# Statistical tests of young radio pulsars with/without supernova remnants:
implying two origins of neutron stars

Xiang-Han Cui1,2, Cheng-Min Zhang1,2,3, Di Li1,2,4, Jian-Wei Zhang1, Bo
Peng1,2,5, Wei-Wei Zhu1,2, Qing-Dong Wu6, Shuang-Qiang Wang6, Na Wang6, De-Hua
Wang7, Yi-Yan Yang8, Zhen-Qi Diao7, Chang-Qing Ye7, and Hsiang-Kuang Chang9
1National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China
2School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China
3School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
4NAOC-UKZN Computational Astrophysics Centre, University of KwaZulu-Natal,
Durban 4000, South Africa
5Guizhou Radio Astronomy Observatory, Chinese Academy of Sciences, Guiyang
550025, China
6Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi,
Xinjiang 830011, China
7School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550001, China
8School of Physics and Electronic Science, Guizhou Education University,
Guiyang 550018, China
9Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan,
China zhangcm@bao.ac.cn(CMZ)

(Accepted XXX. Received YYY; in original form 2021)

###### Abstract

The properties of the young pulsars and their relations to the supernova
remnants (SNRs) have been the interesting topics. At present, 383 SNRs in the
Milky Way galaxy have been published, which are associated with 64 radio
pulsars and 46 pulsars with high energy emissions. However, we noticed that
630 young radio pulsars with spin periods of less than half a second have been
not yet observed the SNRs surrounding or nearby them, which arises a question
of that could the two types of young radio pulsars with/without SNRs hold
distinctive characteristics? Here, we employ the statistical tests on the two
groups of young radio pulsars with (52) and without (630) SNRs to reveal if
they share different origins. Kolmogorov-Smirnov (K-S) and Mann-Whitney-
Wilcoxon (M-W-W) tests indicate that the two samples have the different
distributions with parameters of spin period ($P$), derivative of spin period
($\dot{P}$), surface magnetic field strength ($B$), and energy loss rate
($\dot{E}$). Meanwhile, the cumulative number ratio between the pulsars with
and without SNRs at the different spindown ages decreases significantly after
$\rm 10-20\,Kyr$. So we propose that the existence of the two types of
supernovae (SNe), corresponding to their SNR lifetimes, which can be roughly
ascribed to the low-energy and high-energy SNe. Furthermore, the low-energy
SNe may be formed from the $\rm 8-12\,M_{\odot}$ progenitor, e.g., possibly
experiencing the electron capture, while the main sequence stars of $\rm
12-25\,M_{\odot}$ may produce the high-energy SNe probably by the iron core
collapse.

###### keywords:

pulsars: general - stars: neutron - supernovae: general - methods: statistical

††pubyear: 2021††pagerange: Statistical tests of young radio pulsars
with/without supernova remnants: implying two origins of neutron
stars–Appendix C: Tests of spin period and its derivative with the different
ages

##  1 Introduction

The associations between the pulsars and supernova remnants (SNRs) have been
of considerable interest topics in neutron stars (NSs) astrophysics, since the
radio pulses were firstly observed in the Crab Nebula in 1968 (Staelin &
Reifenstein 1968), as well as the discovery of the Vela pulsar (Large,
Vaughan, & Mills 1968). The story is continuing with identifying the potential
NS in SN 1987A (Page et al. 2020; Greco et al. 2021; Soker 2021). Thanks the
efforts and developments of astronomical facilities in recent years, the
number of pulsars and SNRs has increased significantly. Up to now, there are
more than 3000 pulsars
(ATNF111https://www.atnf.csiro.au/research/pulsar/psrcat/: 2781,
GPPS222http://zmtt.bao.ac.cn/GPPS/: 201 (Han et al. 2021),
CRAFT333https://crafts.bao.ac.cn/: 125) observed in the radio band and 383
SNRs (SNRcat444http://snrcat.physics.umanitoba.ca/SNRtable.php) in the Milky
Way galaxy had been published (Ferrand & Safi-Harb 2012). Among these, there
are 110 pulsars that have been identified in SNRs, including 6 anomalous X-ray
pulsars (AXPs), 5 soft gamma-ray repeaters (SGRs), a total of 13 magnetar
candidates, and 15 central compact objects (CCOs) or CCO candidates. By
comparing with ANTF Pulsar Catalogue, there are also 18 pulsars without radio
emissions (NRAD) in SNRcat, only observed at the infrared or higher
frequencies (Manchester et al. 2005). In short, 64 radio pulsars with SNRs
have been published, as seen in Table 1. Interestingly, the number ratio
between the radio pulsars (64) and SNRs (337) is about 1/5, which is
consistent with the estimation by the beaming fraction of radio pulsars
(Taylor & Manchester 1977; Lorimer et al. 1993; Lorimer & Kramer 2012).

Table 1: List of various types of pulsars with SNRs

Source | Number | Ref.
---|---|---
SNR | 383 | [1]
Pulsara | 110 | [1]
Radio pulsar | 64b | [2]
Magnetar or candidate | 13c | [1, 3, 4]
CCOd or candidate | 15 | [1]
NRADe | 18 | [2]

Note:
a Various pulsars with SNRs.

b The total number of radio pulsars with SNRs is 64, but only 52 of them are
analyzed in this article (selection details in Section 2.1).

c AXP (anomalous X-ray pulsar): 6, SGR (soft gamma-ray repeater): 5.
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html

d CCO: central compact object.

e Pulsars without radio emissions and do not contain the magnetars, CCOs, and
their candidates.

Ref.: [1] Ferrand & Safi-Harb (2012); [2] Manchester et al. (2005); [3]
Olausen & Kaspi (2014); [4] Esposito, Rea, & Israel (2021).

From Table 1, the question of why only a fraction of SNRs have been found with
pulsars can be solved by their beaming cone angles across the earth.
Meanwhile, some other explanations are worthy of mentioning as pointed out as
below. First of all, not every SNe could generate a NS or some NSs might not
be detectable as pulsars (Radhakrishnan & Srinivasan 1980; Srinivasan,
Bhattacharya, & Dwarakanath 1984; Manchester 1987; Narayan & Schaudt 1988).
Next, the flux density or luminosity of young radio pulsars may be
overestimated, implying that some young radio pulsars are too faint to be
observed in some SNRs (Stollman 1987; Lorimer et al. 1993). Finally, the kick
velocity of pulsar may be quite high after birth, which could result in the
pulsars to escape from their SNRs (Frail, Goss, & Whiteoak 1994).

Although some pulsars can be found in the pulsar wind nebulae (Gaensler &
Slane 2006), it is still an open question that so many young radio pulsars
have not seen their SNRs. Nowadays, there exist 630 young radio pulsars (spin
period less than 0.5 s, and details as described in Section 2) without SNRs,
which provides us a new aspect to further study the association between
pulsars and SNRs, as well as the NS origins. In the former studies of X-ray
pulsars (Knigge, Coe, & Podsiadlowski 2011) and double neutron star (DNS)
(Yang et al. 2019) population, researchers suggested that the NSs may birth
from the different origins, i.e., the existence of the electron capture and
iron core collapse SNe (Janka 2012). In the recent work about pulsars and
SNRs, Malov (2021) noticed that the mean values of radio luminosity of pulsars
observed inside and outside SNRs are significantly different with one order of
magnitude.

Inspired by these researches, we conjecture that there may exist two origins
for radio pulsars. Therefore, we attempt to use the statistical methods to
study the properties of radio pulsars and their relations with SNRs. We apply
some physical criteria to select two samples of pulsars, that is, the radio
pulsars with SNR (SNR-PSRs, 52) and without SNR (non SNR-PSRs, 630) (see
Section 2.1 for details) By drawing the cumulative distribution function (CDF)
and utilizing the Kolmogorov-Smirnov (K-S) and Mann-Whitney-Wilcoxon (M-W-W)
tests for these two samples, we find that the obtained results support the
different distributions for two samples. Based on spin period evolution model,
we estimate the spindown age (not characteristic age) of radio pulsars in
these two samples. After a further discussion, it is inferred that the two
samples of pulsars may origin from two types of progenitors, such as the low-
energy and high-energy SNe (e.g., electron capture and iron core collapse),
respectively. However, the energy boundary is still unclear, or there may be
an overlapping part between them, because some researches in SNRs indicated
that SNRs may have a continuous energy distribution like lognormal (Leahy
2017; Leahy, Ranasinghe, & Gelowitz 2020). Meanwhile, we notice that the
cumulative number ratios of SNR-PSRs to non SNR-PSRs are decreased quickly
after $\rm 10-20\,Kyr$. Finally, according to the initial mass function (IMF)
or Salpeter function (Salpeter 1955), it is possible to statistically
distinguish the two types of their progenitor stars at the mass boundary of
$\rm\sim 12\,M_{\odot}$.

The structure of our paper is presented as follows. In Section 2, we describe
the data selection of pulsars and introduce the spin period evolution model.
In Section 3, we apply the statistical tests on the two samples and analyze
the results. Finally, in Section 4, we discuss a possible physical
significance for the two distributions of the pulsars, and main conclusions
are summarized also.

##  2 Data and Model

###  2.1 Data Selection

Our data are taken from ATNF Pulsar Catalogue (Manchester et al. 2005) and
SNRcat (Ferrand & Safi-Harb 2012). Data selections are made to construct the
two samples of the young radio pulsars with and without SNRs, and the
selection criteria are described below.

1) We choose the pulsar samples in the two data bases with the spin periods
($P$) less than 0.5 s. Due to the short duration of SNRs, the pulsars with
SNRs are taken as the young pulsars (Gaensler & Johnston 1995; Malov 2021).
Considering that the characteristic age of pulsar usually has a significant
error compared to the real age (Lai 1996; Kaspi et al. 2001; Tian & Leahy
2006; Lyne & Graham-Smith 2012), we apply the spin period to represent the age
under the spin period evolution model (details in Section 2.2). Generally
speaking, the lifetime of SNRs usually less than 300 Kyr, so the spin period
of pulsars may be around 0.5 s based on the magnetic dipole model (see Section
2.2 and Appendix A for calculation details). Meanwhile, for all radio pulsars,
the median of period is about 0.5 s, which can be known from ATNF database
(Manchester et al. 2005). Additionally, because the periods for most rotating
radio transients (RRATs) and intermittent pulsars are greater than 0.5 s
(McLaughlin et al. 2006; Kramer et al. 2006), we eliminate these special radio
pulsars from our samples. Therefore, we regard the radio pulsars with spin
periods of less than 0.5 s to be the young pulsar, and they are roughly
consistent with the arguments based on that their magnetic fields are
comparable with those measured by cyclotron absorption lines of X-rays (Ye et
al. 2019).

2) The data with the surface magnetic field strength ($B$) ranged from $\rm
10^{11}\,G$ to $\rm 10^{14}\,G$ are used. If the pulsar’s B-field is lower
than $\rm 10^{11}\,G$, it may be a millisecond pulsar (MSP) (Bhattacharya &
van den Heuvel 1991; Lorimer 2008) or a CCO (Halpern & Gotthelf 2010;
Gotthelf, Halpern, & Alford 2013). If the pulsar’s B-field is higher than $\rm
10^{14}\,G$, it may be a magnetar (Duncan & Thompson 1992; Ferrario &
Wickramasinghe 2008; Kaspi & Beloborodov 2017; Esposito, Rea, & Israel 2021).

3) We select the data to satisfy that their spin periods and magnetic fields
are distributed above the spin-up line in B-P diagram (Bhattacharya & van den
Heuvel 1991). While, the pulsars below the spin-up line may have a more
complicated evolutionary tracks, such as experiencing the accretion spin-up in
binary systems (Zhang & Kojima 2006).

4) Some types of the special pulsars are removed from our samples, including
the MSPs, magnetars, recycled pulsars (Zhang & Kojima 2006), CCOs, RRATs,
intermittent pulsars and NRADs. The physical characteristics between the young
radio pulsars and these special pulsars are significantly different, so they
may follow the different birth conditions and evolutionary paths.

After setting these selections ($P<0.5s$, $10^{11}G<B<10^{14}G$, above the
spin-up line, and removing the special pulsars), we obtain the two groups of
samples of 52 SNR-PSRs and 630 non SNR-PSRs, respectively, as illustrated in
Figure 1.

Figure 1: Data distribution graph in the magnetic field versus spin period
(B-P) diagram after the selection criteria. The blue stars and orange dots
stand for the SNR-PSRs and non SNR-PSRs, respectively. The green dashed lines
represent the constraint lines of surface B-field strength (B) upper
($B_{upper}=\rm 10^{14}\,G$) and lower ($B_{lower}=\rm 10^{11}\,G$) limit and
spin period ($P<0.5\,\rm s$). The red solid line represents the spin-up line
of accretion pulsars in binaries (Bhattacharya and van den Heuvel 1991). The
red, green and blue area stand for the approximate ranges of the magnetars,
recycled pulsars and millisecond pulsars (MSPs).

###  2.2 The Model

Because of employing the spin period to represent the pulsar’s age (spindown
age), we briefly introduce the model(Shapiro & Teukolsky 1983; Camilo,
Thorsett, & Kulkarni 1994; Lorimer & Kramer 2012; Lyne & Graham-Smith 2012).
The energy loss rate of a spin-powered pulsar can be expressed as,
$\dot{E}={dE}/{dt}={d(I\Omega^{2}/2)}/{dt}=-I\Omega\dot{\Omega}$, where $E$ is
total energy of pulsar, $\dot{E}$ is energy loss rate, $\Omega$ and
$\dot{\Omega}$ are the spin angular velocity and its derivative, and $I$ is
moment of inertia (Shapiro & Teukolsky 1983). The pulsar kinetic energy and
the radiation energy loss rate are equal, then we have,
$I\Omega\dot{\Omega}=k\Omega^{\rm n+1}$, with $k$ a coefficient, where the
braking index n=3 for the magnetic dipole model. The spin period evolution
equation can be written as (calculation details in Appendix A)

$\begin{split}P(t)=P_{0}\left(\frac{t}{\tau}_{0}+1\right)^{1/(\rm
n-1)},\end{split}$ (1)

where $P_{0}$, $\dot{P}_{0}$, and $\tau_{0}$ are the spin period, the
derivative of spin period and the spindown age at present, respectively. The
spindown age of a pulsar is defined as $\tau={P}/{[({\rm n-1})\dot{P}]}$
(Camilo, Thorsett, & Kulkarni 1994). When $t\gg\tau_{0}$ ($P\gg P_{0}$), the
spin period evolution equation can be expressed as

$\begin{split}P(t)\approx P_{0}\left(\frac{t}{\tau}_{0}\right)^{1/(\rm
n-1)}.\end{split}$ (2)

With n=3 and parameter $k=-{B_{p}^{2}R^{6}}/{(6c^{3})}$ is obtained by the
magnetic dipole model, where $B_{p}$ is polar magnetic, $R$ is NS radius, and
$c$ is speed of light (Shapiro & Teukolsky 1983). Then the Eq.(2) can be
written as $P(t)\approx P_{0}(t/\tau_{0})^{1/2}$. Because n=3 is a theoretical
value and the measured values of many pulsars are generally less than it
(Johnston & Galloway 1999; Kou & Tong 2015), we take n=2.7 to estimate the
ages of pulsars, and the reasons are interpreted as follows. Firstly, Lyne,
Manchester, & Taylor (1985) and Lorimer (2004) estimated that the number of
observable pulsars in Milky Way galaxy is $\sim 2-7\times 10^{4}$, which
corresponds to the relevant life time of radio pulsars to be $\rm\sim
20\,Myr$. If the braking index n is too small, like n=2.1, the corresponding
life time of pulsars will decrease to one or two million years to reach the
observational limit spin period of about 10 s. This will result in too few
observable pulsars, which is inconsistent with the observational facts at
present. Therefore, it is the reason that we do not apply the Crab pulsar’s
breaking index as an average value for the whole pulsar samples, which is from
2.1 to 2.5 or 2.6 (Lyne et al. 2015; Čadež et al. 2016). Secondly, almost all
radio pulsar periods are less than 10s (only two are longer than 10s, e.g.,
J2251-3711 with 12.1s (Morello et al. 2020) and J0250+5854 with 23.5s (Tan et
al. 2018)). By considering the above two reasons, we take the braking index
n=2.7 as a statistical value to estimate the evolutionary time, which can give
the spin period to be about 10s after evolving about 20 Myrs. Then we select
the initial conditions of the Crab-like pulsars to perform the calculations,
e.g., $P_{0}=0.033$ s and $\dot{P}_{0}=4.21\times 10^{-13}\,\rm ss^{-1}$
(Goldreich & Julian 1969; Lyne et al. 2015), implying $\tau_{02.7}=1462$ yr
(n=2.7) and $\tau_{03}=1243$ yr (n=3, characteristic age). The spin period
evolutionary curves are plotted in Figure 2 based on Eq. (2). Here, we point
out that, although initial spin periods of neutron stars may be not like that
of the Crab pulsar’s (Popov & Turolla 2012; Igoshev & Popov 2013), we still
insist to employ the Crab pulsar as a reference sample since it is the sole
pulsar with the known real age.

Figure 2: The diagram of spin period evolution with various braking index.
Different colors label the evolution curves with the breaking index n=2.1
(green), n=2.7 (red), and n=3 (blue), respectively. The orange dashed
horizontal line shows the required time for a pulsar period evolving to 10s.

##  3 Statistics and Results

In the following analysis, we employ the two statistical tests, K-S and M-W-W
(Yang et al. 2019; Cui et al. 2021), to check whether two samples hold the
same distribution. The test results are represented by the parameters,
p-values, and the procedure is described in the following. If p-value is less
than 0.05, it indicates that this test rejects the null hypothesis (the two
samples have the same distribution) at 5% significance level.

Figure 3: The Cumulative distribution function (CDF) of spin period for SNR-
PSRs and non SNR-PSRs. The dashed (solid) line stands for SNR-PSRs (non SNR-
PSRs).

In order to see whether the two groups of samples of SNR-PSRs and non SNR-PSRs
hold the same distribution, we draw the CDF curves in Figure 3, where the two
curves are conspicuously separated to each other. To show the difference of
the two samples more quantitatively, we apply the K-S test and M-W-W test, and
the p-values of these two tests are as low as $1.98\times 10^{-12}$ and
$5.85\times 10^{-13}$. The very low p-values indicate that the distributions
of the two samples should have the different origins. Therefore, with the
results of two CDF curves and p-values, we believe that the two samples may
come from the different statistical distributions. The further test results
for the other parameters are shown in Appendix B.

Figure 4: The cumulative number distribution of SNR-PSRs and non SNR-PSRs in
different ages. The blue (orange) solid lines stand for SNR-PSRs (non SNR-
PSRs) under n=2.7, and dashed lines stand for the cases of n=3.

Next, we draw a cumulative number distribution (Figure 4) for the different
ages that are estimated by Eq.(2). Interestingly, for the age less than $\sim$
10 Kyr the two curves of SNR-PSRs and non SNR-PSRs coincide together, however,
after $\rm 10-20\,Kyr$, the two curves are drifted away. The cumulative number
ratio, expressed as $N_{snr}/N_{nsnr}$, for the two samples with the different
ages probably can infer a fact that the two samples hold the different
origins.

In order to see the variation of this ratio between two samples more
intuitively, we plot the different ratio values from the age of 5 Kyr to 100
Kyr in Figure 5. We find that, before the age of 25 Kyr, we obtain 5 ratio
points for n=2.7 and n=3. While, from 25 Kyr to 100 Kyr, only 2 ratio points
are obtained because of the inadequate data. From the data point of these
ratio values, we obtain that the number ratio between SNR-PSR and non SNR-PSR
is close to unity at the age of $\sim$ 10 Kyr. However, after $\rm
10-20\,Kyr$, there exists a sharp decline in the ratio values. Meanwhile, for
the ratio value after $\sim$ 10 Kyr, with n=2.7 (n=3), we obtain a relation
between the ratio and the age $\mathcal{\phi}_{2.7}=1.1(t/10kyr)^{-1.01}$
($\mathcal{\phi}_{3}=1.2(t/10kyr)^{-0.91}$) and goodness of fit
$\mathcal{R}_{2.7}^{2}=0.998$ ($\mathcal{R}_{3}^{2}=0.996$), as described in
figure 5.

Figure 5: The evolution diagram of cumulative number ratio between SNR-PSRs
($N_{snr}$) and non SNR-PSRs ($N_{nsnr}$). With n=2.7 (n=3), the orange stars
(green triangles) stand for the ratio in different ages, and the solid blue
(red) line is the fitting curve of these ratio points.

##  4 Discussions and Conclusions

On the reasons for a sharp drop of the cumulative number ratio between the
SNR-PSRs and non-SNR-PSRs, after $\rm 10-20\,Kyr$, together with the the
different distributions, we think that there may exist two ways of birth for
radio pulsars. The long aged SNR-PSRs may be involved in the high-energy SNe,
whereas the non SNR-PSRs may be related to the low energy cases with short
duration. For high-mass stars, their SNRs may survive longer time.
Correspondingly, the low-energy SNe may be involved in the explosions of the
low-mass progenitor stars, in which the SNRs may last a shorter duration.

However, can the effect of decline in cumulative number ratio is caused by the
age difference? For example, Leahy, Ranasinghe, & Gelowitz (2020) analysed a
15-Galactic-SNR sample with the best-fit mean energy of $\rm 2.7\times
10^{50}\,erg$, and they concluded that SNRs become incomplete and hard to
identify after 30 Kyr. In order to discuss this question more clearly, we need
to test for samples with the different spindown ages. According to Figures 4
and 5, the two curves are separated after about $\rm 10-20\,Kyr$ for various
braking index. So, we can take 10 Kyr as the critical boundary and divide SNR-
PSRs and non SNR-PSRs into two groups by the ages of less and greater than 10
Kyr. For the pulsar ages less than 10 Kyr, the $P$ distributions of SNR-PSRs
and non SNR-PSRs are same, while their $\dot{P}$ distributions are different.
However, for the pulsar ages more than 10 Kyr, both distributions of $P$ and
$\dot{P}$ are different (details in Appendix C). These results may infer that
their initial periods are independent on the types of pulsars, but their
braking mechanisms are different. The birth of the NSs spin are due to the
transfer of angular momentum of progenitors to NSs (Lyne & Graham-Smith 2012),
then the energy of SN explosions might little effect on their spin periods.
Meanwhile, the difference in $\dot{P}$ will affect the evolution of $P$,
resulting in a different distribution of $P$ after 10 Kyr. The above evidence
support that their intrinsic differences may lead to different distribution
among SNR-PSRs and non SNR-PSRs. Specifically, when radio pulsars are just
born, the differences of progenitor mass or explosion energy may create
differences in the two groups. The effect of age is to amplify the differences
during the evolution, which origin from different initial $\dot{P}$. Thus,
this indicates that the differences between SNR-PSRs and non SNR-PSRs in
Figures 3, 4, and 5 are more likely to be caused by multiple reasons, rather
than only an age difference. The reasons may include different neutron star
generation mechanisms and evolution over time.

We need to emphasize that the duration of $\rm 10-20\,Kyr$ is not a strict
time, but a statistical value. The result based on SNR evolutionary models
(Leahy, Ranasinghe, & Gelowitz 2020) may be slightly different from our
result. Although our results ($\rm 10-20\,Kyr$) are not completely the same
with their (30 Kyr), at least it shows a life time boundary of two type SNRs,
no matter from the perspectives of pulsars in radio and SNRs in X-ray. It can
be inferred from this boundary that two types of pulsars generated by two
types of SNRs can be roughly distinguished. Therefore, the cumulative number
ratio of $\sim 1$ at $\sim$ 10 Kyr may represent the ratio of these two kinds
of pulsar production, that is

$\begin{split}\psi=\frac{N_{snr}}{N_{nsnr}}\sim\frac{N_{high-energy}}{N_{low-
energy}}\sim\frac{N_{high-mass}}{N_{low-mass}}\sim 1,\end{split}$ (3)

where $N_{snr}$ ($N_{nsnr}$), $N_{high-energy}$ ($N_{low-energy}$) and
$N_{high-mass}$ ($N_{low-mass}$) represent the numbers of SNR-PSRs (non SNR-
PSRs), high (low)energy SNe, and high (low) mass stars. Specifically for the
Crab pulsar, it may born from a low-energy SN ($\sim\rm 10^{50}\,erg$) and
low-mass progenitor star (Yang & Chevalier 2015). However, the earlier view
believed that the Crab Nebula has a more extended remnant (Chevalier 1977),
which indicates a higher energy ($\sim\rm 10^{51}\,erg$). Although no medium
have been detected in the surrounding area at radio or X-ray band (Frail et
al. 1995; Seward, Gorenstein, & Smith 2006), it also reminds us that total
energy of the Crab Nebula is still an open question.

The mass range of the progenitor stars for NS formations approximately lies in
$\rm 8-25\,M_{\odot}$ (Arnett & Schramm 1973; Miyaji et al. 1980; Heger et al.
2003). If we consider the initial mass function (IMF) by Salpeter, described
as $\rm dN/dm=\xi_{0}m^{-2.35}$ (Salpeter 1955), where m is star mass in $\rm
M_{\odot}$ units and $\rm\xi_{0}$ is normalization coefficient, to calculate
the boundary mass value for the high and low stellar masses, corresponding to
the SNR-PSRs and non SNR-PSRs, we obtain this critical mass to be be at
$\rm\sim 12\,M_{\odot}$, as shown in Figure 6. So, the SNRs by the low-mass
($\rm 8-12\,M_{\odot}$) progenitor stars may be survived with a shorter time
around $\rm 10-20\,Kyr$ (Braun, Goss, & Lyne 1989), which may be a reason for
the lots of young pulsars without SNRs.

Figure 6: The mass distribution of stars for SNR-PSRs and non SNR-PSRs
according to initial mass function (IMF) (Salpeter 1955). The blue solid line
is the curve of IMF. The left and right orange solid lines are lower ($\rm
8\,M_{\odot}$) and upper ($\rm 25\,M_{\odot}$) mass limit for NS production,
respectively. The area of $\rm 8-12\,M_{\odot}$ ($\rm 12-25\,M_{\odot}$)
stands for the progenitor stars for non-SNR-PSRs (SNR-PSRs).

The physical process of these two types of SNe can be described as the iron
core collapse and electron capture, and the latter is driven by the neutrino
(Janka 2012, 2017). The iron core collapse is the dominant process of high-
energy SNe that is generated from high-mass main sequence stars (Heger et al.
2003). The electron capture may be an explanation for the low-energy SNe,
while the degenerate oxygen-neon core is collapsed to form the NS(Barkat,
Reiss, & Rakavy 1974; Nomoto 1984), and the mass range of these progenitor
stars is about $\rm\sim 8-10\,M_{\odot}$ (Nomoto & Leung 2017; Leung, Nomoto,
& Suzuki 2020).

Moreover, the mass boundary values given by some researchers are similar to
ours, as $\rm\sim 12\,M_{\odot}$ (Sugimoto & Nomoto 1980; Miyaji et al. 1980).
However Nomoto (1984) and Heger et al. (2003) obtained the boundary mass as
$\rm\sim 10\,M_{\odot}$. It is remarked that our result is based on a
statistics, but not a numerical calculation result from a stellar theoretical
model. In addition, because of less pulsar samples at the young age ($<$10
Kyr), the ratio in Eq.(3) may be biased, which will directly affect the mass
boundary.

Finally, the main conclusions are summarized below: The 52 SNR-PSRs and 630
non SNR-PSRs have been tested (K-S and M-W-W) and analyzed (cumulative number
ratio), implying different $\dot{P}$ and other properties for the two sets,
perhaps associated with the different mass ranges of their progenitor masses
for SN explosions. The critical mass of different progenitor stars is
estimated by the Salpeter initial mass function, obtained as $\rm
12\,M_{\odot}$. The low-mass stars (high-mass) with $\rm\sim 8-12\,M_{\odot}$
($\rm\sim 12-25\,M_{\odot}$) will generate the low-energy (high-energy) SNe in
the shorter (longer) SNR duration of about $\rm<10-20\,Kyr$
($\rm>10-20\,Kyr$). These conjectures can explain why many young radio pulsars
are not seen inside SNRs. In the future, with the observations by FAST (Li et
al. 2018) and launch of James Webb Space Telescope (JWST) (Gardner et al.
2006), more fainter and weaker pulsars and SNRs would be discovered, which
will present the better constraints on our conclusions.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant No. 11988101, No. U1938117, No. U1731238, No. 11703003 and No.
11725313), the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008, the National Key R&D Program of China
No. 2016YFA0400702, and the Guizhou Provincial Science and Technology
Foundation (Grant No. [2020]1Y019).

## Data Availability

The data underlying this article are available in the references below: (1)
pulsars data are taken from ATNF Pulsar Catalogue, available at
https://www.atnf.csiro.au/research/pulsar/psrcat/; (2) SNR data are taken from
SNRcat, available at http://snrcat.physics.umanitoba.ca/SNRtable.php.

## References

  * Arnett & Schramm (1973) Arnett W. D., Schramm D. N., 1973, ApJL, 184, L47.
  * Barkat, Reiss, & Rakavy (1974) Barkat Z., Reiss Y., Rakavy G., 1974, ApJL, 193, L21.
  * Bhattacharya & van den Heuvel (1991) Bhattacharya D., van den Heuvel E. P. J., 1991, PhR, 203, 1.
  * Braun, Goss, & Lyne (1989) Braun R., Goss W. M., Lyne A. G., 1989, ApJ, 340, 355.
  * Čadež et al. (2016) Čadež A., Zampieri L., Barbieri C., Calvani M., Naletto G., Barbieri M., Ponikvar D., 2016, A&A, 587, A99.
  * Camilo, Thorsett, & Kulkarni (1994) Camilo F., Thorsett S. E., Kulkarni S. R., 1994, ApJL, 421, L15.
  * Chevalier (1977) Chevalier R. A., 1977, ASSL, 53.
  * Cui et al. (2021) Cui X.-H., Zhang C.-M., Wang S.-Q., Zhang J.-W., Li D., Peng B., Zhu W.-W., et al., 2021, MNRAS, 500, 3275.
  * Duncan & Thompson (1992) Duncan R. C., Thompson C., 1992, ApJL, 392, L9.
  * Esposito, Rea, & Israel (2021) Esposito P., Rea N., Israel G. L., 2021, ASSL, 97.
  * Ferrand & Safi-Harb (2012) Ferrand G., Safi-Harb S., 2012, AdSpR, 49, 1313.
  * Ferrario & Wickramasinghe (2008) Ferrario L., Wickramasinghe D., 2008, MNRAS, 389, L66.
  * Frail, Goss, & Whiteoak (1994) Frail D. A., Goss W. M., Whiteoak J. B. Z., 1994, ApJ, 437, 781.
  * Frail et al. (1995) Frail D. A., Kassim N. E., Cornwell T. J., Goss W. M., 1995, ApJL, 454, L129.
  * Gaensler & Johnston (1995) Gaensler B. M., Johnston S., 1995, MNRAS, 277, 1243.
  * Gaensler & Slane (2006) Gaensler B. M., Slane P. O., 2006, ARA&A, 44, 17.
  * Gardner et al. (2006) Gardner J. P., Mather J. C., Clampin M., Doyon R., Greenhouse M. A., Hammel H. B., Hutchings J. B., et al., 2006, SSRv, 123, 485.
  * Goldreich & Julian (1969) Goldreich P., Julian W. H., 1969, ApJ, 157, 869.
  * Gotthelf, Halpern, & Alford (2013) Gotthelf E. V., Halpern J. P., Alford J., 2013, ApJ, 765, 58.
  * Greco et al. (2021) Greco E., Miceli M., Orlando S., Olmi B., Bocchino F., Nagataki S., Ono M., et al., 2021, ApJL, 908, L45.
  * Halpern & Gotthelf (2010) Halpern J. P., Gotthelf E. V., 2010, ApJ, 710, 941.
  * Han et al. (2021) Han J. L., Wang C., Wang P. F., Wang T., Zhou D. J., Sun J.-H., Yan Y., et al., 2021, RAA, 21, 107.
  * Heger et al. (2003) Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, ApJ, 591, 288.
  * Igoshev & Popov (2013) Igoshev A. P., Popov S. B., 2013, MNRAS, 432, 967.
  * Janka (2012) Janka H.-T., 2012, ARNPS, 62, 407.
  * Janka (2017) Janka H.-T., 2017, hsn..book, 1095.
  * Johnston & Galloway (1999) Johnston S., Galloway D., 1999, MNRAS, 306, L50.
  * Kaspi et al. (2001) Kaspi V. M., Roberts M. E., Vasisht G., Gotthelf E. V., Pivovaroff M., Kawai N., 2001, ApJ, 560, 371.
  * Kaspi & Beloborodov (2017) Kaspi V. M., Beloborodov A. M., 2017, ARA&A, 55, 261.
  * Knigge, Coe, & Podsiadlowski (2011) Knigge C., Coe M. J., Podsiadlowski P., 2011, Natur, 479, 372.
  * Kou & Tong (2015) Kou F. F., Tong H., 2015, MNRAS, 450, 1990.
  * Kramer et al. (2006) Kramer M., Lyne A. G., O’Brien J. T., Jordan C. A., Lorimer D. R., 2006, Sci, 312, 549.
  * Lai (1996) Lai D., 1996, ApJL, 466, L35.
  * Large, Vaughan, & Mills (1968) Large M. I., Vaughan A. E., Mills B. Y., 1968, Natur, 220, 340.
  * Leahy (2017) Leahy D. A., 2017, ApJ, 837, 36.
  * Leahy, Ranasinghe, & Gelowitz (2020) Leahy D. A., Ranasinghe S., Gelowitz M., 2020, ApJS, 248, 16.
  * Leung, Nomoto, & Suzuki (2020) Leung S.-C., Nomoto K., Suzuki T., 2020, ApJ, 889, 34.
  * Li et al. (2018) Li D., Wang P., Qian L., Krco M., Jiang P., Yue Y., Jin C., et al., 2018, IMMag, 19, 112.
  * Lorimer et al. (1993) Lorimer D. R., Bailes M., Dewey R. J., Harrison P. A., 1993, MNRAS, 263, 403.
  * Lorimer (2004) Lorimer D. R., 2004, IAUS, 218, 105
  * Lorimer (2008) Lorimer D. R., 2008, LRR, 11, 8.
  * Lorimer & Kramer (2012) Lorimer D. R., Kramer M., 2012, hpa..book
  * Lyne, Manchester, & Taylor (1985) Lyne A. G., Manchester R. N., Taylor J. H., 1985, MNRAS, 213, 613.
  * Lyne & Graham-Smith (2012) Lyne A., Graham-Smith F., 2012, puas.book
  * Lyne et al. (2015) Lyne A. G., Jordan C. A., Graham-Smith F., Espinoza C. M., Stappers B. W., Weltevrede P., 2015, MNRAS, 446, 857.
  * Malov (2021) Malov I. F., 2021, MNRAS, 502, 809.
  * Manchester (1987) Manchester R. N., 1987, A&A, 171, 205
  * Manchester et al. (2005) Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, AJ, 129, 1993.
  * McLaughlin et al. (2006) McLaughlin M. A., Lyne A. G., Lorimer D. R., Kramer M., Faulkner A. J., Manchester R. N., Cordes J. M., et al., 2006, Natur, 439, 817.
  * Miyaji et al. (1980) Miyaji S., Nomoto K., Yokoi K., Sugimoto D., 1980, PASJ, 32, 303
  * Narayan & Schaudt (1988) Narayan R., Schaudt K. J., 1988, ApJL, 325, L43.
  * Nomoto (1984) Nomoto K., 1984, ApJ, 277, 791.
  * Nomoto & Leung (2017) Nomoto K., Leung S.-C., 2017, hsn..book, 483.
  * Morello et al. (2020) Morello V., Keane E. F., Enoto T., Guillot S., Ho W. C. G., Jameson A., Kramer M., et al., 2020, MNRAS, 493, 1165.
  * Olausen & Kaspi (2014) Olausen S. A., Kaspi V. M., 2014, ApJS, 212, 6.
  * Page et al. (2020) Page D., Beznogov M. V., Garibay I., Lattimer J. M., Prakash M., Janka H.-T., 2020, ApJ, 898, 125.
  * Popov & Turolla (2012) Popov S. B., Turolla R., 2012, Ap&SS, 341, 457.
  * Radhakrishnan & Srinivasan (1980) Radhakrishnan V., Srinivasan G., 1980, JApA, 1, 25.
  * Salpeter (1955) Salpeter E. E., 1955, ApJ, 121, 161.
  * Seward, Gorenstein, & Smith (2006) Seward F. D., Gorenstein P., Smith R. K., 2006, ApJ, 636, 873.
  * Shapiro & Teukolsky (1983) Shapiro S. L., Teukolsky S. A., 1983, bhwd.book
  * Soker (2021) Soker N., 2021, NewA, 84, 101548.
  * Srinivasan, Bhattacharya, & Dwarakanath (1984) Srinivasan G., Bhattacharya D., Dwarakanath K. S., 1984, JApA, 5, 403.
  * Staelin & Reifenstein (1968) Staelin D. H., Reifenstein E. C., 1968, Sci, 162, 1481.
  * Stollman (1987) Stollman G. M., 1987, A&A, 178, 143
  * Sugimoto & Nomoto (1980) Sugimoto D., Nomoto K., 1980, SSRv, 25, 155.
  * Tan et al. (2018) Tan C. M., Bassa C. G., Cooper S., Dijkema T. J., Esposito P., Hessels J. W. T., Kondratiev V. I., et al., 2018, ApJ, 866, 54.
  * Taylor & Manchester (1977) Taylor J. H., Manchester R. N., 1977, ApJ, 215, 885.
  * Tian & Leahy (2006) Tian W. W., Leahy D. A., 2006, A&A, 455, 1053.
  * Yang & Chevalier (2015) Yang H., Chevalier R. A., 2015, ApJ, 806, 153.
  * Yang et al. (2019) Yang Y.-Y., Zhang C.-M., Li D., Chen L., Linghu R.-F., Zhi Q.-J., 2019, PASP, 131, 064201.
  * Ye et al. (2019) Ye C.-Q., Wang D.-H., Zhang C.-M., Diao Z.-Q., 2019, Ap&SS, 364, 198. doi:10.1007/s10509-019-3690-1
  * Zhang & Kojima (2006) Zhang C. M., Kojima Y., 2006, MNRAS, 366, 137.

## Appendix A: Derivation of period evolution equation

Energy loss rate of a spin-powered pulsar can be expressed as

$\begin{split}\dot{E}=\frac{dE}{dt}=\frac{d(I\Omega^{2}/2)}{dt}=-I\Omega\dot{\Omega},\end{split}$
(4)

where $E$ is total energy of pulsar, $\dot{E}$ is energy loss rate, $\Omega$
is spin angular velocity, $\dot{\Omega}$ is rate of spin angular velocity, and
$I$ is moment of inertia. When the pulsar kinetic energy and the radiation
energy loss rate are equal, then there is

$\begin{split}I\Omega\dot{\Omega}=K\Omega^{\rm n+1},\end{split}$ (5)

where $K$ is coefficient, We combined with $\Omega=2\pi/P$ and integrate both
sides of above equation,

$\begin{split}\int_{P_{0}}^{P(t)}P^{\rm
n-2}d\dot{P}=\int_{0}^{t}-\frac{K}{I}(2\pi)^{\rm n-1}dt.\end{split}$ (6)

With assuming that the rate has been constant since pulsar birth, then spin
period evolution equation can be written as

$\begin{split}P(t)=\left[({\rm n-1})P_{0}^{\rm n-2}\dot{P}_{0}t+P_{0}^{\rm
n-1}\right]^{1/(\rm n-1)}.\end{split}$ (7)

In the above equation, $P_{0}$ and $\dot{P}_{0}$ is spin period and rate at
present, respectively. The spindown age of pulsar is $\tau=P/[({\rm
n-1})\dot{P}]$, which is different with characteristic age $\tau_{c}=P/(\rm
2\dot{P})$. Then Eq.(7) can be rewritten as

$\begin{split}P(t)=P_{0}\left(\frac{t}{\tau}_{0}+1\right)^{1/(\rm
n-1)}.\end{split}$ (8)

When $t\gg\tau_{0}$, the equation can be simplified to

$\begin{split}P(t)\approx P_{0}\left(\frac{t}{\tau}_{0}\right)^{1/(\rm
n-1)}.\end{split}$ (9)

## Appendix B: Further tests of derivative of spin period, B-field and energy
loss rate

If the origin of pulsars that with or without SNRs are indeed different, then
the distribution of other physical parameters should also be different. Here
we discuss the derivative of spin period ($\dot{P}$), B-field strength ($B$
with n=3) and energy loss rate ($\dot{E}$) by applying K-S and M-W-W tests,
and the results are shown in Table 2, and the CDFs are shown in Figure 7, 8
and 9, where the distributions for two samples of SNR-PSRs and non SNR-PSRs
are different respect to these three parameters. For $\dot{P}$, the two groups
with the different ages share the significantly different distributions, which
implies that the braking mechanisms of them perhaps are different. The
physical parameter distributions of two samples are quite different, which
possibly could be ascribed to the different origins of radio pulsars.

Figure 7: The Cumulative distribution function (CDF) of derivative of spin
period ($\dot{P}$) of SNR-PSRs and non SNR-PSRs. The dashed line is for SNR-
PSRs, and the solid line is for non SNR-PSRs. Figure 8: The Cumulative
distribution function (CDF) of surface magnetic field strength ($B$ with n=3)
of SNR-PSRs and non SNR-PSRs. The dashed line is for SNR-PSRs, and the solid
line is for non SNR-PSRs. Figure 9: The Cumulative distribution function (CDF)
of spin down energy loss rate ($\dot{E}$) of SNR-PSRs and non SNR-PSRs. The
dashed line is for SNR-PSRs, and the solid line is for non SNR-PSRs. Table 2:
P-values of K-S and M-W-W test for different parameters

Physical parametersa | K-S test | M-W-W test
---|---|---
$P$ | $1.98\times 10^{-12}$ | $5.85\times 10^{-13}$
$\dot{P}$ | $1.55\times 10^{-15}$ | $6.41\times 10^{-21}$
$B$ | $3.24\times 10^{-11}$ | $2.12\times 10^{-14}$
$\dot{E}$ | $1.55\times 10^{-15}$ | $4.33\times 10^{-24}$

a $P$ is spin period, $\dot{P}$ is derivative of spin period, $B$ is surface
magnetic field strength, and $\dot{E}$ is energy loss rate of radio pulsars.

## Appendix C: Tests of spin period and its derivative with the different ages

Figure 10: The Cumulative distribution function (CDF) of spin period ($P$) and
its derivative ($\dot{P}$) of SNR-PSRs and non SNR-PSRs with the spindown ages
of less and over than 10 Kyr. The sub-figures of a and b are CDF of $P$, and
the sub-figures of c and d are CDF of $\dot{P}$. For all sub-figures, the
dashed line represents SNR-PSRs, and the solid line is for non SNR-PSRs. The
text in each sub-figure shows their age ranges.  Table 3: P-values of K-S and
M-W-W test for $P$ and $\dot{P}$ with different ages

Physical parametersa | Spindown age | K-S test | M-W-W test
---|---|---|---
$P$ | ¡10 Kyr | 0.99 | 0.85
¿10 Kyr | $2.75\times 10^{-3}$ | $4.83\times 10^{-4}$
$\dot{P}$ | ¡10 Kyr | $3.41\times 10^{-2}$ |  $5.34\times 10^{-2}$b
¿10 Kyr | $4.38\times 10^{-13}$ | $1.64\times 10^{-13}$

a $P$ is spin period, $\dot{P}$ is derivative of spin period.

b Although this value is slightly larger than 0.05, it may also imply that the
two samples have different statistical distributions at a 90% probability
(e.g. if p-value is less than 0.1, it indicates that this test rejects the
null hypothesis that the two samples have the same distribution at 10%
significance level).

In this part, we plot distributions of $P$ with different age ranges of less
and over than 10 Kyr in Figure 10 (subplots a & b). After K-S and M-W-W tests
(Table 3), we find that when the spindown age is less than 10 Kyr the $P$
distributons of SNR-PSRs and non SNR-PSRs are the same. But the distributions
of $P$ after 10 Kyr are different. Meanwhile, distributions of $\dot{P}$ are
also tested with the same age ranges as that of $P$ (less and more than 10
Kyr) in Figure 10 (subplots c & d). Interestingly, regardless of the age
ranges, the $\dot{P}$ distributions are different under K-S tests in Table 3.
The possible physical explanation is that although the initial $P$
distribution is the same, the initial $\dot{P}$ is different, which makes
pulsars no longer have the same $P$ distribution after evolution over 10 Kyr.
Thus, the above evidence supports that the differences between the SNR-PSRs
and non SNR-PSRs are possibly the result of a combined effect of different
mechanisms and evolution, but not just caused by age difference.


# Two high capacity text steganography schemes based on color coding

Juvet K. Sadié1,2,3,4 Leonel Moyou Metcheka1,2,3,4 René
Ndoundam1,2,3,4,111Corresponding author<EMAIL_ADDRESS>
1Team GRIMCAPE
2Sorbonne Unversity, IRD, UMMISCO, F-93143, Bondy, France
3CETIC, Yaounde, Cameroon
4Department of Computer Science, University of Yaounde I, P.o. Box 812
Yaounde, Cameroon
E.mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Text steganography is a mechanism of hiding secret text message inside another
text as a covering message. In this paper, we propose a text steganographic
scheme based on color coding. This include two different methods: the first
based on permutation, and the second based on numeration systems. Given a
secret message and a cover text, the proposed schemes embed the secret message
in the cover text by making it colored. The stego-text is then sent to the
receiver by mail. After experiments, the results obtained show that our models
performs a better hiding process in terms of hiding capacity as compared to
the scheme of Aruna Malik et al. in which our idea is based.

Keywords : Steganography, text steganography, covert medium, stego-object,
permutation, embedding capacity, numeration systems.

##  1 Introduction

The word steganography is of Greek origin and means covered writing. It is the
hiding of a message within another (cover medium) such as web pages, images or
text, so that the presence of the hidden message is indiscernible. When a
message is hidden in the cover medium, the resulting medium is called a stego-
object. The key concept behind steganography is that the message to be
transmitted should not be detectable with bare eyes. From the definition,
Steganography is used to ensure data confidentiality, like encryption.
However, the main difference between the two method is that with encryption,
anybody can see that both parties are communicating in secret. Steganography
hides the existence of a secret message and in the best case nobody can detect
the presence of the message. When combined, steganography and encryption can
provide more security. Steganography dates back to ancient Greece, where
common practices consisted of etching messages in wooden tablets and covering
them with wax. A number of steganographic methods have been introduced on
different cover media such as images [1, 2, 3], video files [4, 5] and audio
files [6]. In text based steganographic methods, text is used as a cover media
for hiding the secret data. Due to the lack of large scale redundancy of
information in a text file, the human eye is very susceptible to any change
between the original and the modified texts. Therefore, text steganography
seems to be the most difficult kind of steganography [7], as compare to
others.
In this paper, we propose a text steganographic scheme based on color coding,
permutation and numeration systems. Given a secret message and a cover text,
the proposed scheme embed the secret message in the cover text by making it
colored, using a permutation algorithm for the first method and numeration
systems for the second one. After the first section devoted to the
introduction, section 2 presents some preliminaries and related works. Section
3 concerns the presentation of the first method of our scheme. Section 4
labels the second approach of our scheme, and finally conclusion is stated in
section 5.

##  2 Preliminaries and related works

In this section, the focus is to present some preliminaries that lead us to
the comprehension of our scheme. Also, we present related works in the field
of text steganography.

###  2.1 Text Steganography

There are many techniques in text steganography. In Syntactical steganography,
punctuation marks such as full stop (.), comma (,) etc, are used to hide bits
in cover text. The problem with this method is that it requires identification
of correct places to insert punctuation [8, 9]. In lexical steganography,
words are used to hide secret bits. A word could be replaced by its synonyms
and the choice of word to be chosen from the list of synonyms would depend on
secret bits. Sms texting is a combination of abbreviated words used in sms
[10]. This technique proposes to hide binary data by using full form or its
abbreviated form. For instance, to hide 0, full form of the word is used and
to hide 1, abbreviated form of word is used [10]. The CSS technique encrypts a
message using RSA public key cryptosystem and cipher text is then embedded in
a cascading style Sheet (CSS) by using End of Line on each CSS style
properties, exactly after a semi-colon. A space after a semi-colon embeds bit
0 and a tab after a semicolon embeds bit 1 [11]. Anandaprova Majumder and al
[12] proposed an approach for text steganography through a technique that uses
reflection symmetry of the English alphabet. Ekodeck and Ndoundam [13]
proposed different approaches of PDF file based steganography, essentially
based on the Chinese Remainder Theorem. Here, after a cover PDF document has
been released from unnecessary characters of ASCII code A0, a secret message
is hidden in it using one of the proposed approaches, making it invisible to
common PDF readers, and the file is then transmitted through a non-secure
communication channel. Aruna Malik and al [14], proposed a high capacity text
steganography scheme based on LZW compression and color coding. Their scheme
uses the forward mail platform to hide secret data. The algorithm first
compresses secret data and then hide the compressed data into the email
addresses and also, in the cover message of email. The secret data is embedded
in the message by making it colored using a color table. Here below, some
limits of that scheme will be presented.

###  2.2 Critic and limits

LZW is a lossless compression technique that performs high compression ratio
when the source contains repetition pattern. In the LZW based steganographic
scheme propose by Aruna Malik [14], they apply this lossless compression on
the secret message to increase the embedding capacity. But in the example
proposed, there is no compression. In other words, the size of the compressed
text is much greater than the size of the secret. To show this, we will give
three different implementations of LZW algorithm applied to the secret
message.

####  2.2.1 The LZW Algorithm with initial dictionary fixed and known

This algorithm [15] starts by initializing the dictionary with the 256
characters of the ASCII code from 0 to 255. The output codes start at a
minimum bit size equal to 9 and in general, as long as the indexes considered
are strictly inferior to n = $2^{k}$ \- 1, we can represent them on k bits.
When the first integer greater than or equal to $2^{k}$ \- 1 is met, the
sequence 1. . . 1 (k times bit 1) and continue with coding the integers on k +
1 bits. Applying this method to the following secret message: "underlying
physiological mechanisms", we obtain the outputs presented in Table 1. The
binary compressed text is obtained by converting the indexes of the output
column of the array to 9 bits :
001110101 001101110 001100100 001100101 001110010 001101100 001111001
001101001 001101110 001100111 000100000 001110000 001101000 001111001
001110011 001101001 001101111 001101100 001101111 001100111 001101001
001100011 001100001 001101100 000100000 001101101 001100101 001100011
001101000 001100001 001101110 001101001 001110011 001101101 001110011. Hence,
the size of the output is 35*9 = 315 bits.

Buffer | input-char | Output | New Item | Buffer | input-char | Output | New Item
---|---|---|---|---|---|---|---
u | n | 117 | 256=un | l | o | 108 | 273=lo
n | d | 110 | 257=nd | o | g | 111 | 274=og
d | e | 100 | 258=de | g | i | 103 | 275=gi
e | r | 101 | 259=er | i | c | 105 | 276=ic
r | l | 114 | 260=rl | c | a | 99 | 277=ca
l | y | 108 | 261=ly | a | l | 97 | 278=al
y | i | 121 | 262=yi | l |  | 108 | 279=l
i | n | 105 | 263=in |  | m | 32 | 280= m
n | g | 110 | 264=ng | m | e | 109 | 281=me
g |  | 103 | 265=g | e | c | 101 | 282=ec
| p | 32 | 266= p | c | h | 99 | 283=ch
p | h | 112 | 267=ph | h | a | 104 | 284=ha
h | y | 104 | 268=hy | a | n | 97 | 285=an
y | s | 121 | 269=ys | n | i | 110 | 286=ni
s | i | 115 | 270=si | i | s | 105 | 287=is
i | o | 105 | 271=io | s | m | 115 | 288=sm
o | l | 111 | 272=ol | m | s | 109 | 289=ms
|  |  |  | s |  | 115 |
Table 1: LZW Algorithm output with initial dictionary fixed and known

####  2.2.2 The LZW algorithm with sharing of the initial dictionary

In this version [15], initial dictionary contains only the character of the
secret message. The output code is represented on height bits. The
particularity of this implementation is from the initial dictionary which must
be shared between the two parties in order to be able to decompress the binary
code. Table 2 presents the initial dictionary for the same secret message:
Here are the output code : 1 2 3 4 5 6 7 8 2 9 10 11 12 7 13 8 14 6 14 9 8 15
16 6 10 17 4 15 12 16 2 8 13 17 13 and in binary we have :
00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000
00000010 00001001 00001010 00001011 00001100 00000111 00001101 00001000
00001110 00000110 00001110 00001001 00001000 00001111 00010000 00000110
00001010 00010001 00000100 00001111 00001100 00010000 00000010 00001000
00001101 00010001 00001101. Hence, the size of the output is the sum of the
size of initial dictionary and the output code: 17+35 = 52 bytes = 416 bits.

Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
char | u | n | d | e | r | l | y | i | g | space | p | h | s | o | c | a | m
Table 2:  Initial Dictionnary

####  2.2.3 The Unix compress command

The ncompress package [17] is a compression utility available on Linux which
contains the compress command for fast compression and decompression using LZW
algorithm. The algorithm behind this command is explained at page 153 of the
data compression book [15]. The initial dictionary size is 512 and the minimum
output code size is 9 bits. This package can be installed by using the command
"sudo apt-get install ncompress". Based on the Ubuntu 16.04 platform, this
command produces a file with .Z extension as compress file. By Applying this
command "compress -v source.txt" to the secret message contained in text
source file with the -v option, the given output indicates that there is no
compression and the .Z file size is 44 bytes = 352 bits.
Finally the table 3 shows the comparison in bit between the original text size
and the output size after the compression using the three different approaches
of LZW implementation: The LZW Algorithm with initial dictionary fixed and
known, The LZW algorithm with sharing of the initial dictionary and The Unix
compress command.

Secret message size | Output size 1 | Output size 2 | Output size 3
---|---|---|---
280 | 315 | 416 | 352
Table 3: Secret message size comparison

From Aruna and al. paper [14], the size obtained was 264 bits, but we have
proven above that there is no compression for this example. This is the
principal limit of this steganographic scheme, where for some messages the
reduction of the message size will not be possible. Our paper uses:

  * •

The idea of color coding contained in the paper of Aruna Malik and al [14];

  * •

The permutation generation method of W. Myrvold and F. Ruskey [19];

  * •

The numeration systems;

to present a new scheme where the secret message embedding capacity is better
than the scheme of Aruna Malik and al [14].

###  2.3 Permutation Generation Methods

Permutation is one of the most important combinatorial object in computing,
and can be applied in various applications, for example, the scheduling
problems. Permutation generation can form the basis of a backtracking program
to solve any problem involving reordering a set of items. It is well-known
that, for n distinct items, the total number of permutations is n!.
Permutation generation has a long history. Surveys in the field have been
published in 1960 by D.H. Lehmer [20]. Several authors [19, 21, 22, 23] have
since developed many methods to generate all the possible permutations of n
elements. Also, several works [13, 24, 25, 26, 27] in steganography taking
advantage of permutations have been done. In particular, H. Hioki [26] in
2013, proposed a permutation steganography, which is an effective method for
hiding messages provided where the contents of cover objects are not affected
by the rearrangement of their elements. In their paper, W. Myrvold and F.
Ruskey [19] proposed a ranking function for the permutations on n symbols
which assigns a unique integer in the range [0, $n$! - 1] to each of the $n$!
permutations. Also, they proposed an unranking function for which, given an
integer $r$ between 0 and $n$! - 1, the value of the function is the
permutation of rank r. Their algorithms is presented below [19].

####  2.3.1 Unranking function

First of all, recall that a permutation of order $n$ is an arrangement of $n$
symbols. An array $\pi[0\cdots n-1]$ is initialized to the identity
permutation $\pi[i]=i$, for $i=0,1,\cdots n-1$.

Procedure $unrank(n,r,\pi)$[19]
begin
if $n>0$ then
$\phantom{salutsal}swap(\pi[n-1],\pi[r$ mod $n])$;
$\phantom{salutsal}unrank(n-1,\left\lfloor r/n\right\rfloor,\pi)$;
end;
end;

Note: $swap(a,b)$ exchanges the values of variables $a$ and $b$.

####  2.3.2 Ranking function

To rank, first compute $\pi^{-1}$. This can be done by iterating
$\pi^{-1}[\pi[i]]=i$, for $i=0,1,\cdots,n-1$.
In the algorithm below, both $\pi$ and $\pi^{-1}$ are modified.

function $rank(n,\pi,\pi^{-1})$:integer[19]
begin
if $n=1$ then return(0) end;
$\phantom{salutsal}s:=\pi[n-1]$ ;
$\phantom{salutsal}swap(\pi[n-1],\pi[\pi^{-1}[n-1]])$ ;
$\phantom{salutsal}swap(\pi^{-1}[s],\pi^{-1}[n-1])$ ;
return$(s+n.rank(n-1,\pi,\pi^{-1}))$ ;
end;

##  3 Scheme design based on permutation

In this section, we present the first method of our scheme.

###  3.1 Embedding Algorithm

Input:
$\phantom{saluts}C$: the cover text;
$\phantom{saluts}M$: the secret message to embed;
The key $\pi$: the initial permutation of $n$ colors;
$\phantom{saluts}e$: the e-mail address of the receiver
Output:
$\phantom{saluts}C^{\prime}$: the stego-message;
begin:
1\. Compute $m$, the binary representation of $M$;
2\. Compute $t=\left\lfloor log_{2}(n!)\right\rfloor$
3\. Divide $m$ into $p$ blocks of $t$ bits each, $b_{1},b_{2},\cdots,b_{p}$;
4\. Divide $C$ into $k$ blocks of $n$ characters each
$c_{1},c_{2},\cdots,c_{k}$;
5\. For each block $b_{i},1\leq i\leq p$:
a. compute $Nperm=(b_{i})_{10}$, the decimal representation of $b_{i}$;
b. compute $\pi^{\prime}=unrank(n,Nperm,\pi)$, the permutation corresponding
to the number $Nperm$. $\pi^{\prime}$ can be considered as
$\pi^{\prime}(1),\pi^{\prime}(2),\cdots,\pi^{\prime}(n)$;
c. color each character of $c_{i}$ by the corresponding color given by the
permutation $\pi^{\prime}$ and obtain the string $c^{\prime}_{i}$;
d. compute $C^{\prime}\leftarrow C^{\prime}||c^{\prime}_{i}$; where a$||$b is
the concatenation of a and b.
6\. If the next character is EOF (End of File) then
begin
a. Use e to send C’ by mail to the receiver;
end
Else
begin
a. Colour the next character with a color different of permutation colors.
This color is shared by the sender and the receiver. However, this color will
not be very distant from the others.
b. Randomly color the rest of characters of C by the colors of colors table,
until the EOF character is obtained;
c. Use e to send C’ by mail to the receiver;
end;
end;

###  3.2 Retrieval Algorithm

Input:
$\phantom{saluts}C^{\prime}$: the stego-text;
The key $\pi$: the initial permutation of $n$ colors;
Output:
$\phantom{saluts}M$: the secret message;
begin:
1\. Retrieve all characters coloured by the permutation colors, until a color
different from the colors in the colors table, or the EOF character is
obtained. Lets call them $C^{\prime\prime}$;
2\. Divide $C^{\prime\prime}$ into $p$ blocks of $n$ characters each
$c_{1},c_{2},\cdots,c_{p}$;
3\. For each block $c_{k},1\leq k\leq p$:
a. use the color order of characters to compute the relative permutation, that
we call $\pi^{\prime}$. $\pi^{\prime}$ can be considered as
$\pi^{\prime}(1),\pi^{\prime}(2),\cdots,\pi^{\prime}(n)$;
b. compute the number $Nperm=rank(n,\pi^{\prime},\pi^{\prime-1})$.
c. compute $m^{\prime}=(Nperm)_{2}$, the binary representation of $Nperm$;
d. compute $M\leftarrow M||m^{\prime}$;
end;

###  3.3 Experimentation

In this subsection, we present some experimentations of this method. First, we
propose a theoretical estimation of our embedding capacity for $n$ colors.
Secondly, we present a practical experimentation in the case of 10, 16, 32 and
64 colors, based on example 1 and figure 5 of [14]. These colors are given in
figure 1, figure 2, figure 3 and figure 4.

Figure 1: The table of 10 colors

.

Figure 2: The table of 16 colors

.

Figure 3: The table of 32 colors

.

Figure 4: The table of 64 colors

.

In order to present our embedding capacity, we use as cover text and secret
message those of example 1 and figure 5 of Aruna Malik and al [14].

####  3.3.1 Theoretical estimations

The table 4 presents the embedding capacity of our scheme for some different
values of n: 10, 16, 20 32,60, 64. This theoretical estimation is based on our
embedding algorithm.
More generally, in a set of n colors, the number of permutation of n distinct
colors is n !. According to the stirling formula [28] we have:

$n!\sim\left(\frac{n}{e}\right)^{n}\times\sqrt{2\pi n}$ (1)

Where $\pi=3.14$ is the area of the circle with unit radius, $e=2.718$ is the
base of the natural logarithm, and $\sim$ means approximate equality.

we know that:

$n=2^{log_{2}(n)}$

By replacing the value of n in equation 1 we have:

$\displaystyle n!$
$\displaystyle\sim\left(\frac{2^{log_{2}(n)}}{2^{1.442695}}\right)^{n}\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{log_{2}(n)-1.442695}\right)^{n}\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times
2^{log_{2}(\sqrt{2\pi n})}$
$\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times
2^{\frac{1}{2}log_{2}(2\pi n)}$
$\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n+{\frac{1}{2}log_{2}(2\pi
n)}}\right)$

Proposition : the embedding capacity (E) using n colors to hide a secret is :

$E=\frac{M\times 100}{n\times 8}$

where $M={n(log_{2}(n)-1.442695)+{\frac{1}{2}log_{2}(2\pi n)}}$, and $n$ the
number of colors.

$\blacksquare$

n | M= $\left\lfloor log(n!)\right\rfloor$ | P=M/8 | 100*(P/n),(embedding capacity)
---|---|---|---
10 | 21 | 2.6 | 26.25%
16 | 44 | 5.5 | 34.37%
20 | 61 | 7.6 | 38%
32 | 117 | 14.6 | 45.63%
60 | 272 | 34 | 56.67%
64 | 295 | 36.9 | 57.66%
Table 4: Theoretical estimations of the proposed scheme

Remark: As far as the space characters of the stego-text are not coloured, the
embedding capacity can decrease in the experimentations.

####  3.3.2 Experimentation 1

Here, the secret message is : underlying physiological mechanisms
and the cover text is:
Only boats catch connotes of the islands sober wines only ships wrap the slips
on the cleats of twining lines only flags flap in tags with color that assigns
only passage on vessels

Here we present the embedding process.

  1. 1.

We compute the binary representation of the secret and obtain the following
result:

01110101 01101110 01100100 01100101 01110010 01101100 01111001 01101001
01101110 01100111 00100000 01110000 01101000 01111001 01110011 01101001
01101111 01101100 01101111 01100111 01101001 01100011 01100001 01101100
00100000 01101101 01100101 01100011 01101000 01100001 01101110 01101001
01110011 01101101 01110011

  2. 2.

We compute $t=\left\lfloor log_{2}(10!)\right\rfloor=21$ ;

  3. 3.

The binary secret is then divided into blocks of 21 bits each. For instance,
the first block $b_{1}$ = 011101010110111001100 and the second block $b_{2}$ =
100011001010111001001, …

  4. 4.

We divide the cover text into blocks of 10 characters. For instance, the first
block $c_{1}=$ Only boats c, the second block $c_{2}=$ atch connot, …;

  5. 5.

We color the cover text:

     * •

For the block $b_{1}$ = 011101010110111001100, Nperm = 961996, its decimal
representation.

     * •

The permutation relative to 961996 is given by $\pi^{\prime}$ = unrank(10,
961996, $\pi)$ = 3 8 5 2 1 4 9 0 7 6, $\pi$ = 0 1 2 3 4 5 6 7 8 9, with the
corresponding colors given by figure 1.

     * •

The block $c_{1}=$ Only boats c is coloured relatively to the permutation
$\pi^{\prime}$. We then obtain the color text given in figure 5

Figure 5: The Stego-Text

.

     * •

For the block $b_{2}$ = 100011001010111001001, Nperm = 1152457, its decimal
representation.

     * •

The permutation relative to 1152457 is given by $\pi^{\prime}$ = unrank(10,
1152457, $\pi)$ = 2 9 1 6 3 8 4 5 0 7, $\pi$ = 0 1 2 3 4 5 6 7 8 9, with the
corresponding colors given by figure 1.

     * •

The block $c_{2}=$ atch connot is coloured relatively to the permutation
$\pi^{\prime}$. We then obtain the color text given in figure 5

  6. 6.

The process is the same, and finally we obtain the stego-text given by the
figure 6. That stego-text is then send by mail to the receiver.

Figure 6: The Stego-Text

.

With this example :

  * •

in the case of 10 colors, the embedding capacity is 20.58 %;

  * •

with 16 colors, the embedding capacity is 25.5 %;

  * •

With 32 colors, the embedding capacity is 29.5 %;

  * •

with 64 colors, the embedding capacity is 45.45 %.

####  3.3.3 Experimentation 2

In the example of figure 5 [14], the secret message is : behind using a cover
text is to hide the presence of secret messages the presence of embedded
messages in the resulting stego-text cannot be easily discovered by anyone
except the intended recipient.

and the cover-text is:

in the research area of text steganography, algorithms based on font format
have advantages of great capacity, good imperceptibility and wide application
range. However, little work on steganalysis for such algorithms has been
reported in the literature. based on the fact that the statistic features of
font format will be changed after using font-format-based steganographic
algorithms, we present a novel support vector machine-based steganalysis
algorithm to detect whether hidden information exists or not. this algorithm
can not only effectively detect the existence of hidden information, but also
estimate the hidden information length according to variations of font
attribute value. as shown by experimental results, the detection accuracy of
our algorithm reaches as high as 99.3 % when the hidden information length is
at least 16 bits. Our scheme present experimentation based on different colors
number.

In the case of 10 colors, We apply our embedding algorithm and obtain the
following stego-text, given by figure 7.

Figure 7: The Stego-Text

.

With this example:

  * •

In the case of 10 colors, the embedding capacity is 22.32 %;

  * •

With 16 colors, the embedding capacity is 29.64 %;

  * •

With 32 colors, the embedding capacity is 38 %;

  * •

With 64 colors, the embedding capacity is 44 %.

##  4 Scheme design based on numeration systems

In this new approach, we improve the method of the first scheme with the
assertion that each color can be repeated as many times on some positions of a
given group of characters. Unlike the previous scheme in which each color
could only appear once in a group of precise characters.

###  4.1 The Scheme description

we give a brief description of how this new scheme works by following these
steps:

  1. 1.

Choose a base $B$ such that $2\leq B\leq 2^{24}$. where $2^{24}$ is the number
of existing colors ;

  2. 2.

choose B colors from the set of $2^{24}$ colors number from 0 to $B-1$;

  3. 3.

convert the secret m to base B such that : $m=(m_{q-1}...m_{1}m_{0})_{B}$,
where $0\leq m_{i}\leq B-1$;

  4. 4.

We assume that the number of characters of the covert text is : $n$ and $q\leq
n$;

  5. 5.

For $i=0$ to $q-1$ do
The character $c_{i}$ is coloured with the color relative to $m_{i}$

  6. 6.

The text coloured is then send to the receiver.

The reverse procedure consists to extract the secret conceal in the colors
distribution. These steps must be performed by the receiver of the stego-text
:

  1. 1.

Take the text with the first $q$ characters which has been coloured;

  2. 2.

For $i=0$ to $q-1$ do

Find the color number $z_{i}$ associated to the character $c_{i}$ by using the
reference color table shared between the sender and the receiver;

  3. 3.

Convert $z=(z_{q-1}...z_{1}z_{0})_{B}$ to binary and get the secret message.

###  4.2 Embedding Algorithm

Input

C: the cover text;

M: the secret message to embed;

$\beta$ : The base;

T : a table of $\beta$ color;

e : the e-mail address of the receiver;
Output

C’: the stego-message;
Begin

  1. 1.

Convert the secret M to base B such that : $m=(m_{n-1}...m_{1}m_{0})_{B}$,
where $0\leq m_{i}\leq B-1$;

  2. 2.

For $i=n-1$ to $0$ do

    1. (a)

Find in the color table, the color $a_{i}$ associated to the value $m_{i}$;

    2. (b)

Coloured the character $c_{i}$ of $C$ with the color $a_{i}$ and obtain
$c^{\prime}_{i}$ ;

    3. (c)

Compute $C^{\prime}\longleftarrow C^{\prime}\mid\mid c^{\prime}_{i}$; where
$a\mid\mid b$ is the concatenation of a and b.

  3. 3.

If the next character is not EOF (End of File) then

    1. (a)

Colour the next character with a color different from the colors table T. This
color is shared by the sender and the receiver. However, this color will not
be very distant from the others;

    2. (b)

Randomly color the rest of characters of C by the colors from the colors
table, until obtain the EOF character;

    3. (c)

Compute $C^{\prime}\longleftarrow C^{\prime}\mid\mid c^{\prime}_{j}$ : $n\leq
j\leq m$, where $m$ is the position of the last character of $C$;

  4. 4.

Use e to send $C^{\prime}$ by mail to the receiver;

End

###  4.3 Retrieving Algorithm

Input

C’: the stego-text;

T : a table of $\beta$ color;

$\beta$ : The base;
Output

M: the secret message;
Begin

  1. 1.

Retrieve all characters coloured with the table colors, until obtain a color
different from those of the colors table, or obtain the EOF character. Lets
call them $C^{\prime\prime}$ and $\mid C^{\prime\prime}\mid=n$;
($C^{\prime\prime}=c_{n-1}c_{n-2}...c_{1}c_{0}$);

  2. 2.

For $i=n-1$ to $0$ do

    1. (a)

get the color $a_{i}$ associated to the color of the character $c_{i}$ of
$C^{\prime\prime}$;

    2. (b)

Find in the color table, the value $m_{i}$ associated to the color $a_{i}$ ;

    3. (c)

compute $M\longleftarrow M\mid\mid m_{i}$;

    4. (d)

Compute $M_{2}$, the binary representation of the secret $M$;

End

###  4.4 Experimentation

This subsection presents some experimentations for this method. We first
propose a theoretical estimation of our embedding capacity for $B$ colors.
Secondly, we present a practical experimentation in the case of 10, 16 and 32
colors, based on example 1 and figure 5 of [14]. These colors are given in
figure 1, figure 2 and figure 3.

####  4.4.1 Theoretical Estimation

We want to color a block of text with $\eta$ characters. Each character is
coloured with a single color. The number of colors used is $B$. Knowing that a
color can appear as many times on some positions, the total number of
colouring possibilities for each character is : $B$. For the $\eta$
characters, the total number of colouring possibilities is : $B^{\eta}$.

The number of bits used to color the $\eta$ characters is :
$log_{2}(B^{\eta})$

The embedding capacity [14, 18] is define as the ratio of the secret bits
message by the stego cover bits :

$\displaystyle Capacity$
$\displaystyle=\frac{Bits\;of\;secret\;message}{Bits\;of\;stego\;cover}$ (2)
$\displaystyle Capacity$ $\displaystyle=\frac{log_{2}(B^{\eta})}{\eta\times
8}$ (3) $\displaystyle Capacity$ $\displaystyle=\frac{log_{2}(B)}{8}$ (4)

The following table gives a theoretical estimate of the capacity as a function
of the base $B$ used:

Table 5: Embedding capacity estimation as a function of $B$ $B$ | Capacity $\times 100$
---|---
2 | 12.5%
4 | 25%
8 | 37.5%
10 | 41.5%
16 | 50%
32 | 62.5%
64 | 75%

####  4.4.2 Experimentation 1

This experimentation is based on example 1 of [14], where the number of color
$B$ is equal to 10. The figure 8 presents the results of the embedding process
based on this second method for 10 colors.

Figure 8: The Stego-Text for a table of 10 colors

.

With this example :

  * •

in the case of 10 colors, the embedding capacity is 34.31 %;

  * •

with 16 colors, the embedding capacity is 41.17 %;

  * •

With 32 colors, the embedding capacity is 52.23 %;

####  4.4.3 Experimentation 2

The figure 9 presents the results of our embedding process for 10 colors,
based on the example gives by figure 5 of [14].

Figure 9: The Stego-Text for a table of 10 colors

.

With this example :

  * •

in the case of 10 colors, the embedding capacity is 35.29 %;

  * •

with 16 colors, the embedding capacity is 42.85 %;

  * •

With 32 colors, the embedding capacity is 53.22 %;

The table 6 recapitulates the embedding capacity of our schemes in comparison
with the scheme of Aruna and al [14], in the case of 10 colors.

| First Method | Second Method | The scheme of Aruna and al [14]
---|---|---|---
example 1 [14] | 20.58 % | 34.31 % | 6.03 %
example of figure 5 [14] | 22.32 % | 35.29 % | 13.43%
Table 6: Comparison between our scheme and the scheme of Aruna [14], in terms
of embedding capacity, for 10 colors

##  5 Conclusion

In this paper, two text steganographic schemes based on color coding have been
proposed. The first based on permutation and the second based on numeration
systems. Given a secret message and a cover text, the proposed schemes embed
the secret message in the cover text by making it coloured. Using 32 colors,
the first scheme achieves a theoretical and practical embedding capacity of
45.63 % and 38 % respectively. While with the second scheme the theoretical
and practical embedding capacity are 62.5% and 53.22% respectively with the
same number of colors. These two high capacity text steganographic scheme
significantly improve the existing work of Aruna and al.

##  6 Acknowledgments

This work was supported by UMMISCO, CETIC and the University of Yaounde 1.

## References

  * [1] R. Chandramouli, N. Memon, "Analysis of LSB Based Image Steganography Techniques", IEEE pp. 1019-1022, 2001.
  * [2] D. Artz, "Digital Steganography: Hiding Data within Data", IEEE Internet Computing, pp. 75-80, May-Jun 2001.
  * [3] J. Chen, T. S. Chen, M. W. Cheng, "A New Data Hiding Scheme in Binary Image", in Proc. Fifth Int. Symp. on Multimedia Software, Engineering. Proceedings, pp. 88-93 (2003).
  * [4] G. Doerr and J.L. Dugelay, "A Guide Tour of Video Watermarking", Signal Processing: Image Communication, vol. 18, Issue 4, 2003, pp. 263-282.
  * [5] G. Doerr and J.L. Dugelay, "Security Pitfalls of Frameby- Frame Approaches to Video Watermarking", IEEE Transactions on Signal Processing, Supplement on Secure Media, vol. 52, Issue 10, 2004, pp. 2955-2964.
  * [6] K. Gopalan, "Audio steganography using bit modification", Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing,(ICASSP ’03), vol. 2, 6-10 April 2003, pp. 421-424.
  * [7] J.T. Brassil, S. Low, N.F. Maxemchuk, and L.O’Gorman, "Electronic Marking and Identification Techniques to Discourage Document Copying", IEEE Journal on Selected Areas in Communications, vol. 13, Issue. 8, October 1995, pp. 1495-1504.
  * [8] M. H. S. Shahreza, and M. S. Shahreza, "A new approach to Persian/Arabic text steganography," In Proceedings of 5th IEEE/ACIS Int. Conf. on Computer and Information Science and 1st IEEE/ACIS Int. Workshop on Component-Based Software Engineering, Software Architecture and Reuse, 2006, pp. 310-315.
  * [9] M. H. S. Shahreza, and M. S. Shahreza, "A new synonym text steganography," Int. Conf. on Intelligent Information Hiding and Multimedia Signal Processing, 2006, pp. 1524-1526.
  * [10] M. S. Shahreza, and M. H. S. Shahreza, "Text steganography in SMS," 2007 Int. Conf. on Convergence Information Technology, 2007, pp. 2260-2265.
  * [11] H. Kabetta, B. Y. Dwiandiyanta, and Suyoto, "Information hiding in CSS: a secure scheme text steganography using public key cryptosystem," Int. Journal on Cryptography and Information Security, vol.1, pp. 13-22, 2011.
  * [12] Anandaprova Majumder, Suvamoy Changder, "A Novel Approach for Text Steganography: Generating Text Summary using Reflection Symmetry", International Conference on Computational Intelligence: Modeling Techniques and Applications, pp. 112-120, 2013.
  * [13] S.G.R. Ekodeck, R. Ndoundam, "PDF steganography based on Chinese Remainder Theorem", Journal of Information Security and Applications, Vol. 29, No. 1, pp. 1-15, 2016.
  * [14] Aruna Malik, Geeta Sikka, Harsh K. Verma, "A high capacity text steganography scheme based on LZW compression and color coding", Engineering Science and Technology, an International Journal 20, 72, 2017.
  * [15] Zhi-Hui Wang, Hai-Rui Yang, Ting-Fang Cheng, Chin-Chen Chang , "A high-performance reversible data-hiding scheme for LZW codes", Journal of Systems and Software, (2013) 2771-2778.
  * [16] Khalid Sayood, Introduction to Data Compression, "The Morgan Kaufmann Series in Multimedia Information and Systems", 5th Edition,2017.
  * [17] Linux Compress Command Examples for Files and Directory, https://linux.101hacks.com/unix/compress/
  * [18] E. Satir, H. Isik, A compression-based text steganography method, J. Syst. Softw. 85 (2012) 238594.
  * [19] W. Myrvold & F. Ruskey, "Ranking and Unranking Permutations in Linear Time", Information Processing Letters,Vol. 79, Issue 6, pp. 281-284, 2001.
  * [20] D.H. Lehmer ,"Teaching Combinatorial Tricks to a Computer", Proceedings of Symposium in Applied Mathematics, Combinatorial Analysis, American Mathematical Society, Vol. 10, pp. 179-193, 1960.
  * [21] A. Nijenhuis & H.S. Wilf, "Combinatorial Algorithms: For Computers and Calculators", $2^{nd}$ Edition, Academic Press, New York, 1978.
  * [22] C.D. Savage, "Generating Permutations with k-differences", SIAM Journal Discrete Mathematics, Vol.3, No. 4, pp. 561-573, 1990.
  * [23] C.T. Djamégni & M. Tchuente, "A Cost-Optimal Pipeline Algorithm for Permutation Generation in Lexicographic Order", Journal of Parallel Distribed Computing, Vol.44, No. 2, pp. 153-159, 1997.
  * [24] A.M. Shihab, R.K. Mohammed & W.M. Abed, "Evaluating the Performance of the Secure Block Permutation Image Steganography Algorithm", International Journal of Network Security & its Applications, Vol. 5, No. 5, 2013.
  * [25] H. Al-Bahadili, "A Secure Block Permutation Image Steganography Algorithm", International Journal on Cryptography and Information Security, Vol. 3, No. 3, pp. 11-22, 2013.
  * [26] H. Hioki, "Data Embedding Methods Not Based on Content Modification", Multimedia Information Hiding Technologies and Methodologies for Controlling Data, Chapter 7, pp. 272-294, 2012.
  * [27] H.-C. Lin & S.-J Lin, "Piecewise Permutation Steganography for 3D Humanoid Mesh Models", 2nd International Conference on Computer Science and its Applications, pp. 355-360, 2009.
  * [28] Leversha G, L. Lovacz, J. Pelikan and K. Vesztergombi, "Discrete mathematics, elementary and beyond", The Mathematical Gazette (Springer), 2004 Jul 88(512):378-9, pp. 290.