∎

11institutetext:  1 Ioffe Institute, Politekhnicheskaya 26, 194021 Saint
Petersburg, Russia
2 Franco-Chilean Laboratory for Astronomy, Camino El Observatorio 1515, Las
Condes, Santiago, Chile
3 Institut d’Astrophysique de Paris, CNRS-SU, UMR 7095, 98bis bd Arago, 75014
Paris, France
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Molecular hydrogen in absorption at high redshifts.

Science cases for CUBES.

S.A. Balashev1 P. Noterdaeme2,3

(Received: date / Accepted: date)

###### Abstract

Absorption lines from molecular hydrogen ($\rm H_{2}$) in the spectra of
background sources are a powerful probe of the physical conditions in
intervening cold neutral medium. At high redshift, $z>2$, $\rm H_{2}$ lines
are conveniently shifted in the optical domain, allowing the use of ground-
based telescopes to perform high-resolution spectroscopy, which is essential
for a proper analysis of the cold gas. We describe recent observational
progress, based on the development of efficient pre-selection techniques in
low-resolution spectroscopic surveys such as the Sloan Digital Sky Survey
(SDSS). The next generation of spectrographs with high blue-throughput, such
as CUBES, will certainly significantly boost the efficiency and outcome of
follow-up observations. In this paper, we discuss high priority science cases
for CUBES, building on recent $\rm H_{2}$ observations at high-z: probing the
physical conditions in the cold phase of regular galaxies and outflowing gas
from active galactic nucleus.

###### Keywords:

Molecular hydrogen Quasar absorption lines Active galactic nuclei

##  1 Introduction

Molecular hydrogen ($\rm H_{2}$) – the most abundant molecule in the Universe
– traces the cold ($T\sim 100$ K) phase of the neutral gas. Since this phase
is inclined to gravitational collapse through Jeans instability, understanding
its physical and chemical properties is fundamental for a complete theory of
star-formation throughout the Universe’s history. Molecular hydrogen can be
detected directly through electronic absorption lines in the Lyman and Werner
bands ($\sim 91-110$ nm rest-frame) Carruthers1970 ; Spitzer1974 . At high
redshift ($z>2$), $\rm H_{2}$ lines are conveniently shifted into optical
domain and become accessible for the ground telescopes. Because of their
brightness, quasars are generally used as background sources, although there
is growing interest in the use of gamma-ray burst afterglows as well
Prochaska2009 ; Kruhler2013 ; Bolmer2019 . The large collecting area of
ground-based telescopes, combined with high-resolution spectrographs, permits
to resolve narrow $\rm H_{2}$ lines that typically arise from several
rotational levels in the cold $\sim 100$ K neutral medium (see e.g.
Balashev2019 ). The ability to detect rotational levels is key to probe the
physical conditions in the gas by modelling the relative population of H2
levels (e.g. Klimenko2020 ).

In this paper, we briefly describe an historical perspective and recent
progress in studying diffuse molecular and translucent gas at high-$z$,
through $\rm H_{2}$ absorption lines, mostly with Ultraviolet-Visual Echelle
Spectrograph (UVES, Dekker2000 ) and X-shooter Vernet2011  instruments on the
Very Large Telescope (VLT) in Paranal. We discuss the tremendous boost in this
field that would be enabled by Cassegrain U-Band Efficient Spectrograph
(CUBES, CUBES ), in particular thanks to its efficient coverage of $\sim
300-400$ nm wavelengths. Key benefits will be: (i) an increase of the sample
sizes of reachable systems by an order of magnitude. The corresponding systems
are already available from pre-selection in the Sloan Digital Sky Survey
(SDSS, York2000 ) but other surveys will also contribute to discover new ones;
(ii) the access to more extreme and elusive environments (e.g. more
dusty/molecular rich). This will allow us not only to access the thermal state
of the cold phase of the neutral medium of the overall population of high-$z$
galaxies but also to investigate the active galactic nuclei (AGN) feedback and
quasar environment as probed by a recently discovered population of proximate
$\rm H_{2}$ absorption systems111”Proximate” here refers to absorption systems
with redshift $z_{\rm abs}$ similar to that of the quasar emission, $z_{em}$,
typically with $\Delta v\approx c\cdot|z_{\rm abs}-z_{\rm em}|/(1+z_{\rm
em})<\rm few\times 1000\,km\,s^{-1}$, where $c$ is the speed of light. Such
absorption systems can be physically associated with the quasar and/or its
host galaxy..

Figure 1: Sketch of the studies of H2-bearing damped Ly-$\alpha$ systems
(DLAs) at high redshift towards quasars. The red lines mark absorption lines
associated with DLAs, that are likely produced by interstellar or circum-
galactic neutral gas located on the line of sight. These lines include H i
Lyman series, metal absorption lines as well as lines from H2, HD and CO
molecules.

##  2 Observations of H2 absorption

Molecular hydrogen being a homonuclear molecule, it has no dipole moment in
the ground electronic state. Therefore, in emission, $\rm H_{2}$ can be
observed only under specific conditions in the interstellar medium (ISM) –
mainly in photo-dissociated regions, which represent dense and warm gas, with
temperatures $\gtrsim 500$ K – the typical distances between rotational energy
levels of orto- or para-H2. In turn, the electronic Lyman ($\rm
B^{1}\Sigma^{+}_{u}-\rm X^{1}\Sigma^{+}_{g}$) and Werner ($\rm
C^{1}\Pi_{u}-\rm X^{1}\Sigma^{+}_{g}$) bands transition of $\rm H_{2}$ are
permitted. These transitions were first detected in space via rocket flight
observations by Carruthers1970  and then extensively analysed using the
Copernicus satellite Spitzer1974 , mostly towards bright nearby stars. These,
as well as following observations by the Far Ultraviolet Spectroscopic
Explorer (FUSE, Sahnow1996 ) satellite, concerned the local group Shull2000 ;
Tumlinson2002 . In total, about 140 sightlines with H2 detection where
identified in FUSE data, through a recent reanalysis by Shull2021 .

###  2.1 High redshifts

In principle, and somehow paradoxically, H2 in remote galaxies is easier to
observe than locally, owing to the Lyman and Werner bands of H2 being
redshifted into the optical domain. This permits the use of large optical
telescopes to collect high-resolution spectra ($R\gtrsim
50000$222Corresponding to full width half maximum (FWHM) of the instrument
function, ${\rm FWHM}=c/R\approx 6$ km s-1, where $c$ is the speed of light.)
of the background source. In comparison, low-$z$ observations have been
predominantly performed at $R\lesssim 20000$. The gain in resolution for
high-$z$ observations is an important advantage, since it allows to fully
resolve the H2 bands and to perform a proper de-blending from the
Lyman-$\alpha$ forest as well as between H2 velocity components. The
instrumental profile is then also getting much closer to the intrinsic width
of individual H2 components that typically have Doppler parameters of only a
few km s-1 or less, allowing much more precise measurements. At high
redshifts, quasars and gamma ray burst (GRB) afterglows can be used as bright
background sources, albeit with some limiting capabilities in case of GRBs due
their transient (a rapid decrease in luminosity) nature. In practice, H2
absorption lines are always found to be associated with high column densities
of neutral hydrogen, with the corresponding H i Lyman series lines being
located in the same wavelength region (see Fig. 1) The overall measured H i
column densities333Here and after the column densities, $N$, are expressed in
particles (atoms or molecules) per cm2 throughout H2-bearing systems are
typically $\log N(\text{H\,{\sc i}})>19$ and possibly much above. H2
absorption systems hence belong to either the so-called Damped Lyman-$\alpha$
systems (DLAs), with $\log N(\text{H\,{\sc i}})\geq 20.3$), or to strong sub-
DLAs with $19\leq\log N(\text{H\,{\sc i}})<20.3$. A third sub-class has
recently been introduced for the highest column densities, with $\log
N(\text{H\,{\sc i}})>21.7$ and dubbed extremely strong DLAs (ESDLAs,
Noterdaeme2014 ).

The first $\rm H_{2}$ absorption system at high $z$ was detected
serendipitously by Levshakov1985  towards PKS $0528-250$ using only 2 Å
resolution spectrum obtained by Anglo-Australian 3.9m telescope Morton1980 .
The second high-$z$ H2 absorption system was detected only a decade later
Ge1997  using the Multiple Mirror Telescope, indicating a very slow progress
in this field in pre-VLT/Keck era.

###  2.2 VLT/Keck era

A new epoch of high-z H2 absorption studies began with the advent of high-
resolution spectrographs, High Resolution Echelle Spectrometer (HIRES,
Vogt1994 ) and UVES on the 8-m class telescopes Keck and VLT, respectively.
This resulted in a increase of the number of detected molecular hydrogen
absorption systems, with the most important contribution arising from the VLT,
thanks to dedicated and regular science programs to search for H2
Petitjean2000 ; Ledoux2003 ; Srianand2005 . The UVES database for H2 systems
comprised thirteen H2 systems Noterdaeme2008 . On Keck telescope, in spite of
a large observational program focusing on DLAs ($\gtrsim 100$ DLA were studied
at $z\gtrsim 2$ Prochaska2003b ), H2 was reported in only a handful of systems
Prochaska2003a ; Jorgenson2009 , mostly due a wavelength coverage not
optimised from H2 during observations. The semi-blind surveys on these
telescopes (and also Magellan, see Jorgenson2014 ) indicated that the
incidence rate of H2 in DLAs is pretty low $\sim 10$% for H2 column densities
above $10^{17}$ cm-2 Noterdaeme2008 . An independent estimate of the incidence
rate in the overall population of DLAs, $\sim 4.0\pm 0.5(\rm stat.)\pm 1.0(\rm
syst.)$% for saturated H2 absorption systems ($\log N(\rm H_{2})>18$) was
later obtained from the detection of H2 lines in the composite DLA spectrum
Balashev2018  constructed with SDSS.

Figure 2: Portions of spectra of J $1237+0647$ featuring H2 absorption lines
from a system at $z=2.625$, see Noterdaeme2010 . The top, middle and bottom
panels correspond to the spectrum obtained using SDSS ($R\approx 2000$),
X-shooter ($R\approx 6000$) and UVES ($R\approx 50000$), respectively. The
black and colored thick lines show the observed spectrum and total model
profile of H2 absorption system, respectively. The colored lines and regions
in each panel show the profiles of individual velocity components, that are
resolved and fitted using the UVES spectrum Noterdaeme2010 . The horizontal
line with marks show the positions of H2 absorption lines of the Lyman
(denoted as $L$) and Werner (denoted as $W$) bands from different rotational
levels, with the number above each tick indicate the rotational level.

##  3 Pre-selection with SDSS

The low detection rate of H2 among DLAs, together with the $\sim 15$%
incidence rate of high-$z$ DLA in quasar spectra results in a very low
efficiency of blind searches, making this approach time-costly and impractical
on highly over-subscribed telescopes. Fortunately, the availability of the
SDSS, that collected several 105 quasar spectra with $R\approx 2000$, not only
permitted the detection of many thousands high-$z$ DLAs/sub-DLAs
Noterdaeme2012 ; Parks2018 , but also triggered the development of efficient
techniques for pre-selecting H2-bearing systems. Indeed, it was shown that
strong H2 absorption systems (with $\log N(\rm H_{2})\gtrsim 18$) can be
either detected directly via their characteristic Lyman-Werner band signature
in SDSS spectra (provided their quality is high enough and the Lyman-$\alpha$
forest not too dense) Balashev2014 ; Noterdaeme2019  or using C i as a good
tracer of H2 Srianand2008 ; Noterdaeme2018 , in particular at high
metallicity. Several C i lines are generally located redwards of the
Lyman-$\alpha$ emission, allowing their automated detection in SDSS Ledoux2015
. An example of the H2-bearing DLA that has been initially found in SDSS and
lately followed-up with both X-shooter and UVES is shown in Fig. 2. Finally,
it was found that the incidence rate of H2 significantly increases (up to
$\sim 50$%) in the ESDLA sub-sample from QSO Noterdaeme2015 ; Balashev2018 ;
Ranjan2020  as well as GRB afterglows sightlines Bolmer2019 . This is most
likely due to the corresponding lines of sight intercepting galaxies at small
galacto-centric distances Arabsalmani2015 ; Lyman2017 ; Ranjan2018 ;
Ranjan2020 , where the prevailing physical conditions, and in particular the
gas pressure, favour the existence of the cold phase and the H i/H2 conversion
Guimaraes2012 ; Noterdaeme2015 ; Balashev2017 ; Bolmer2019 .

A comparative study of these three pre-selection techniques Balashev2019
indicates that they have different selection function in terms of the
properties (such as metallicity, $N(\text{H\,{\sc i}})$, $N(\rm H_{2})$,
presence of CO, etc.) and therefore should complement each other. For example,
regarding the metallicity, the CI-selected systems have on average an order of
magnitude higher metallicities than ESDLA-selected ones, while direct search
for H2 lines pick up absorption systems in between Balashev2019 . It is also
important to note, that these pre-selections were obtained based solely on
SDSS owing to its statistical power. Current samples therefore inherit from
the quasar selection functions of the different SDSS spectroscopic surveys
Krogager2019 . In particular, optical selection tends to exclude highly
reddened quasars and hence dusty and large H2 column systems. This, together
with the small cross-section of dense molecular clouds implies that the H2
absorption systems currently probe mostly diffuse molecular gas (with
exception of a few cases, see Noterdaeme2010 ; Balashev2017 ).

Notwithstanding, these techniques allowed us to significantly increase the
number of confirmed H2 absorption systems from $\sim 15$ to $\approx 50$ in
the last decade, with an extremely efficient use of the observational time on
the largest optical telescopes. This also permitted the study of systems
spanning a very large range of properties of the underlying galaxy population.
Among them, we can note the detection of: (i) Perseus-like cloud at $z\approx
2$ Noterdaeme2017 ; (ii) CO and 2175Å dust bump in high-metallicity H2-bearing
DLAs Srianand2008 ; Noterdaeme2009 ; Noterdaeme2011 ; (iii) large H2 column
density and low-metallicity absorbers, representing CO-dark molecular clouds
Balashev2017 ; Ranjan2018 ; (iv) low impact parameter ESDLA with large H2
column Ranjan2018 . In short, the study of H2 absorbers at high-$z$ towards
quasars and GRB afterglows opened a unique window to the understanding of the
ISM properties in the overall galaxy population, similar to what has been done
in the local ISM towards bright nearby stars.

##  4 CUBES science case

In spite of significant progress in recent years, the observations are now
reaching a bottleneck, mostly due to the limiting instrumental capabilities to
follow-up faint targets (typically $m_{\rm G}\gtrsim 19$) that make the bulk
of the new and most promising candidates. In fact, while observations were
first obtained with UVES (since high resolution has a high priority for H2
studies), the preference has then shifted to X-shooter to optimize the
observing time for $m_{\rm G}\gtrsim 19$ targets. The drawback for choosing
X-shooter is the loss in spectral resolution $R\sim 6000$444H2 absorption
lines mostly falls in UVB arm of X-shooter, which quite complicates the
analysis and limits the derivation of physical quantities from the data (see
e.g. Noterdaeme2017 ; Balashev2019  and comparison of the spectra in Fig. 2).
Currently, most of the bright targets accessible from VLT have already been
observed, and available candidates in SDSS are already out of reach for UVES
and at the limit for X-shooter within reasonable observing time, see Fig. 5.

CUBES CUBES  will exactly fill this gap, with a high efficiency in the blue,
while keeping a high-resolution to maximise the scientific outcome. This is
essential to continue investigating the cold neutral phase in and around
galaxies at high redshifts. Indeed the efficiency of CUBES is about an order
of magnitude higher than UVES, and the declared resolution ($R\sim 20000$) is
enough to resolve the H2 lines much better than X-shooter currently can do.
Using CUBES exposure time calculator CUBES_ETC , we estimate that for a QSO
with G-band magnitude around 20, $\rm S/N\approx 10$ can be reached in only
one hour of observing time. This will allow one to increase the available
samples by several fold, as needed for the relevant science goals.
Additionally, the wavelength coverage of CUBES is already optimized for H2
identified in SDSS (see Fig. 5). In the following three subsections, we will
briefly describe the most important CUBES science goals to our eyes, dealing
with intervening and proximate H2-bearing DLAs, i.e. cold gas in,
respectively, the overall population (hence mostly faint) galaxies and the
environment of massive active galaxies.

###  4.1 Intervening DLAs: probing galaxies at high-z

Figure 3: The measured thermal pressure versus number density in intervening
H2 absorption systems at high redshifts. The panels from left to right
correspond to three samples of selected metallicities in $\log
Z=0.5,0.0,-0.5,-1.0,-1.5$ bins, shown on the top of each panel. The curves on
each plot show the calculation of the phase diagram at median metallicity of
the bin. The color of the lines and symbols indicate a measured (and assumed)
UV field (in Draine field units). For calculations we also co-scaled cosmic
ray ionization rate with UV field (assuming that they are both associated with
star formation), which is indicated by the color bar at the top of each panel.

While the first detected H2 absorption system at high-$z$ was found at the
quasar redshift (see recent re-analysis by Balashev2020a ), studies of H2
initially focused on intervening systems – absorption systems at cosmological
remote distances from the quasar, i.e. that are not physically associated with
the later. Typically, these are securely chosen by a velocity difference
$\Delta v\approx(z_{\rm abs}-z_{\rm qso})/(1+z_{\rm qso})c>\rm few\times 1000$
km s-1. This permits a large redshift path to be probed and therefore, such
systems were preferentially selected in the first blind surveys.555However,
since the cross section of associated systems are found to be much larger, as
was found later, the incidence rate of intervening and proximate systems are
not much different, see Section 4.2

The determination of the physical conditions was one of the primary goal of
studies of intervening H2 absorption systems Srianand2005 ; Noterdaeme2007 ;
Balashev2010 , since excitation of H2 rotational levels (e.g. Noterdaeme2007b
; Balashev2019 ), as well as associated C i fine-structure (e.g. Jorgenson2010
; Balashev2017 ), and rotational levels of HD (e.g. Balashev2010 ) and CO
molecules Srianand2008 ; Noterdaeme2011  provide a straight way to do it. The
later two molecules were found in a handful of H2-bearing DLAs systems, but
provide valuable independent constraints on the physical conditions for
molecular gas with high column, thanks to different sensitivities of the
molecules on the various excitation processes. Indeed, the rotational levels
of molecules as well as fine-structure levels of atomic species are populated
by a competition between collisional excitation, radiative pumping (following
the electronic UV transitions) and direct radiative excitation. Therefore the
measured populations on these levels are sensitive to the rates of these
processes and provide the way to constrain the kinetic temperature, number
density, UV flux and Cosmic Microwave Background (CMB) temperature (mostly
using CO, Noterdaeme2011 ).

Originally, H2-bearing DLAs were analysed mostly on a case-by-case basis. The
main outcome of these studies were: (i) The measurement of the gas kinetic
temperature, estimated using the ortho-to-para ratio666In the case of H2, this
corresponds to the ratio of J=1 and J=0 rotational levels. of H2 molecules,
and giving $T_{k}\sim$100 K, i.e. H2 probe the cold phase of ISM; (ii) The
measurement of the number densities, found in the range $n\approx 10-\rm 300$
cm-3, i.e. also consistent with what is expected for the cold neutral ISM;
(iii) The derivation of the strength of the UV field, found to span a large
range from slightly lower than the Draine field up to two orders of magnitude
higher. Recently performed systematic analyses of the emergent samples of
H2-bearing DLAs Klimenko2020 ; Kosenko2021  start to show interesting
dependencies between estimated physical conditions. For example: (i) The
thermal pressure measured in the cold phase of the ISM increases with
increasing neutral gas column density Balashev2017 ; Balashev2019 . This is
quite expected from analogy with local measurements as well as simulations,
which shows that the high column density systems probe galaxies at the
smallest impact parameters, and hence expected to have enhanced pressures;
(ii) The ratio of UV field to number density correlates with the gas
temperature Klimenko2020 . This correlation is also metallicity-dependent but
it seems not to contradict with heating-cooling balance of the neutral ISM
(see Fig. 3). (iii) The cosmic ray ionization rate estimated using HD/H2 ratio
is found to depend quadratically on the strength of the UV field Kosenko2021 .
Since cosmic ray ionization ray also determine the thermal balance especially
at low metallicities Bialy2019 , this should also be take into account for
future studies. To summarize, we anticipate many progresses in our
understanding of the physics of the cold diffuse gas at high-$z$, and hence in
our understanding of how star-formation occurs, when larger samples (probing
wider ranges of the physical conditions and chemical enrichment) will get
observed with CUBES.

###  4.2 Proximate DLAs: probing AGNs outflowing gas

Figure 4: Sketch representing studies of AGN outflowing gas using
VLT/X-shooter observations (reproduced from Noterdaeme2021a ). The left and
right bottom panels show portions of J $0015+1842$ spectrum covering H2
absorption lines and Ly-$\alpha$ emission/absorption line, respectively. The
detection of extended [O III] and Ly-$\alpha$ emission (shown in 2d spectrum
in the bottom panel) and surprisingly high excitation of H2 rotational levels,
indicate that this absorption system is associated with outflowing gas.

A recent study of the SDSS quasar database by Noterdaeme2019  uncovered a
population of proximate (i.e. at sensibly the quasar redshift) H2 absorption
systems. It was found that the incidence rate of proximate H2 is actually up
to a factor of 10 higher than what is expected from the statistics of
intervening ones. This is opposite to naive expectations that the strong
radiation field of quasar should suppress the presence of H2 in its vicinity,
due to efficient photo-dissociation of the molecules. However, the observed
strong excess hypothetically can be explained by either clustering of the
matter and satellite galaxies around quasars or the presence of galactic-scale
outflows, which, as current local observations indicate, frequently contains a
large amount of molecular gas Veilleux2013 ; Cicone2014 . The detection of
leaking Ly-$\alpha$ emission (seen as Ly-$\alpha$ emission inside the damped
Ly-$\alpha$ absorption) in a roughly half of the candidates from SDSS
Noterdaeme2019 , as well as follow-up observations Noterdaeme2021a  indicate
that substantial fraction of proximate H2 absorption systems may indeed arise
from outflowing gas. In that sense, molecular hydrogen served as the species
that allow the pre-selection of these systems in low-resolution spectra, but
also provides a key diagnostic of the physical conditions in the diffuse
molecular gas associated with outflow. For example, in the first and only one
system of such class studied in details so far, J $0015+1842$ at $z=2.631$, it
was found, using the excitation of H2 rotational levels, that the number
density is $n\sim 10^{4}$ cm-3 and that the UV field is $\sim 10^{3}$ times
higher than the Draine field. This is significantly higher than what is
typically measured in intervening DLAs. Assuming that the quasar produces most
of the UV radiation, this permitted to constrain the distance between the AGN
central engine and H2-bearing medium to be $\sim 10-15$ kpc, see Fig. 4. The
analysis of J $0015+1842$ urges for observing a sample to enable statistical
studies and address open questions about the nature of these systems,
constraining key parameters of AGN outflows, such as the mass rate, fraction
of active AGNs, opening angles etc. Additionally, the detection of such kind
of systems opens wide opportunities for the multi-wavelength observations of
AGN outflows at high-$z$, including near IR (to study H2 in emission), sub-mm
(to study cold molecular phase by CO emission Noterdaeme2021b  as well as
other molecules) and makes valuable quantitative input for the modelling of
outflowing gas.

###  4.3 Probing dense gas

As we already mentioned above, SDSS has, by construction, significant
selection bias against dusty and red quasars Krogager2019 . Indeed, if dust is
present anywhere along the line of sight, then the background quasar will
appear fainter and redder than it intrinsically is (e.g. Pei1991 ; Pontzen2009
). Quasars with such dusty systems are therefore more likely to fall outside
the color-selection and/or below the flux-limit of spectroscopic surveys.
Since, as local measurements indicate, the amount of dust obscuration scales
with the H2 column density (e.g. Savage1977 ; Roman-Duval2010 ; Shull2021 ),
this implies a bias against absorption systems with high H2 column densities.
These elusive systems correspond to translucent or dense molecular gas and
therefore their detailed study is very important for a complete theory of the
baryon cycle in the ISM over a wide range of metallicities. The next
generation spectroscopic survey telescopes (such as 4-m Multi-Object
Spectroscopic Telescope (4MOST), Dark Energy Spectroscopic Instrument, Mauna
Kea Spectroscopic Explorer, William Hershel Telescope Enhanced Area Velocity
Explorer, etc) will likely push further the quasar selection parameter space
and obtain samples with significantly less selection bias against dusty
absorbers. Follow-up observations of reddened quasars in the blue will
naturally require a very good efficiency as can be brought by CUBES.

###  4.4 Astrophysical probes of fundamental physics

Molecular hydrogen absorption systems also provide unique probes of
fundamental physics and cosmology. By comparing the observed relative
wavelengths of the numerous absorption lines from H2 with those measured in
the laboratory, it is possible to constrain the variation of the electron-to-
proton mass ratio, $\mu$, over cosmological time scales Thompson1975 ;
Varshalovich1993 ; Bagdonaite2012 , which gives typical limits of
$|\Delta\mu/\mu|<5\times 10^{-6}$ Ubachs2016 . This value is limited by
systematics (including instrumental ones, Rahmani2013 ), the sensitivity of H2
transitions to variation of $\mu$, as well as a lack of systems with suitable
characteristics. While it was found that some molecules (e.g. CH3OH, NH3 and
HC3N) are much more sensitive to variation of $\mu$ Henkel2009 ; Kanekar2015 ,
these are very rarely detected in the distant Universe777This is due to
combination of two factors: (i) the cross-section of dense molecular gas (that
is needed to detect aforementioned complex molecules) is much smaller than
that of H2-bearing gas (associated mostly with more diffuse cold medium) (ii)
the small fraction of radio bright QSOs, which are needed for the analysis
sensitive transition. Therefore, H2 will remain a main prove for variations of
$\mu$. By surveying H2, CUBES will bring new targets at different redshifts,
which are essential to decrease statistical errors as well as alleviating a
number of possible systematics related to the systems characteristics.
Additionally, accompanying detections of HD and CO molecules in H2 absorption
systems can also provide independent constrain on $\mu$ Ivanov2008 ; Darpa2016
; Dapra2017 ; Ubachs2019 . Moreover, as we mentioned above, the detection of
CO associated with H2 Srianand2008  provides a direct measurement of the CMB
temperature at high redshift Noterdaeme2011 , a fundamental test for the
adiabatic cooling of the Universe predicted by standard Big-Bang theory.

In short, while CUBES’s design will not permit direct use of these
astrophysical probes (because of a relatively short wavelength range and
because it is not intended to have exquisite wavelength calibration), CUBES
will be extremely effective in characterizing a large number of appropriate
H2-bearing DLAs, very much easing the selection for follow-up studies with
higher resolution and stable spectrographs.

###  4.5 Candidates

Figure 5: G-band magnitude and redshifts of H2 absorption systems at high
redshift. The blue and violet circles represent systems already confirmed and
analysed using UVES/HIRES and X-shooter, respectively. The red and orange
squares represent repectively the intervening and proximate H2 candidates,
pre-selected in the SDSS. The blue and red dashed lines indicate the typical
magnitudes of the quasars that can be reached with UVES 10 hrs and CUBES 1hr
exposure time, respectively, to achieve $\rm SNR\sim 10$ in the spectrum. The
redshift range accessible with CUBES is marked by the black horizontal solid
lines with arrows.

Fig. 5 shows the G-band magnitude - redshift diagram for the confirmed
(already followed-up) and candidates H2 absorption systems (preselected in
SDSS and that can be observed from Paranal). The intervening candidates were
compiled based on DR14 using the direct-H2 and C i pre-selection techniques
(for descriptions of techniques, see Ledoux2015 ; Balashev2014 ) and proximate
candidates were taken from Noterdaeme2019 . The improved efficiency of the
CUBES in comparison with UVES will allow to collect S/N$\sim 10$ spectra for
all available candidates in less than one hour per object, when UVES requires
already 10h for 2 magnitude fainter objects In addition, the wavelength range
of CUBES corresponds to redshifts for H2 lines that almost perfectly match the
redshift distribution of the candidates. This shows that we already have a
substantial number of the targets that can be straightforwardly follow-up with
CUBES with high efficiency to multiply the available samples of intervening
and proximate H2-bearing DLAs.

##  5 Conclusions

Investigating molecular hydrogen absorption systems at high-z provides an
extremely powerful probe of the physical conditions of the cold phase of the
interstellar medium in the overall population of remote galaxies. The stepped-
like progress in these studies has always been determined by observational
capabilities, where two evident boosts were associated with ($i$) the advent
of 8-m class telescopes (KECK and VLT) and $(ii)$ the development of efficient
pre-selection techniques in massive spectroscopic surveys (SDSS). We are now
facing a bottleneck since follow-up studies of the many targets identified in
the SDSS are faint and already reach the capabilities of current instruments.
This issue can be overcome with CUBES - the next generation efficient blue
spectrograph that is planned to be mounted on the VLT. We briefly discussed
three H2-related top-priority science cases for CUBES in our opinion: ($i$)
Studies of the thermal state of the cold ISM in the wide range of the
environments sampled by intervening DLAs, (ii) Detailed investigations of the
molecular gas in AGN outflows at high redshifts (iii) Probes of translucent
and dark molecular gas in absorption via hunting for the dusty and high-column
H2-bearing DLAs that can be exclusively be reached with CUBES. This, together
with the forthcoming generation of spectroscopic surveys, such as 4MOST (which
will provide better exploration of the sky accessible from Paranal) will
definitely provide the next boost in understanding the small-scale physics in
the gas over a wide range of redshifts and environments.

## Conflict of interests

The authors declare that they have no conflict of interest.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

###### Acknowledgements.

SB is supported by Russian Science Foundation grant 18-12-00301. PN is
supported by the French Agence Nationale de la Recherche under grant
ANR-17-CE31-0011 (“HIH2”).

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# Matrix Based Adaptive Short Block Cipher ††thanks: Citation: Authors. Title.
Pages…. DOI:000000/11111.

Awnon Bhowmik
Department of Mathematics
University of Central Florida
<EMAIL_ADDRESS>
###### Abstract

Every day, millions of credit cards are swiped and transactions are carried
out across the world. Due to numerous forms of unethical digital activities,
users are vulnerable to credit card fraud, phishing, identity theft, etc. This
paper outlines a novel block encryption algorithm involving multiple private
keys and a resilient trapdoor function that ensures data security while
maintaining an optimal run time and space complexity. The proposed scheme
consists of an irrepressible trapdoor based on a depressed cubic function and
a unique key generation algorithm that uses Fibonacci sequences and invertible
square matrices for improved security. The paper involves data obtained from
comprehensive crypt analysis exploiting the strengths and weaknesses of the
system and comments on its potential large-scale industry applications.

_K_ eywords Credit Card Security $\cdot$ block cipher $\cdot$ block encryption
algorithm

##  1 Introduction

Over the years, the new cryptosystems have gradually increased in obscurity
thus providing more diffusion and confusion. The essence of any cryptosystem
relies on some special mathematical trapdoor function that makes it
practically impossible for an unwelcome interceptor to gain access to
secretive information. This has brought about better security protocols that
are being used in everyday digital systems today. The wonderful fact is that
with time it has been realized that any mathematical concept can be used to
create trapdoors for cryptosystems as long as they are used appropriately.
These functions also ensure that the authorized parties (who know the secret
key) can continue sharing data among themselves. The rate of data exchange in
the present era requires trapdoor functions that are extremely fast and take
up the minimum memory resources while operating on data.

Credit card encryption is a security measure used to minimize the likelihood
of card information being stolen as it may lead to countless fraudulent
practices [1]. In the past, banks used the magnetic strip on the back of
credit cards to accept purchases. However, these security features became
heavily corrupted with malware during the fraud crisis of 2014. Once the
information is stolen, it can be put into a new strip and used to make
fraudulent purchases. Additionally, magnets can tamper with the magnetic strip
on credit cards, erasing the information stored there and rendering them
useless [2]. During an ordinary credit card transaction, there are mainly 3
steps where the card is vulnerable:

  1. 1.

The electronic terminal where the card is being scanned.

  2. 2.

The transmission of information between that terminal and its bank’s back-end
servers.

  3. 3.

The card itself, as flaws in internal components of the card, such as
electronic chip

##  2 Linear Algebra in Cryptography

Modern cryptography began only in the 1970s, and over the years many
fascinating cryptosystems have been developed by involving it with the various
mathematical concepts we are aware of, predominantly by Number Theory since
this is where we get to learn about prime numbers and cryptography stands on
the foundation of prime number distribution. Anyone finding the pattern to the
distribution of primes can erase the concept of cryptography from textbooks
and modern times forever. Not far behind number theory; linear algebra
concepts are no slouch themselves. Ranging from the connection between the
determinant and invertibility of a matrix to eigenvalues and eigenvectors as
components of potential trapdoor functions, linear algebra provides a wide
range of tools up for grabs [3]. The following talks about how it is used in
cryptography.

##  3 Encryption in credit card transactions

Credit cards constitute a straightforward and fast method of digital payment
scheme that allows the transfer of massive proportions of money of various
currencies and denominations. This makes credit cards an intrinsic component
of the payment process. Businesses generally have electronic terminals where
consumers can simply swipe or scan the cards and the appropriate amount of
money gets seamlessly transferred. The card has unique identification digits
stored on the onboard chip that gets sent to authorized computer servers where
the balance amount is re-evaluated after deducting the appropriate amount and
verification is carried out. There are a couple of mechanisms used by banks to
ensure secure transactions. Some of these include a private PIN number for
authentication, CVV numbers for online transactions and even encrypting the
magnetic strip at the back of the card. All these methods are effective,
however, there is still a need for a more efficient encryption scheme to
ensure that phishing, identity theft, and other forms of online money-based
fraud are. Credit card encryption makes it impossible for an unethical hacker
to access the card credentials without a special private key that ensures
seamless transactions between various authorized parties. As it turns out, the
proposed algorithm was observed to deliver superior performance in terms of
security and run time for shorter string lengths. This attribute makes this
system a feasible candidate for credit card encryption that could with
multiple applications in different stages of the process.

###  3.1 Summary of applications in Linear Algebra

  1. 1.

A matrix can be used as a cipher to encrypt a message.

     * •

The matrix must be invertible for use in decrypting.

  2. 2.

Cipher matrix can be as simple as a $3\times 3$ matrix composed of random
integers.

  3. 3.

To encrypt plaintext, each character in the plaintext must be denoted with a
numerical value and placed into a matrix.

     * •

These numbers can range in value, but an example is using $1-26$ to represent
A to Z and $27$ to represent a space.

  4. 4.

This matrix is then multiplied with the cipher matrix to form a new matrix
containing the ciphertext message.

###  3.2 Encrypting a message

  1. 1.

Each character of the message is given a numerical value as mentioned above.
These values are then separated into vectors, such that the number of rows of
each vector is equivalent to the number of rows of the cipher matrix.

  2. 2.

Values are placed into each vector one at a time, going down a row for each
value. Once a vector is filled the next vector is created. If the last vector
does not get filled by the plaintext then the remaining entries will hold the
value for a space.

  3. 3.

The vectors are then augmented to form a matrix that contains the plaintext.

  4. 4.

The plaintext matrix is then multiplied with the cipher matrix to create the
ciphertext matrix

###  3.3 Decrypting a message

  1. 1.

To decrypt a ciphertext matrix the original cipher matrix must be used. The
cipher matrix must be inverted to decrypt the ciphertext. This cipher matrix
can be used as the private key for decryption.

  2. 2.

This inverted cipher matrix is then multiplied with the ciphertext matrix.

  3. 3.

The product produces the original plaintext matrix.

  4. 4.

The plaintext can be found again by taking this product and splitting it back
up into its separate vectors and then converting the numbers back into their
corresponding alphabetic form.

##  4 Confusion and Diffusion

In cryptography, confusion and diffusion are two properties of the operation
of a secure cipher which were identified by Claude Shannon in his paper
Communication Theory of Secrecy Systems, published in 1949 [4]. In Shannon’s
original definitions, confusion refers to making the relationship between the
key and the ciphertext as complex and involved as possible [5]; diffusion
refers to the property that the redundancy in the statistics of the plaintext
is "dissipated" in the statistics of the ciphertext. In other words, the non-
uniformity in the distribution of the individual letters (and pairs of
neighboring letters) in the plaintext should be redistributed into the non-
uniformity in the distribution of much larger structures of the ciphertext,
which is much harder to detect. In diffusion, the statistical structure of the
plaintext is dissipated into long-range statistics of the ciphertext.
Diffusion means that the output bits should depend on the input bits in a very
complex way. In a cipher with good diffusion, if one bit of the plaintext is
changed, then the ciphertext should change completely, in an unpredictable or
pseudorandom manner. AES protocol is a perfect example that exhibits both
confusion and diffusion.

##  5 Trapdoor Function

The essence of any cryptosystem relies on some special mathematical trapdoor
function [6] that makes it practically impossible for an unwelcome interceptor
to gain access to secretive information. Simultaneously, these functions also
ensure that the authorized parties (who know the secret key) can continue
sharing data among themselves. A trapdoor function is a mathematical
transformation that is easy to compute in one direction, but extremely
difficult (practically impossible) to compute in the opposite direction in a
feasible time unless some special information is known (private key).
Analogously, this can be thought of as the lock and key in modern cryptography
where until and unless someone has access to the exact key, they can’t open
the lock. In mathematical terms, if $f$ is a trapdoor function, then $y=f(x)$
easy to calculate but $x=f^{-1}(y)$ is tremendously hard to compute without
some special knowledge $k$ (called key). In case $k$ is known, it becomes easy
to compute the inverse $x=f^{-1}(y,k)$.

##  6 Initial Encoding Procedure

The encoding procedure depends on a few results which are modified according
to fit the needs of this paper.

Cantor Pairing Function The Cantor Pairing Function is a primitive recursive
pairing function $\pi:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by

$\pi(k_{1},k_{2})=\dfrac{1}{2}(k_{1}+k_{2})(k_{1}+k_{2}+1)+k_{2}$ (1)

######  Theorem 1.

Product of $n$ consecutive integers is divisible by $n$.

###### Proof.

By contradiction. Suppose that none of the $n$ consecutive integers is
divisible by $n$. There are $n-1$ different remainders upon division by =
while $n$ remainders are needed. Therefore, by the pigeonhole principle, two
of the divisions have the same remainder. But that would imply that the
difference in those dividends is divisible by $n$ and hence is $n$. But the
largest difference is $n-1$. Contradiction. ∎

The algorithm starts off with a list of ASCII values and a list of unique
primes. The process is as follows.

Encode

  1. 1.

Let $x$ be the ASCII value and $y$ is a prime. Compute

$t=\dfrac{(x+y-1)(x+y)(x+y+1)}{6}$ (2)

  2. 2.

Setting $n=x+y$ yields $t=\dfrac{n^{3}-n}{6}$ and $n^{3}-n\equiv 0\pmod{6}$ by
Theorem 1.

Decode

  1. 1.

Given $t$, we solve $n^{3}-n-6t=0$ for $n$.

  2. 2.

The prime number $y$ is known, so $x=n-y$ returns the ASCII data.

##  7 Remarks on the cubic equation

A cubic equation

$ax^{3}+bx^{2}+cx+d=0$ (3)

can be converted to a _depressed_ cubic equation via small substitutions to
attain the form of

$x^{3}+mx+n=0$ (4)

. For this equation, it is required that $27n^{2}+4m^{3}\neq 0$ to avoid
singularities. In our case, $n^{3}-n-6t=0$, we require

$\displaystyle 27(-6t)^{2}+4(-1)^{3}$ $\displaystyle\neq 0$ $\displaystyle
27\cdot 36t^{2}-4$ $\displaystyle\neq 0$ $\displaystyle 243t^{2}-1$
$\displaystyle\neq 0$ $\displaystyle t$
$\displaystyle\neq\dfrac{\sqrt{3}}{27}$

Since $t\in\mathbb{Z}$, this is impossible.

##  8 An example demonstrating the initial encoding procedure

Suppose the input character is ’A’ which has an ASCII value of $65$, and the
corresponding prime is $13$. The following calculations take place

_Encode_

$t=\dfrac{1}{6}(65+13-1)(65+13)(65+13+1)=79079$

_Decode_

$\displaystyle n^{3}-n-6t$ $\displaystyle=0$ $\displaystyle n^{3}-n-6(79079)$
$\displaystyle=0$ $\displaystyle n^{3}-n-474474$ $\displaystyle=0$

Let $n=\alpha+\beta$, then

$\displaystyle n^{3}$ $\displaystyle=(\alpha+\beta)^{3}$
$\displaystyle=\alpha^{3}+\beta^{3}+3\alpha\beta(\alpha+\beta)$
$\displaystyle=\alpha^{3}+\beta^{3}+3\alpha\beta n$ $\displaystyle
n^{3}-3\alpha\beta n-(\alpha^{3}+\beta^{3})$ $\displaystyle=0$

Comparing coefficients, we have

$\begin{split}3\alpha\beta&=1\\\
\alpha\beta&=\dfrac{1}{3}\end{split}\qquad\begin{split}\alpha^{3}+\beta^{3}&=474474\end{split}$

Generating a quadratic equation with the roots $\alpha^{3}$ and $\beta^{3}$

$\displaystyle s^{2}-(\alpha^{3}+\beta^{3})s+\alpha^{3}\beta^{3}$
$\displaystyle=0$ $\displaystyle s^{2}-474474s+\dfrac{1}{27}$
$\displaystyle=0$ $\displaystyle s$
$\displaystyle=\dfrac{474474\pm\sqrt{474474^{2}-\dfrac{4}{27}}}{2}$
$\displaystyle s$ $\displaystyle=(\alpha^{3},\beta^{3})=474474,7.9\times
10^{-8}$ $\displaystyle n$
$\displaystyle=\alpha+\beta=\sqrt[3]{474474}+\sqrt[3]{7.9\times 10^{-8}}=78$

Here is a Mathematica notebook demonstration.

⬇

In: x:=65

y:=13

n:=x+y

t:=(n^3-n)/6

Solve[r^3-r-6t==0,r,Integers]

Out:{{r->78}}

In: 78-y

Out: 65

In: ClearAll["Global‘*"]

In: f[x_]:=x^3-x-6t

In: f’[x]

Out: -1+3x^2

In: Reduce[f’[x]>0,x]

Out:

$x<-\frac{1}{\sqrt{3}}\lor x>\frac{1}{\sqrt{3}}$

This implies that the notebook demonstration also showed that a given function

$f(x)=x^{3}-x-6t$

exhibits an increasing behavior on
$x\in\left(\dfrac{1}{\sqrt{3}},\infty\right)$. Since $x,y>0$, and $n=x+y$, we
have $n>0$, and so $f(n)=n^{3}-n-6t$ will always be increasing on
$\mathbb{Z}^{+}$.

##  9 Proposed algorithm

###  9.1 Encryption

  1. 1.

Input a message string

  2. 2.

Generate an array $A$ of ASCII values and another array $P$ containing unique
pseudorandom primes.

  3. 3.

Take the ASCII values and their corresponding mapped prime and apply the
encoding scheme described in section 6.

  4. 4.

Put these values into matrix $B$ and split them into blocks of $4$. Apply
’$0$’ padding to whichever $1\times 4$ array isn’t completely filled with
entries. Convert each $1\times 4$ array into a $2\times 2$ matrix. Suppose
each block matrix is $B_{i}$.

  5. 5.

Generate a $2\times 2$ Fibonacci matrix $Q^{n}$ [7], where

$Q^{n}=\begin{bmatrix}F_{n+1}&F_{n}\\\ F_{n}&F_{n-1}\end{bmatrix}$

and perform

$C=B\cdot Q^{n}$

  6. 6.

Choose an integer $n$ such that $\theta=\dfrac{n\pi}{2}$ to generate a
rotation matrix $R$ such that

$R=\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix}$

and perform

$\displaystyle D$ $\displaystyle=C\cdot R$ $\displaystyle D$
$\displaystyle=D^{T}$

  7. 7.

Choose a $2\times 2$ _invertible_ secret key $K$ and perform encryption

$E=D\cdot K$

After the final step, the result will alter drastically depending on the
contents of the original string, thus demonstrating _confusion_. On the other
hand, the choice of a different key for the same message string will
completely alter the result, thus demonstrating _diffusion_.

###  9.2 Decryption

  1. 1.

Perform $D=E\cdot K^{-1}$

  2. 2.

Perform transpose $(D^{T})^{T}=D$

  3. 3.

Perform $C=D\cdot R^{-1}$

  4. 4.

Perform $B=C\cdot(Q^{n})^{-1}$

  5. 5.

Apply the decoding scheme described in section 6 to generate the ASCII data.

  6. 6.

Convert each $2\times 2$ block to $1\times 4$ array. Remove padding.

This system can probably be extended to a $n\times n$ block size with numerous
tweaks in the proposed algorithm while dealing with exponential runtime
complexity.

##  10 Properties of determinants

Determinants play an important part in this cryptosystem so it is essential to
recall their properties. For any arbitrary square matrix $A$

  1. 1.

$\det(A)=\det(A^{T})$

  2. 2.

For any arbitrary constant $c$, $\det(cA)=c^{n}\det(A)$

##  11 Few remarks on R and K

The $2D$ rotation matrix is given by

$R=\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix}$

It is to be noted that $det(R)\neq 0$. It means the sender can choose any
$\theta\in\mathbb{R}$ to encrypt the data. The only drawback is that the sine
and cosine ratios can provide only three integral values from
$\left\\{-1,0,1\right\\}$. Any other angles will raise floating point
precision errors which will amount to incorrect decryption of the data.

A rotation matrix is a square matrix with real entries, and also an orthogonal
matrix with a determinant of $1$. A matrix $R$ is orthogonal iff

$R^{T}R=RR^{T}=I$

which consequently means that

$R^{-1}=R^{T}$

This can make things a bit simpler while coding up the algorithm.

An arbitrarily chosen matrix $K$ should be checked for invertibility since it
is used in the decryption procedure. There can be a special case where $K$
itself is a rotation matrix. Then

$K^{-1}=K^{T}$

can speed up the process.
It is not necessary to choose $K$ as a square matrix at all. Suppose we choose
a $m\times n$ matrix $M$, so $M^{T}$ would be of order $n\times m$. The
product $M^{T}M$ will be a square symmetric matrix of order $m\times m$.

If $M$ is chosen as skew-symmetric, then

$\displaystyle M$ $\displaystyle=M^{T}\qquad\text{By property 1}$
$\displaystyle\det(M)$ $\displaystyle=\det(M^{T})$ $\displaystyle\det(M)$
$\displaystyle=\det(-M)$ $\displaystyle\det(M)$
$\displaystyle=(-1)^{n}\det(M)$ $\displaystyle\det(M)$
$\displaystyle=\det(M)\qquad\text{for odd }n$ $\displaystyle 2\det(M)$
$\displaystyle=0$ $\displaystyle\det(M)$ $\displaystyle=0$

Hence, for an odd-order square matrix, $M$ shouldn’t be chosen as skew
symmetric to avoid the singularity.

##  12 Automated generation of key matrix K

The user inputting a $2\times 2$ matrix that has to always account for its
invertibility by calculating its determinant. A singular matrix has a
determinant of $0$ and this must be avoided since $K^{-1}$ is required during
decryption. To overcome this hassle, a random $2\times 2$ matrix with integer
entries can be checked for non-singularity by checking its column
independence.

Given an arbitrary matrix with integer entries

$K=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$

For arbitrary scalars $k_{1},k_{2}$, we are required to solve the following
system

$k\begin{bmatrix}a\\\ c\end{bmatrix}+k_{2}\begin{bmatrix}b\\\
d\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}$

which is equivalent to the linear system

$\begin{cases}k_{1}a+k_{2}b&=0\\\ k_{1}c+k_{2}d&=0\end{cases}$

If $k_{1}=k_{2}=0$, the matrix $K$ is column independent and thereby non-
singular.

##  13 Conclusion

This paper introduced a new encryption scheme that was inspired by the Hill
Cipher. A couple of edge cases were disclosed and further study is required to
make this system even better. The runtime complexity can be safely assumed to
be exponential or that it would reach exponential complexity as the length of
the string increases or the other parameters are varied. Currently, this
system may not be perfect for use in the industry but it is a system
nevertheless. Further study might give insights into other problems or
drawbacks of the system and appropriate countermeasures can be taken to make
it good enough to be used in the industry. Our work so far has been compiled
into a GitHub repository available here [8].

## References

  * [1] Brian Whitworth.  Fraud resistant credit card using encryption, encrypted cards on computing devices, October 25 2001.  US Patent App. 09/764,369.
  * [2] Ralph J Oldenkamp.  Method for tamper-proofing magnetic stripe card reader, March 30 1982\.  US Patent 4,322,613.
  * [3] Othmar Staffelbach and Willi Meier.  Cryptographic significance of the carry for ciphers based on integer addition.  In Conference on the Theory and Application of Cryptography, pages 602–614. Springer, 1990.
  * [4] Claude E Shannon.  Communication theory of secrecy systems.  The Bell system technical journal, 28(4):656–715, 1949.
  * [5] William Stallings.  Cryptography and network security, 4/E.  Pearson Education India, 2006.
  * [6] Andrew C Yao.  Theory and application of trapdoor functions.  In 23rd Annual Symposium on Foundations of Computer Science (SFCS 1982), pages 80–91. IEEE, 1982.
  * [7] Henry W Gould.  A history of the fibonacci q-matrix and a higher-dimensional problem.  Fibonacci Quart, 19(3):250–257, 1981.
  * [8] Awnon Bhowmik and Unnikrishnan Menon.  https://github.com/awnonbhowmik/adaptive-matrix-based-short-block-cipher, Aug 2020\.