwent some form of
instabilities shortly before core-collapse.

##  4 Conclusion

In this Letter, we present an unprecedented dataset of progenitor activity of
a Type Ibn SN almost 150 days before core-collapse occurs. The observed
luminosity increases exponentially during this time, while the object
maintained an almost constant photospheric radius. Pre-SN spectra reveal a
complex evolving He I profile. These observations of SN 2023fyq and the final
moments of the progenitor, highlight that the progenitors to CCSNe can undergo
some extreme instabilities shortly before their final demise.

Progenitor analysis typically occurs after the star has been destroyed, by
searching through archival images, and measuring the photometric properties of
the assumed progenitor. Although this area of transient astronomy is in its
infancy, the repercussions of detecting precursor activity are immense,
highlighting that the progenitor is not in a equilibrium state, and may not be
represented well by standard stellar evolutionary models. SN 2023fyq and
similar transients, highlight that the pre-SN appearance of the progenitor is
non-trivial, and without careful consideration, may produce misleading results
in supernova progenitor studies.

## Data availability

The spectroscopic and photometric data underlying this article are available
in the Weizmann Interactive Supernova Data Repository
(WISeREP999https://wiserep.weizmann.ac.il/; Yaron & Gal-Yam, 2012).

## Acknowledgements

SJB would like to thank M. Fraser and A. Guinness for their insights into late
time stellar evolution and precursor events. SJB and RL acknowledges their
support by the European Research Council (ERC) under the European Union’s
Horizon Europe research and innovation programme (grant agreement No. 10104229
- TransPIre). S. Schulze is partially supported by LBNL Subcontract NO.
7707915 and by the G.R.E.A.T. research environment, funded by Vetenskapsrådet,
the Swedish Research Council, project number 2016-06012. Based on observations
obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at
the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF
is supported by the National Science Foundation under Grant No. AST-2034437
and a collaboration including Caltech, IPAC, the Weizmann Institute of
Science, the Oskar Klein Center at Stockholm University, the University of
Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO
Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity
College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of
Warwick, Ruhr University Bochum and Northwestern University. Operations are
conducted by COO, IPAC, and UW. SED Machine is based upon work supported by
the National Science Foundation under Grant No. 1106171. This work was
supported by the GROWTH project funded by the National Science Foundation
under Grant No 1545949. The Oskar Klein Centre is funded by the Swedish
Research Council. The data presented here were obtained in part with ALFOSC,
which is provided by the instituto de Astrofisica de Andalucia (IAA) under a
joint agreement with the University of Copenhagen and NOT. The ZTF forced-
photometry service was funded under the Heising-Simons Foundation grant
#12540303 (PI: Graham). The Gordon and Betty Moore Foundation, through both
the Data-Driven Investigator Program and a dedicated grant, provided critical
funding for SkyPortal. The Liverpool Telescope is operated on the island of La
Palma by Liverpool John Moores University in the Spanish Observatorio del
Roque de los Muchachos of the Instituto de Astrofisica de Canarias with
financial support from the UK Science and Technology Facilities Council. Based
on observations obtained at the international Gemini Observatory, a program of
NSF’s NOIRLab, which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with the National
Science Foundation on behalf of the Gemini Observatory partnership: the
National Science Foundation (United States), National Research Council
(Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio
de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência,
Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space
Science Institute (Republic of Korea). The Gemini observations were obtained
through program GN-2023A-Q-122 and processed using DRAGONS (Data Reduction for
Astronomy from Gemini Observatory North and South). This work was enabled by
observations made from the Gemini North telescope, located within the Maunakea
Science Reserve and adjacent to the summit of Maunakea. We are grateful for
the privilege of observing the Universe from a place that is unique in both
its astronomical quality and its cultural significance.

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∎ 11institutetext: Juan Carlos Ku-Cauich 22institutetext: Computer Science,
CINVESTAV-IPN, Mexico City, Mexico, 22email<EMAIL_ADDRESS>33institutetext: Javier Arturo Díaz-Vargas 44institutetext: Facultad de
Matemáticas, UADY, Mérida Yucatán, Mexico, 44email<EMAIL_ADDRESS>55institutetext: Sara Mandujano-Velazquez 66institutetext: ESFM, IPN, Mexico
City, Mexico, 66email<EMAIL_ADDRESS>
# Bent functions construction using extended Maiorana-McFarland’s class

Juan Carlos Ku-Cauich  Javier Arturo Díaz-Vargas  Sara Mandujano-Velazquez The
authors acknowledge the support of Mexican Conacyt

(Received: / Accepted: date)

###### Abstract

In a particular case, we consider the extended Maiorana-McFarland class to
obtain bent functions with balance in its images, restricted to vectors with
even Hamming weight. Additionally, we see that all bent functions are balanced
when we restrict to vectors of even Hamming weight or to vectors with odd
Hamming weight. Given the necessary tools, we provide a simpler algorithm to
obtain new bent functions using Maiorana-McFarland.

Kewords: Bent functions, Maiorana-McFarland, Balancedness

##  1 Introduction

The functions $f:{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$ are called
boolean functions. They are important in cryptography and code theory because
they have important properties such as non-linearity, balance, low auto-
correlation and high algebraic immunity. The search space of these functions
is very large, $2^{2^{n}}$, and different methods exist to search for these
functions: random search, algebraic and heuristic methods, see for example
CarletEvolutionary , GeneticAlgorithm .

We are interested in the non-linearity of a function, defined as the distance
between a boolean function and the set of affine functions. Functions with
maximum non-linearity are called bent functions; Rothaus introduced the name
in 1976 Rothaus . These functions have been classified and constructed, as
examples we have the Maiorana-McFarland class DillonMaio  and Rothaus Rothaus
.

In this work, we use a particular case of the extended Maiorana-McFarland
class CarletMaioranaExtendido , i.e. $f$ bent given by
$f:{\mathbb{F}}_{2}^{s+1}\rightarrow{\mathbb{F}}_{2}$, $x\mapsto
x\cdot\phi(y)\oplus g_{e}(y),$ where
$\phi(y):{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2}$ is such that
$\phi^{-1}(a)$ is an affine space of dimension $s-1$, and
$g_{e}(y):{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2}$,
${g_{e}}_{|\phi^{-1}(a)}$ is a bent function.

First, $\phi$ and $g_{e}$ are defined and, after a series of results, we find
the direct relationship between a bent function and a bent function with a
higher dimensional domain. Finally, we provide a pair of algorithms that
simplify this relationship.

Additionally, we show that any bent function is balanced when restricted to an
even Hamming weight or to an odd Hamming weight in its domain. This result is
crucial, since it allows us to obtain a bent function
$g_{e}:{\mathcal{C}}_{0}\rightarrow{\mathbb{F}}_{2}$ from a bent function
$g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$, where
${\mathcal{C}}_{0}:=\phi^{-1}(0)$ is a linear code with a vector of even
Hamming weight and ${\mathcal{C}}_{1}:=\phi^{-1}(1)$ the affine space of odd
Hamming weight.

##  2 Background

Definitions and results about boolean functions, particularly bent functions,
are recalled in this section. These can be found, for example, in MacWilliams
, TokarevaBent , BFunctionsRepresentationCarlet .

###### Definition 1

A function $f:{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$ is called a
boolean function. ${\mathcal{B}}_{n}$ is the set of all boolean functions with
domain ${\mathbb{F}}_{2}^{n}.$

All boolean functions $f\in{\mathcal{B}}_{n}$ have an algebraic normal form
(ANF):

$f(x)={\bigoplus}_{u\in{\mathbb{F}}_{2}^{n}}a_{u}x^{u},$

$a_{u}\in{\mathbb{F}}_{2},$ $x^{u}=x_{1}^{u_{1}}\cdots x_{n}^{u_{n}},$
$x=(x_{1}\ldots,x_{n}),u=(u_{1},\ldots,u_{n}).$

###### Example 1

The boolean function $f(x)\in{\mathcal{B}}_{3},$ $f(x_{1},x_{2},x_{3})=1\oplus
x_{1}x_{2}\oplus x_{1}x_{2}x_{3}$ is in its ANF.

###### Theorem 2.1

Let $f\in{\mathcal{B}}_{n}$. Then,

$f(x)={\bigoplus}_{u\in{\mathbb{F}}_{2}^{n}}a_{u}x^{u},$

$a_{u}={\oplus}_{x\leq u}f(x)$, $x\leq u\Leftrightarrow x_{i}\leq u_{i},$
$x=(x_{1},\ldots,x_{n})$, $u=(u_{1},\ldots,u_{n}).$

###### Definition 2

The set of all affine boolean functions with domain ${\mathbb{F}}_{2}^{n}$,
denoted ${\mathcal{A}}_{n}$, is defined as

${\mathcal{A}}_{n}:=\\{a\cdot x\oplus
a_{0}~{}|~{}a,x\in{\mathbb{F}}_{2}^{n},~{}a_{0}\in{\mathbb{F}}_{2}\\},$

where $\cdot$ is the dot product.

Note that the number of affine functions is $2^{n+1}$ and the number of linear
functions is $2^{n}$.

###### Definition 3

The non-linearity of a boolean function $f\in{\mathcal{B}}_{n}$ is defined as
the Hamming distance between $f$ and the affine functions:

$Nl(f):={\rm{\mbox{ m\'{i}n }}}_{g\in{\mathcal{A}}_{n}}d_{H}(f,g).$

We define the following function to characterize the non-linearity:

###### Definition 4

The Walsh-Hadamard Transform of a boolean function, $f$ , is defined as

$\widehat{{\mathcal{W}}}_{f}(a)=\sum_{x\in{\mathcal{F}}_{2}^{n}}(-1)^{f(x)\oplus
a\cdot x},~{}~{}a\in{\mathbb{F}}_{2}^{n}.$

###### Theorem 2.2

The non-linearity of the boolean function $f\in{\mathcal{B}}_{n}$ is
characterized as

$Nl(f)=2^{n-1}-\frac{1}{2}{\rm\mbox{max}}_{a\in{\mathbb{F}}_{2}^{n}}|\widehat{{\mathcal{W}}}_{f}(a)|.$

The boolean functions with maximum non-linearity are called bent functions and
their non-linearity is $2^{n-1}-2^{n/2-1}.$

##  3 Bent functions balancedness, restricted to even or odd Hamming weight

We will need bent functions over an affine space to find new bent functions
when using the Extended Maiorana-McFarland class CarletMaioranaExtendido .
Following this idea, given a bent function with the traditional definition, we
want to find a bent function over an affine space with the same dimension.

For that purpose, the main characteristic that we need is: for all bent
functions $f\in{\mathcal{B}}_{n}$, we have that
${f}_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}$ is balanced or
${f}_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}$ is balanced.

In the following, we prove this claim. For this, we first give the particular
cases $n=2$ and $n=4.$

###### Example 2

In the case ${\mathcal{B}}_{2},$ all the bent functions satisfy this property
because these must have three images $0$ and one image $1$, or three images
$1$ and one image $0$.

$f(x_{1},x_{2})=x_{1}x_{2},$

$\begin{array}[]{||l|l|l||}\hline\cr\hline\cr x_{1}&x_{2}&ev(f)\\\ \hline\cr
0&0&0\\\ \hline\cr 1&1&1\\\ \hline\cr 1&0&0\\\ \hline\cr 0&1&0\\\
\hline\cr\hline\cr\end{array}.$

###### Remark 1

  1. 1.

Let $g:{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$ be a bent function
such that ${g}_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}$ is balanced.

    1. (a)

If $\widehat{{\mathcal{W}}}_{g}(a)=2^{\frac{n}{2}}$, then

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}^{-1}(0)|=2^{n-2}$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}^{-1}(1)|=2^{n-2}$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}^{-1}(0)|=2^{n-2}+2^{\frac{n-2}{2}}.$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}^{-1}(1)|=2^{n-2}-2^{\frac{n-2}{2}}.$

    2. (b)

If $\widehat{{\mathcal{W}}}_{g}(a)=-2^{\frac{n}{2}}$, then

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}^{-1}(0)|=2^{n-2}$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}^{-1}(1)|=2^{n-2}$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}^{-1}(0)|=2^{n-2}-2^{\frac{n-2}{2}}.$

$|({g})_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}^{-1}(1)|=2^{n-2}+2^{\frac{n-2}{2}}.$

  2. 2.

Similar observations if ${g}_{|\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}odd\\}}$
is balanced.

###### Remark 2

Let ${\mathcal{A}}:=\\{x\in{\mathbb{F}}_{2}^{n}~{}|~{}w_{H}(x)\mbox{ even}\\}$
and ${\mathcal{B}}:=\\{x\in{\mathbb{F}}_{2}^{n}~{}|~{}w_{H}(x)\mbox{ odd}\\}.$
We can write,

${\mathbb{F}}_{2}^{n}=\left[{\begin{array}[]{c}{\mathcal{A}}\\\
{\mathcal{B}}\\\
\end{array}}\right]=\left[{\begin{array}[]{c}{{\mathcal{A}}^{\prime}}~{}~{}\bar{0}\\\
{{\mathcal{B}}^{\prime}}~{}~{}\bar{1}\\\
{{\mathcal{A}}^{\prime}}~{}~{}\bar{1}\\\
{{\mathcal{B}}^{\prime}}~{}~{}\bar{0}\\\
\end{array}}\right]=\left[{\begin{array}[]{c}{\mathbb{F}}_{2}^{n-1}|\begin{array}[]{c}\bar{0}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{n-2}-entries\\\
\bar{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{n-2}-entries\\\ \end{array}\\\
{\mathbb{F}}_{2}^{n-1}|\begin{array}[]{c}\bar{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{n-2}-entries\\\
\bar{0}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{n-2}-entries\\\ \end{array}\\\
\end{array}}\right]=\left[{\begin{array}[]{c}I\\\ II\\\ III\\\ 1V\\\
\end{array}}\right]$

Note that ${\mathcal{A}}$ is a MDS linear code of dimension $n-1$ and
${{\mathcal{B}}}$ is an affine space. Also,
${{\mathcal{A}}^{\prime}}=\\{x\in{\mathbb{F}}_{2}^{n-1}~{}|~{}w_{H}(x)\mbox{
even}\\}$ and
${{\mathcal{B}}^{\prime}}=\\{x\in{\mathbb{F}}_{2}^{n-1}~{}|~{}w_{H}(x)\mbox{
odd}\\}$.

The following result is the particular case $n=4.$ We use the notation of the
Observation 2.

###### Theorem 3.1

Every bent function $f:{\mathbb{F}}_{2}^{4}\rightarrow{\mathbb{F}}_{2}$ is
such that $f_{|{\mathcal{A}}}$ is balanced or $f_{|{\mathcal{B}}}$ is
balanced.

###### Proof

Let $f$ be a bent function and $l_{a}$ an affine function. Observe that,
$w_{H}(f\oplus l_{a})=6$ or $w_{H}(f\oplus l_{a})=10.$ In particular, the
linear functions $l_{0}:=0$ and $l_{\bar{1}}(x):=x_{1}\oplus x_{2}\oplus
x_{3}\oplus x_{4}$ satisfy the above observation. Additionally, notice that
$l_{\bar{1}}({\mathcal{A}})=\\{0\\}$ and $l_{\bar{1}}({\mathcal{B}})=\\{1\\}.$

Since $f$ is a bent function:

Case 1. If ${f}_{|{\mathcal{A}}}$ has zero images $1$, then
${f}_{|{\mathcal{B}}}$ has six images $1$. Hence, $(f\oplus
l_{\bar{1}})({\mathcal{A}})$ has zero images $1$ and $(f\oplus
l_{\bar{1}})({\mathcal{B}})$ has four images $1$. Therefore, $f\oplus
l_{\bar{1}}$ is not a bent function; consequently, $f$ is not a bent function.

Case 2. If ${f}_{|{\mathcal{A}}}$ has two images $1$, then
${f}_{|{\mathcal{B}}}$ has four or eight images $1$. If ${f}_{|{\mathcal{B}}}$
has four images $1$, that means ${f}_{|{\mathcal{B}}}$ is balanced. Hence, the
theorem is proven. If ${f}_{|{\mathcal{B}}}$ has eight images $1$, $(f\oplus
l_{\bar{1}})({\mathcal{A}})$ has two images $1$ and $(f\oplus
l_{\bar{1}})({\mathcal{B}})$ has zero images $1$. Therefore, $f\oplus
l_{\bar{1}}$ is not a bent function; consequently, $f$ is not a bent function.

Case 3. If ${f}_{|A}$ has four images $1$, that means ${f}_{|{\mathcal{A}}}$
is balanced. Hence, the theorem is proven.

Case 4. If ${f}_{|A}$ has six images $1,$ ${f}_{|{\mathcal{A}}}$ has two
images $0$, then ${f}_{|{\mathcal{B}}}$ has four or eight images $0$. If
${f}_{|{\mathcal{B}}}$ has four images $0$, that means ${f}_{|{\mathcal{B}}}$
is balanced. Hence, the theorem is proven. If ${f}_{|{\mathcal{B}}}$ has eight
images $0$, $(f\oplus l_{\bar{1}})({\mathcal{A}})$ has two images $0$ and
$(f\oplus l_{\bar{1}})({\mathcal{B}})$ has zero images $0$. Therefore,
$f\oplus l_{\bar{1}}$ is not a bent function; consequently, $f$ is not a bent
function.

Case 5. If ${f}_{|A}$ has eight images $1$, ${f}_{|{\mathcal{A}}}$ has zero
images $0$, then ${f}_{|{\mathcal{B}}}$ has six images $0$. Hence, $(f\oplus
l_{\bar{1}})({\mathcal{A}})$ has zero images $0$ and $(f\oplus
l_{\bar{1}})({\mathcal{B}})$ has four images $0$. Therefore, $f\oplus
l_{\bar{1}}$ is not a bent function; consequently, $f$ is not a bent function.

In all the sub-cases where $f_{|{\mathcal{A}}}$ is not balanced and
$f_{|{\mathcal{B}}}$ is not balanced, we obtain a contradiction. Indeed, if
$f$ is a bent function, it must satisfy, $f_{|{\mathcal{A}}}$ is balanced or
$f_{|{\mathcal{B}}}$ is balanced.

∎

We note the symmetry of the cases $1$ and $2$ with cases $5$ and $4$,
respectively, interchanging the number of images $1$ by number of images $0$.
This will be very important to the general case.

The following result is a general case, $n$ even. We use the notation of the
Observation 2.

###### Theorem 3.2

Every bent function $f:{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$,
$n\geq 6$, is such that $f_{|{\mathcal{A}}}$ is balanced or
$f_{|{\mathcal{B}}}$ is balanced.

###### Proof

Let $f$ be a bent function and $l_{a}$ an affine function. Observe that,
$w_{H}(f\oplus l_{a})=2^{n-1}-2^{\frac{n-2}{2}}$ or $w_{H}(f\oplus
l_{a})=2^{n-1}+2^{\frac{n-2}{2}}$. In particular, the linear functions
$l_{0}(x):=0$ and $l_{\bar{1}}(x):=x_{1}\oplus\dots\oplus x_{n}$ satisfy the
observation. Additionally, notice that $l_{\bar{1}}({\mathcal{A}})=\\{0\\}$
and $l_{\bar{1}}({\mathcal{B}})=\\{1\\}.$

Case 1. If ${f}_{|{\mathcal{A}}}$ has $c$ images $1$, $0\leq c\leq
2^{\frac{n-2}{2}}$.

Case 1a. ${f}_{|{\mathcal{B}}}$ has $2^{n-1}-(2^{\frac{n-2}{2}}+c)$ images
$1$. Then, $(f\oplus l_{\bar{1}})({\mathcal{A}})$ has $c$ images $1$ and
$(f\oplus l_{\bar{1}})({\mathcal{B}})$ has $2^{\frac{n-2}{2}}+c$ images $1$.

Case 1a1 $(f\oplus l_{\bar{1}})$ has
$c+(2^{\frac{n-2}{2}}+c)=2^{n-1}-2^{\frac{n-2}{2}}$ images $1$. Therefore,
$c=2^{n-2}-2^{\frac{n-2}{2}},$ and ${f}_{|{\mathcal{B}}}$ is balanced. But,
remember that, $n\geq 6,$ then $2^{\frac{n-2}{2}}<2^{n-2}-2^{\frac{n-2}{2}}.$

Case 1a2 $(f\oplus l_{\bar{1}})$ has
$c+(2^{\frac{n-2}{2}}+c)=2^{n-1}+2^{\frac{n-2}{2}}$ images $1$. Therefore,
$c=2^{n-2},$ and ${f}_{|{\mathcal{A}}}$ is balanced. But, remember that,
$n\geq 6,$ then $2^{\frac{n-2}{2}}<2^{n-2}.$

Case 1b ${f}_{|{\mathcal{B}}}$ has $2^{n-1}+(2^{\frac{n-2}{2}}-c)$ images $1$.
Hence, $c=2^{\frac{n-2}{2}}$. Then, $(f\oplus l_{\bar{1}})({\mathcal{A}})$ has
$2^{\frac{n-2}{2}}$ images $1$ and $(f\oplus l_{\bar{1}})({\mathcal{B}})$ has
zero images $1$. Therefore, $(f\oplus l_{\bar{1}})$ is not a bent function
when $n\geq 6$; consequently, $f$ is not a bent function.

Case 2. If ${f}_{|{\mathcal{A}}}$ has $c+2^{\frac{n-2}{2}}$ images $1$,
$0<c\leq 2^{n-2}-2^{\frac{n-2}{2}}$.

Case 2a. ${f}_{|{\mathcal{B}}}$ has $2^{n-2}+(2^{n-2}-22^{\frac{n-2}{2}}-c)$
images $1$. $(f\oplus l_{\bar{1}})({\mathcal{A}})$ has $c+2^{\frac{n-2}{2}}$
images $1$ and $(f\oplus l_{\bar{1}})({\mathcal{B}})$ has
$22^{\frac{n-2}{2}}+c$ images $1$.

Case 2a1 $(f\oplus l_{\bar{1}})$ has $(a)$
$c+2^{\frac{n-2}{2}}+22^{\frac{n-2}{2}}+c=2^{n-1}-2^{\frac{n-2}{2}}$ images
$1$. Therefore, $c=2^{n-2}-2^{\frac{n}{2}},$ then, ${f}_{|{\mathcal{B}}}$ has
$2^{n-2}$ images $1$. Hence, ${f}_{|{\mathcal{B}}}$ is balanced.

Case 2a2 $(f\oplus l_{\bar{1}})$ has
$c+2^{\frac{n-2}{2}}+22^{\frac{n-2}{2}}+c=2^{n-1}+2^{\frac{n-2}{2}}$ images
$1$. Therefore, $c=2^{n-2}-2^{\frac{n-2}{2}},$ then, ${f}_{|{\mathcal{A}}}$
has $2^{n-2}$ images $1$. Hence, ${f}_{|{\mathcal{A}}}$ is balanced.

Case 2b ${f}_{|{\mathcal{B}}}$ has $2^{n-2}+(2^{n-2}-c)$ images $1$. Hence,
$(f\oplus l_{\bar{1}})({\mathcal{A}})$ has $c+2^{\frac{n-2}{2}}$ images $1$
and $(f\oplus l_{\bar{1}})({\mathcal{B}})$ has $c$ images $1$.

Case 2b1 $(f\oplus l_{\bar{1}})$ has
$c+(c+2^{\frac{n-2}{2}})=2^{n-1}-2^{\frac{n-2}{2}}$ images $1$. Therefore,
$c=2^{n-2}-2^{\frac{n-2}{2}}.$ Hence, ${f}_{|{\mathcal{A}}}$ is balanced.

Case 2b2 $(f\oplus l_{\bar{1}})$ has
$c+(c+2^{\frac{n-2}{2}})=2^{n-1}+2^{\frac{n-2}{2}}$ images $1$. Therefore,
$c=2^{n-2}.$ But, $0<c\leq 2^{n-2}-2^{\frac{n-2}{2}}$. Thus, we don’t consider
this case.

When ${f}_{|{\mathcal{A}}}$ has more than $2^{n-2}$ images $1$, the proof is
similar to the previous cases, but we use the number of images $0$ of
${f}_{|{\mathcal{A}}}$ in place of number of images $1$ of
${f}_{|{\mathcal{A}}}$. Also, we use the fact that, $w_{H}(\bar{1}\oplus
f)=2^{n-1}-2^{\frac{n-2}{2}}$ or $w_{H}(\bar{1}\oplus
f)=2^{n-1}+2^{\frac{n-2}{2}}$. That means, the number of zeros of $f$ is
$2^{n-1}-2^{\frac{n-2}{2}}$ or $2^{n-1}+2^{\frac{n-2}{2}}$.

In all the cases where $f_{|{\mathcal{A}}}$ is not balanced and
$f_{|{\mathcal{B}}}$ is not balanced, we obtain a contradiction. Indeed, if
$f$ is a bent function, it must satisfy, $f_{|{\mathcal{A}}}$ is balanced or
$f_{|{\mathcal{B}}}$ is balanced.

∎

##  4 Construction of a particular family from the extended Maiorana-McFarland
class

We extend the definition of bent functions over ${\mathbb{F}}_{2}^{n}$ to an
affine subspace included in ${\mathbb{F}}_{2}^{n}$ as suggested in the
extended Maiorana–McFarland’s Proposition 1 CarletMaioranaExtendido .

###### Definition 5

A function $f:{\mathcal{C}}\rightarrow{\mathbb{F}}_{2}$,
${\mathcal{C}}\subset{\mathbb{F}}_{2}^{n}$ affine space, $m\leq n$,
$\dim{\mathcal{C}}=m$, is bent if ${Nl}(f):=d_{H}(f,{\mathcal{A}}_{n})$ is a
maximum.

The following two results are easily obtained, similarly to the traditional
bent function proofs.

###### Theorem 4.1

Let a function $f:{\mathcal{C}}\rightarrow{\mathbb{F}}_{2}$. Then
${Nl}(f)=2^{m-1}-\frac{1}{2}\max_{a\in{\mathbb{F}}_{2}^{n}}|\widehat{{\mathcal{W}}}_{f}(a)|,$

where
$\widehat{{\mathcal{W}}}_{f}(a):=\sum_{x\in{\mathcal{C}}}(-1)^{f(x)\oplus
a\cdot x}.$

∎

Observe that all $a\in{\mathbb{F}}_{2}^{n}$ are considered.

###### Theorem 4.2

If $f:{\mathcal{C}}\subset{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$,
$\dim{\mathcal{C}}=m,$ is a bent function, then
$\widehat{{\mathcal{W}}}_{f}(a)=\pm 2^{m/2}.$

∎

The following theorem corresponds to a class of bent functions: the extended
Maiorana–McFarland class.

###### Theorem 4.3

CarletMaioranaExtendido  Let the function
$\phi(y):{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2}^{r}$ such that for
all $a\in{\mathbb{F}}_{2}^{r},$ $\phi^{-1}(a)$ is an affine space of dimension
$s-r$. Also, let a function
$g_{e}(y):{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2}$, where
${g_{e}}_{|\phi^{-1}(a)}$ is a bent function. Then, the function
$f:{\mathbb{F}}_{2}^{r+s}\rightarrow{\mathbb{F}}_{2}$, $(x,y)\mapsto
x\cdot\phi(y)\oplus g_{e}(y)$, $x\in{\mathbb{F}}_{2}^{r}$, is a bent function.

Here in after, according to Theorem 4.3, particular case $r=1,$ we denote
$\phi(y):{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2},$
$g_{e}:{\mathbb{F}}_{2}^{s}\rightarrow{\mathbb{F}}_{2},$ and define

$g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ a bent function,

${\mathcal{C}}_{0}=\phi^{-1}(0):=\\{x\in{\mathbb{F}}_{2}^{s}~{}|~{}w_{H}(x)\mbox{
even}\\}$ and
${\mathcal{C}}_{1}=\phi^{-1}(1):=\\{x\in{\mathbb{F}}_{2}^{s}~{}|~{}w_{H}(x)\mbox{
odd}\\},$

$g_{e_{0}}:{\mathcal{C}}_{0}\rightarrow{\mathbb{F}}_{2},$ defined
$g_{e_{0}}(x|x_{s}):=g(x)$, $x\in{\mathbb{F}}_{2}^{s-1},$
$x_{s}\in{\mathbb{F}}_{2}$,

$g_{e_{1}}:{\mathcal{C}}_{1}\rightarrow{\mathbb{F}}_{2},$ defined
$g_{e_{1}}(x|x_{s}):=g(x)$, $x\in{\mathbb{F}}_{2}^{s-1},$
$x_{s}\in{\mathbb{F}}_{2}$,

${g_{e}}_{|{\mathcal{C}}_{0}}:=g_{e_{0}}$ and
${g_{e}}_{|{\mathcal{C}}_{1}}:=g_{e_{1}}.$

###### Remark 3

  1. 1.

We can see that, ${\mathcal{C}}_{0}=\phi^{-1}(0)$ is a linear code of
dimension $s-1.$

  2. 2.

$b\oplus{\mathcal{C}}_{0}={\mathcal{C}}_{1},$ for any
$b\in{\mathbb{F}}_{2}^{s}$ with odd Hamming weight.

  3. 3.

${\mathcal{C}}_{0}$ is an MDS linear code.

  4. 4.

In ${\mathcal{C}}_{0},$ if $x\in{\mathbb{F}}_{2}^{s-1}$ has an even Hamming
weight, then $x_{s}$ is $0$ and, if $x\in{\mathbb{F}}_{2}^{s-1}$ has and odd
Hamming weight, then $x\in{\mathbb{F}}_{2}$ is $1$.

  5. 5.

In ${\mathcal{C}}_{1},$ if $x\in{\mathbb{F}}_{2}^{s-1}$ has an even Hamming
weight, then $x_{s}$ is $1$ and, if $x\in{\mathbb{F}}_{2}^{s-1}$ has an odd
Hamming weight, then $x_{s}$ is $0$.

In the proof of the following theorem, it is essential that if $g$ is a bent
function, then $g_{|\\{x|w_{H}(x)~{}even\\}}$ is balanced or
$g_{|\\{x|w_{H}(x)~{}odd\\}}$ is balanced.

Using the above notation.

###### Theorem 4.4

Let $g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ be a bent function.
Then, $g_{e_{0}}:{\mathcal{C}}_{0}\rightarrow{\mathbb{F}}_{2}$ is a bent
function and $g_{e_{1}}:{\mathcal{C}}_{1}\rightarrow{\mathbb{F}}_{2}$ is a
bent function.

###### Proof

We will rely on the fact that, if $g$ is a bent function, then
$g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}even\\}}$ is balanced or
$g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}odd\\}}$ is balanced.

Let $b\in{\mathbb{F}}_{2}^{s}$ and
$\bar{x}=(x_{1},\ldots,x_{s-1},x_{s})\in{\mathcal{C}}_{0}.$

$\displaystyle\widehat{{\mathcal{W}}}_{g_{e_{0}}}(b)$ $\displaystyle=$
$\displaystyle\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},x_{s})+(x_{1}\ldots\cdot
x_{s-1},x_{s})\cdot b}$ $\displaystyle=$
$\displaystyle\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},x_{s})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}+x_{s}b_{s}}.$

If $b_{s}=0,$

$\widehat{{\mathcal{W}}}_{g_{e_{0}}}(b)=\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g(x_{1},\ldots,x_{s-1})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}=\widehat{{\mathcal{W}}}_{g}(b_{1},\ldots,b_{s-1}).$

If $b_{s}=1$

$\displaystyle\widehat{{\mathcal{W}}}_{g_{e_{0}}}(b)$ $\displaystyle=$
$\displaystyle\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},x_{s})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}+x_{s}}$
$\displaystyle=$
$\displaystyle\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},0)+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}+\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},1)+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}+1}$
$\displaystyle=$
$\displaystyle\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},0)+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}+(-1)\sum_{\bar{x}\in{\mathcal{C}}_{0}}(-1)^{g_{e_{0}}(x_{1},\ldots,x_{s-1},1)+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}$
$\displaystyle=$
$\displaystyle\sum_{\\{x\in{\mathbb{F}}_{2}^{s-1}~{}|~{}w_{H}(x)~{}even\\}}(-1)^{g(x_{1},\ldots,x_{s-1})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}$
$\displaystyle+$
$\displaystyle(-1)\sum_{\\{x\in{\mathbb{F}}_{2}^{s-1}~{}|~{}w_{H}(x)~{}odd\\}}(-1)^{g(x_{1},\ldots,x_{s-1})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}$

The last equality, by Observation 3, Item 4.

If $g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}even\\}}$ is balanced,

$\widehat{{\mathcal{W}}}_{g_{e_{0}}}(b)=(-1)\sum_{\\{x\in{\mathbb{F}}_{2}^{s-1}~{}|~{}w_{H}(x)~{}odd\\}}(-1)^{g(x_{1},\ldots,x_{s-1})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}=\widehat{{\mathcal{W}}}_{g}((b_{1},\ldots,b_{s-1})).$

If $g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}odd\\}}$ is balanced,

$\widehat{{\mathcal{W}}}_{g_{e_{0}}}(b)=\sum_{\\{x\in{\mathbb{F}}_{2}^{s-1}~{}|~{}w_{H}(x)~{}even\\}}(-1)^{g(x_{1},\ldots,x_{s-1})+x_{1}b_{1}+\cdots+x_{s-1}b_{s-1}}=\widehat{{\mathcal{W}}}_{g}((b_{1},\ldots,b_{s-1})).$

In both cases, since $g$ is a bent function, then $g_{e_{0}}$ is a bent
function.

Proceeding similarly, $g_{e_{1}}:{\mathcal{C}}_{1}\rightarrow{\mathbb{F}}_{2}$
is a bent function.

∎

###### Corollary 1

Let $g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ be a bent function.
For all $a\in{\mathbb{F}}_{2}^{s-1}:$

  1. 1.

$\widehat{{\mathcal{W}}}_{g}(a)=\widehat{{\mathcal{W}}}_{g_{e_{0}}}(a,0)=\widehat{{\mathcal{W}}}_{g_{e_{1}}}(a,0).$

  2. 2.

If $g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}even\\}}$ is balanced,

$\widehat{{\mathcal{W}}}_{g}(a)=2^{\frac{s-1}{2}}~{}\Leftrightarrow~{}\widehat{{\mathcal{W}}}_{g_{e_{0}}}(a,1)=-2^{\frac{s-1}{2}}$
and $\widehat{{\mathcal{W}}}_{g_{e_{1}}}(a,1)=2^{\frac{s-1}{2}}.$

  3. 3.

If $g_{|\\{x\in{\mathbb{F}}_{2}^{s-1}|w_{H}(x)~{}odd\\}}$ is balanced,

$\widehat{{\mathcal{W}}}_{g}(a)=-2^{\frac{s-1}{2}}~{}\Leftrightarrow~{}\widehat{{\mathcal{W}}}_{g_{e_{0}}}(a,1)=2^{\frac{s-1}{2}}$
and $\widehat{{\mathcal{W}}}_{g_{e_{1}}}(a,1)=-2^{\frac{s-1}{2}}.$

∎

Now, we are ready to use the extended Maiorana-McFarland, particular case
$r=1$.

In the following theorem, we always have that
${f}_{|\\{(x,y)|w_{H}((x,y))~{}even\\}}$ is uniquely balanced.

###### Theorem 4.5

Let $g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ be a bent function,
$g_{e}$, and $\phi$ defined as above. Then, the function

$f:{\mathbb{F}}_{2}^{1+s}\rightarrow{\mathbb{F}}_{2},~{}(x,\bar{y})\mapsto
x\phi(\bar{y})\oplus
g_{e}(\bar{y}),~{}x\in{\mathbb{F}}_{2},~{}\bar{y}\in{\mathbb{F}}_{2}^{s},$

is a bent function and ${f}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is
balanced.

###### Proof

Since, $\phi$ y $g_{e}$ satisfy the conditions of Theorem 4.3 (Observation 3
and Theorem 4.4), then $f$ is a bent function.

We will rely on the fact that,
${g}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}even\\}}$ is
balanced or
${g}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}}$ is
balanced.

Let’s see the balance. Let ${g}_{|\\{\bar{x}|w_{H}(\bar{x})~{}even\\}}$ be
balanced and $\widehat{{\mathcal{W}}}_{g}(0)=2^{\frac{s-1}{2}}.$ If we
consider $\bar{x}=(y_{1},\ldots,y_{s-1})$, we can write $f$ as

$f(x,y_{1},\ldots,y_{s-1},y_{s})=x\phi(y_{1},\ldots,y_{s-1},y_{s})\oplus
g_{e}(y_{1},\ldots,y_{s-1},y_{s}).$

Let $x=0$ and $y_{s}=0.$

Case 1. If $w_{H}(y_{1},\ldots,y_{s-1})$ is even,

$f(0,y_{1},\ldots,y_{s-1},0)=0\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)=g_{e_{0}}(y_{1},\ldots,y_{s-1},0)=0,~{}~{}2^{s-3}\mbox{
times}.$

Case 2. If $w_{H}(y_{1},\ldots,y_{s-1})$ is odd,

$f(0,y_{1},\ldots,y_{s-1},0)=0\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)=g_{e_{1}}(y_{1},\ldots,y_{s-1},0)=0,~{}~{}2^{s-3}+2^{\frac{s-3}{2}}\mbox{
times}.$

Let $x=0$ and $y_{s}=1.$

Case 3. If $w_{H}(y_{1},\ldots,y_{s-1})$ is even,

$f(0,y_{1},\ldots,y_{s-1},1)=0\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)=g_{e_{1}}(y_{1},\ldots,y_{s-1},1)=0,~{}~{}2^{s-3}\mbox{
times}.$

Case 4. If $w_{H}(y_{1},\ldots,y_{s-1})$ is odd,

$f(0,y_{1},\ldots,y_{s-1},1)=0\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)=g_{e_{0}}(y_{1},\ldots,y_{s-1},1)=0,~{}~{}2^{s-3}+2^{\frac{s-3}{2}}\mbox{
times}.$

Let $x=1$ and $y_{s}=0.$

Case 5. If $w_{H}(y_{1},\ldots,y_{s-1})$ is even,

$f(1,y_{1},\ldots,y_{s-1},0)=1\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)=g_{e_{0}}(y_{1},\ldots,y_{s-1},0)=0,~{}~{}2^{s-3}\mbox{
times}.$

Case 6. If $w_{H}(y_{1},\ldots,y_{s-1})$ is odd,

$f(1,y_{1},\ldots,y_{s-1},1)=1\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)=1\oplus
g_{e_{1}}(y_{1},\ldots,y_{s-1},0)=0,~{}~{}2^{s-3}-2^{\frac{s-3}{2}}\mbox{
times}.$

Let $x=1$ and $y_{s}=1.$

Case 7. If $w_{H}(y_{1},\ldots,y_{s-1})$ is even,

$f(1,y_{1},\ldots,y_{s-1},1)=1\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)=1\oplus
g_{e_{1}}(y_{1},\ldots,y_{s-1},1)=0,~{}~{}2^{s-3}\mbox{ times}.$

Case 8. If $w_{H}(y_{1},\ldots,y_{s-1})$ is odd,

$f(1,y_{1},\ldots,y_{s-1},1)=1\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)=0\oplus
g_{e_{0}}(y_{1},\ldots,y_{s-1},1)=0,~{}~{}2^{s-3}+2^{\frac{s-3}{2}}\mbox{
times}.$

The elements with even Hamming weight in ${\mathbb{F}}_{2}^{s+1}$ are in: Case
1, Case 4, Case 6, Case 7. Hence,

$|f_{|\\{(x,\bar{y})\in{\mathbb{F}}_{2}^{s+1}|w_{H}(x)~{}even\\}}^{-1}(0)|=(2^{s-3})+(2^{s-3}+2^{\frac{s-3}{2}})+(2^{s-3}-2^{\frac{s-3}{2}})+(2^{s-3})=2^{s-1}.$

Therefore, ${f}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is balanced.

Similarly,

When ${g}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}even\\}}$ is
balanced and $\widehat{{\mathcal{W}}}_{g}(0)=-2^{\frac{s-1}{2}}:$

${f}_{|\\{(x,\bar{y})\in{\mathbb{F}}_{2}^{s+1}|w_{H}((x,\bar{y}))~{}even\\}}$
is balanced.

When ${g}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}}$ is
balanced and $\widehat{{\mathcal{W}}}_{g}(0)=2^{\frac{s-1}{2}}:$

${f}_{|\\{(x,\bar{y})\in{\mathbb{F}}_{2}^{s+1}|w_{H}((x,\bar{y}))~{}even\\}}$
is balanced.

When ${g}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}}$ is
balanced and $\widehat{{\mathcal{W}}}_{g}(0)=-2^{\frac{s-1}{2}}:$

${f}_{|\\{(x,\bar{y})\in{\mathbb{F}}_{2}^{s+1}|w_{H}((x,\bar{y}))~{}even\\}}$
is balanced.

∎

The additional cases, from the above theorem, with odd Hamming weight in
${\mathbb{F}}_{2}^{s+1}$ will be necessary for the development of the
subsequent algorithm, to obtain $f$ from $g$.

Now, we consider the addition of a linear function.

###### Theorem 4.6

Let $g:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ be a bent function
and $f:{\mathbb{F}}_{2}^{s+1}\rightarrow{\mathbb{F}}_{2},~{}(x,\bar{y})\mapsto
x\phi(\bar{y})\oplus g_{e}(\bar{y})$ be a Maiorana-McFarland bent function.
The function $f\oplus l_{b}$ with respect to $g\oplus l_{\bar{a}}$,
$l_{\bar{a}}(\bar{x})={\bar{a}}\cdot\bar{x}$,
$\bar{x}\in{\mathbb{F}}_{2}^{s-1}$, ${\bar{b}}=(a_{0},{\bar{a}},a_{s})$,
$a_{0},a_{s}\in{\mathbb{F}}_{2}$, ${\bar{a}}\in{\mathbb{F}}_{2}^{s-1}$ is as
follows:

$\mbox{If }~{}a_{0}=1,~{}{(f\oplus
l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}~{}\mbox{ is
balanced.}$ $\mbox{If }~{}a_{0}=0,~{}{(f\oplus
l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}~{}\mbox{ is
balanced.}$

###### Proof

Let $\bar{x}=(y_{1},\ldots,y_{s-1}),\bar{y}=(\bar{x},y_{s}).$ Given $g$ a bent
function, then ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{(\bar{x},y_{s})|\bar{x}\in{\mathbb{F}}_{2}^{s-1},w_{H}(\bar{x})~{}even\\}}$
is balanced or ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{(\bar{x},y_{s})|\bar{x}\in{\mathbb{F}}_{2}^{s-1},w_{H}(\bar{x})~{}odd\\}}$
is balanced.

Suppose that ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{(\bar{x},y_{s})|\bar{x}\in{\mathbb{F}}_{2}^{s-1},w_{H}(\bar{x})~{}even\\}}$
is balanced and
$\widehat{{\mathcal{W}}}_{g_{e}}({\bar{a}},a_{s})=2^{\frac{s-1}{2}}.$

We can write $f\oplus l_{\bar{b}}$ as,

$f(x,y_{1},\ldots,y_{s-1},y_{s})\oplus
l_{\bar{b}}=x\phi(y_{1},\ldots,y_{s-1},y_{s})\oplus
g_{e}(y_{1},\ldots,y_{s-1},y_{s})\oplus
a_{0}x\oplus({\bar{a}},a_{s})\cdot\bar{y},~{}~{}\bar{y}=(y_{1},\ldots,y_{s}).$

Let $a_{0}=1.$ The elements with odd Hamming weight in
${\mathbb{F}}_{2}^{s+1}$ are in the following cases:

Case 1. Let $x=0$, $y_{s}=0$, and $w_{H}(y_{1},\ldots,y_{s-1})$ odd. Then

$f(0,y_{1},\ldots,y_{s-1},0)\oplus l_{\bar{b}}=0\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)\oplus
l_{(\bar{a},a_{s})}=g_{e_{1}}(y_{1},\ldots,y_{s-1},0)\oplus
l_{({\bar{a}},a_{s})}=0,~{}~{}2^{s-3}+2^{\frac{s-3}{2}}\mbox{ times}.$

Case $2$. Let $x=0$, $y_{s}=1$, and $w_{H}(y_{1},\ldots,y_{s-1})$ even. Then,

$f(0,y_{1},\ldots,y_{s-1},1)\oplus l_{\bar{b}}=0\cdot 1\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)\oplus
l_{({\bar{a}},a_{s})}=g_{e_{1}}(y_{1},\ldots,y_{s-1},1)\oplus
l_{({\bar{a}},a_{s})}=0,~{}~{}2^{s-3}\mbox{ times}.$

Case $3$. Let $x=1$, $y_{s}=0,$ and $w_{H}(y_{1},\ldots,y_{s-1})$ even. Then,

$f(1,y_{1},\ldots,y_{s-1},0)\oplus l_{\bar{b}}=1\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},0)\oplus l_{({\bar{a}},a_{s})}\oplus
1=g_{e_{0}}(y_{1},\ldots,y_{s-1},0)\oplus l_{({\bar{a}},a_{s})}\oplus
1=0,~{}~{}2^{s-3}\mbox{ times}.$

Case $4$. Let $x=1$, $y_{s}=1,$ and $w_{H}(y_{1},\ldots,y_{s-1})$ odd. Then,

$f(1,y_{1},\ldots,y_{s-1},1)\oplus l_{\bar{b}}=1\cdot 0\oplus
g_{e}(y_{1},\ldots,y_{s-1},1)\oplus l_{({\bar{a}},a_{s})}\oplus 1=0\oplus
g_{e_{0}}(y_{1},\ldots,y_{s-1},1)\oplus l_{({\bar{a}},a_{s})}\oplus
1=0,~{}~{}2^{s-3}-2^{\frac{s-3}{2}}\mbox{ times}.$

Hence,

$|f_{|\\{(x,\bar{y})\in{\mathbb{F}}_{2}^{s+1}|w_{H}(x,\bar{y})~{}odd\\}}^{-1}(0)|=(2^{s-3}+2^{\frac{s-3}{2}})+(2^{s-3})+(2^{s-3})+(2^{s-3}-2^{\frac{s-3}{2}})=2^{s-1}.$

Therefore, ${(f\oplus
l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}$ is balanced.

The demonstration when $a_{0}=0$ is similar to Case 1, Case 4, Case 6 and Case
7 of Theorem 4.5, and we obtain that ${(f\oplus
l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is balanced.

Similarly,

When ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}}$
is balanced and
$\widehat{{\mathcal{W}}}_{g_{e}}({\bar{a}},a_{s})=2^{\frac{s-1}{2}}:$

If $a_{0}=1$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}$ is
balanced.

If $a_{0}=0$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is
balanced.

When ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}even\\}}$
is balanced and
$\widehat{{\mathcal{W}}}_{g_{e}}({\bar{a}},a_{s})=-2^{\frac{s-1}{2}}:$

If $a_{0}=1$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}$ is
balanced.

If $a_{0}=0$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is
balanced.

When ${(g_{e}\oplus
l_{({\bar{a}},a_{s})})}_{|\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}}$
is balanced and
$\widehat{{\mathcal{W}}}_{g_{e}}({\bar{a}},a_{s})=-2^{\frac{s-1}{2}}:$

If $a_{0}=1$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}$ is
balanced.

If $a_{0}=0$,

${(f\oplus l_{\bar{b}})}_{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}$ is
balanced.

∎

We can use the demonstration of Theorem 4.6 to have a simple way to obtain
bent functions of any even dimension in its domain, greater than dimension of
given the bent function. These new functions are specifically balanced for
even Hamming weight in its domain.

In Algorithms 1 and 2 we are considering
${\mathcal{A}}:=\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}even\\}$,
${\mathcal{B}}:=\\{\bar{x}\in{\mathbb{F}}_{2}^{s-1}|w_{H}(\bar{x})~{}odd\\}.$

Algorithm 1 Extended Maiorana-McFarland $r=1$

1:$s-1\geq 2$ odd, $g_{s-1}(\bar{y})$,
$g_{s-1}:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ be a bent
function,
$\\{(x,\bar{y},y)\in{\mathbb{F}}_{2}^{s+1}~{}|~{}x,y\in{\mathbb{F}}_{2},\bar{y}\in{\mathbb{F}}_{2}^{s-1}\\}$

2:$g_{new}(x,\bar{y},y)$ a bent function, ${g_{new}}_{|{\mathcal{A}}}$
balanced

3:Integer $end$;

4:$new:=s-1$

5:while $new\neq end$ do

6: for $x,y$ from $0$ to $1$ do

7: if $\bar{y}$ is even, $x=1$, $y=1$ or $\bar{y}$ is odd, $x=1$, $y=0$ then

8: $g_{new+2}(x,\bar{y},y)=1\oplus g_{new}(\bar{y});$

9: else

10: $g_{new+2}(x,\bar{y},y)=g_{new}(\bar{y});$

11: $new:=new+2;$

12: $\bar{y}=(x,\bar{y},y);$

13: end if

14: end for

15:end while

Algorithm 2 Extended Maiorana-McFarland $\oplus l_{b}~{}r=1$

1:$s-1\geq 2$ odd, $g_{s-1}(\bar{x})$,
$g_{s-1}:{\mathbb{F}}_{2}^{s-1}\rightarrow{\mathbb{F}}_{2}$ a bent function,
$b=(a_{0},\bar{a},a)\in{\mathbb{F}}_{2}^{s+1},$
$\\{(x,\bar{x},y)\in{\mathbb{F}}_{2}^{s+1}~{}|~{}x,y\in{\mathbb{F}}_{2},\bar{x}\in{\mathbb{F}}_{2}^{s-1}\\}$

2:$(g_{new}\oplus l_{0,\bar{a},a})(x,\bar{x},y)$ a bent function,
$({g_{new}}\oplus l_{0,\bar{a},a})_{|{\mathcal{A}}}$ balanced $(g_{new}\oplus
l_{1,\bar{a},a})(x,\bar{x},y)$ a bent function, $({g_{new}}\oplus
l_{1,\bar{a},a})_{|{\mathcal{B}}}$ balanced

3:Integer $end;$

4:$new:=s-1$

5:while $new\neq end$ do

6: if $a_{0}=0$ and $a=0$ then

7: for $x,y$ from $0$ to $1$ do

8: if ($\bar{y}$ is even, $x=1$, $y=1$) or ($\bar{y}$ is odd, $x=1$, $y=0$)
then

9: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=1\oplus(g_{new}\oplus
l_{a})(\bar{x});$

10: else

11: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=(g_{new}\oplus l_{a})(\bar{x});$

12: end if

13: $new:=new+2;$ $\bar{y}=(x,\bar{y},y);$

14: end for

15: end if

16: if $a_{0}=0$ and $a=1$ then

17: for $x,y$ from $0$ to $1$ do

18: if ($\bar{y}$ is even, $x=0$, $y=1$) or {($\bar{y}$ is odd, [($x=0$,
$y=1$) or ($x=1$, $y=0$) or ($x=1$, $y=1$)]} then

19: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=1\oplus(g_{new}\oplus
l_{a})(\bar{x});$

20: else

21: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=(g_{new}\oplus l_{a})(\bar{x});$

22: end if

23: $new:=new+2;$ $\bar{y}=(x,\bar{y},y);$

24: end for

25: end if

26: if $a_{0}=1$ and $a=0$ then

27: for $x,y$ from $0$ to $1$ do

28: if ($\bar{y}$ is even, $x=1$, $y=0$) or ($\bar{y}$ is odd, $x=1$, $y=1$)
then

29: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=1\oplus(g_{new}\oplus
l_{a})(\bar{x});$

30: else

31: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=(g_{new}\oplus l_{a})(\bar{x});$

32: end if

33: $new:=new+2;$ $\bar{y}=(x,\bar{y},y);$

34: end for

35: end if

36: if $a_{0}=1$ and $a=1$ then

37: for $x,y$ from $0$ to $1$ do

38: if {($\bar{y}$ is even, [($x=0$, $y=1$) or ($x=1$, $y=0$) or ($x=1$,
$y=1$)]} or ($\bar{y}$ is odd, $x=0$, $y=1$) then

39: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=1\oplus(g_{new}\oplus
l_{a})(\bar{x});$

40: else

41: $(g_{new+2}\oplus l_{0,a,0})(x,\bar{x},y)=(g_{new}\oplus l_{a})(\bar{x});$

42: end if

43: $new:=new+2;$ $\bar{y}=(x,\bar{y},y);$

44: end for

45: end if

46:end while

##  5 Conclusions

We found, using the Maiorana-McFarland class, a new subclass of bent functions
such that it is uniquely balanced on the set
${\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}((x)~{}even\\}}$. For this, we see, first,
that all bent functions in ${\mathcal{B}}_{n}$ satisfy
${\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}$ is balanced or
${\\{x\in{\mathbb{F}}_{2}^{n}|w_{H}(x)~{}even\\}}$ is balanced. Also, if we
have a bent function $f$ obtained by the Maiorana-McFarland construction, to
see the behaviour when we consider the addition of a linear function $l_{b}$,
$b=(a_{0},\bar{a},a)$, we now know that, if $a_{0}=0$, $(f\oplus
l_{b})_{|{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}even\\}}}$ is balanced and if
$a_{0}=1$, $(f\oplus l_{b})_{|{|\\{(x,\bar{y})|w_{H}((x,\bar{y}))~{}odd\\}}}$
is balanced.

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