# An axially symmetric spacetime with causality violation in Ricci-inverse
gravity

J. C. R. de Souza<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this paper, Ricci-inverse gravity is investigated. It is an alternative
theory of gravity that introduces into the Einstein-Hilbert action an anti-
curvature scalar that is obtained from the anti-curvature tensor which is the
inverse of the Ricci tensor. An axially symmetric spacetime with causality
violation is studied. Two classes of the model are discussed. Different
sources of matter are considered. Then a direct relation between the content
of matter and causality violation is shown. Our results confirm that Ricci-
inverse gravity allows the existence of Closed Time-like Curves (CTCs) that
lead to the violation of causality. Furthermore, a comparison is made between
the results of general relativity and Ricci-inverse gravity. Other spacetimes,
such as Gödel and Gödel-type universes, which are exact solutions of general
relativity and allow for causality violations, are also explored in Ricci-
inverse gravity framework.

##  I Introduction

General relativity is an extraordinary gravitational model that describes
physical reality very well and has undergone numerous tests since its
formulation in 1915 [1, 2]. Although the theory and observational data confirm
the success of general relativity, it is not a complete theory. There are two
fundamental problems that require attention and solution. The classic version
of gravity is complete, but a consistent quantum version has yet to be built.
This fact shows that the gravitational interaction is different from other
fundamental interactions described by the standard model of particle physics,
since they have a well-known quantum version. Another open question in general
relativity is: how to explain the recent accelerated expansion of the
universe? The current accelerated cosmic expansion is a widely accepted fact
in the scientific community and confirmed by several observational sources [3,
4, 5, 6, 7, 8, 9]. In an attempt to find solutions to these problems, two
different ways have been proposed in the literature: (i) introducing an exotic
component of energy, called dark energy, into general relativity or (ii)
modifying the Lagrangian of general relativity without resorting to dark
energy. In this work, the second option is considered.

Alternative theories to general relativity have been a hot topic since
Einstein proposed his model. A historical review of the first attempts to
extend general relativity, for different motivations, is presented in
reference [10]. For an overview of modified gravity theories, see references
[11, 12]. Here, Ricci-inverse gravity [13] is considered the alternative
theory of gravity to be investigated. In this model, the modification consists
of adding the anti-curvature tensor $\displaystyle A^{\mu\nu}$ in the
Einstein-Hilbert action. The tensor $\displaystyle A^{\mu\nu}$ is defined as
the inverse of the Ricci tensor $\displaystyle R_{\mu\nu}$. With this tensor
the anti-curvature scalar $\displaystyle A=g_{\mu\nu}A^{\mu\nu}$ is defined.
It is important to emphasize that $\displaystyle A\neq R^{-1}$. In the
reference [14] the Ricci-inverse theory is generalized and two classes of
Ricci-inverse gravity are defined: Class I and Class II. In Class I, the
Lagrangian is proportional to a function $\displaystyle f(R,A)$ that depends
on the Ricci scalar $\displaystyle R$ and the anti-curvature scalar
$\displaystyle A$, while in Class II the function takes the form
$\displaystyle f(R,A^{\mu\nu}A_{\mu\nu})$ which is a function of Ricci scalar
and square of anti-curvature tensor. Ricci-inverse gravity has been
investigated in several contexts, for example, anisotropic compact structures
has been explored [15], the matter-antimatter asymmetry through baryogenesis
in the realm of $\displaystyle f(R,A)$ theory has been analyzed [16], a non-
relativistic static and spherically symmetric cosmic structure embedded into a
de Sitter cosmology has been investigated [17] and no-go theorem for inflation
in an extended Ricci-inverse gravity model has been studied [18, 19]. Among
the various topics investigated in this theory, an investigation into
causality, its violation, and the existence of Closed Time-like Curves (CTCs)
is missing. Here such a study is carried out.

To explore causality and the presence of CTCs in Ricci-inverse gravity, an
axially symmetric metric is considered [20, 21, 22]. The main feature of this
spacetime is the possibility of CTCs which are trajectories that allows
objects to return to a point in their past. Another important point of this
spacetime is that it satisfies the energy conditions. In such a study,
different matter contents are analyzed. Furthermore, it is interesting to say
that the existence of CTCs is not exclusive to axially symmetric metrics.
Other widely studied solutions that lead to the violation of causality are the
Gödel metric [23] and Gödel-type metric [24], which are general relativity
solutions with rotating matter and cosmological constant. These metrics are
also analyzed in the context of Ricci-inverse gravity.

The present paper is organized as follows. In section II, an introduction to
Ricci-inverse gravity is made. In section III, an axially symmetric metric in
Ricci-inverse gravity is studied. The conditions for the existence of CTCs are
presented. The set of field equations is solved in both general relativity and
Ricci-inverse gravity. The results of the two theories are compared. It is
shown that this alternative theory allows for the causality violation.
Different matter contents are considered, such as a scalar field and an
electromagnetic field. A discussion about Gödel-type universes in Ricci-
inverse gravity is made. In section IV, remarks and conclusions are presented.

##  II Ricci-inverse gravity

In this section, a brief introduction to Ricci-inverse gravity is presented.
This is a new way to modify the Einstein-Hilbert action proposed in [13]. This
modification in Einstein’s theory consists of the introduction of the anti-
curvature tensor ($\displaystyle A^{\mu\nu}$) defined as $\displaystyle
A^{\mu\nu}R_{\nu\sigma}=\delta^{\mu}_{\sigma}$. In the reference [14] two
classes of Ricci-inverse gravity are introduced: Class I: the gravitational
action is characterized by the function $\displaystyle f(R,A)$ which depends
on Ricci ($\displaystyle R$) and anti-curvature ($\displaystyle A$) scalars.
Class II: the theory is described by the function $\displaystyle
f(R,A^{\mu\nu}A_{\mu\nu})$ which is a function of Ricci scalar and square of
anti-curvature tensor. Here, Class I is considered. Then the action describing
Ricci-inverse gravity is given as

$\displaystyle\displaystyle S=\int d^{4}x\sqrt{-g}\left[f(R,A)-2\Lambda+{\cal
L}_{m}\right].$ (1)

Taking $\displaystyle f(R,A)=R+\kappa A$, Eq. (1) becomes

$S=\int d^{4}x\sqrt{-g}\left[(R+\kappa A-2\Lambda)+{\cal L}_{m}\right],$ (2)

where $\displaystyle g$ is the metric determinant, $\displaystyle\kappa$ is
the coupling constant, $\displaystyle R$ is the Ricci scalar, $\displaystyle
A=g_{\mu\nu}A^{\mu\nu}$ is anti-curvature scalar, $\displaystyle\Lambda$ is
the cosmological constant and $\displaystyle{\cal L}_{m}$ is the matter
Lagrangian. This action can be written as

$\displaystyle\displaystyle S=\int
d^{4}x\sqrt{-g}\left[(g_{\mu\nu}R^{\mu\nu}+\kappa
g_{\mu\nu}A^{\mu\nu}-2\Lambda)+{\cal L}_{m}\right].$ (3)

Varying the action with respect to the metric leads to the field equations
given by

$\displaystyle\displaystyle R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}+\Lambda
g^{\mu\nu}-\kappa
A^{\mu\nu}-\frac{\kappa}{2}Ag^{\mu\nu}+\frac{\kappa}{2}\Bigl{[}2g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}(A^{\sigma\alpha}A^{\nu}_{\sigma})$
$\displaystyle\displaystyle-\nabla^{2}(A^{\sigma\mu}A^{\nu}_{\sigma})-g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(A^{\sigma\alpha}A^{\beta}_{\sigma})\Bigl{]}=T^{\mu\nu},$
(4)

where $\displaystyle\nabla_{\mu}$ denotes the covariant derivative and
$\displaystyle T^{\mu\nu}$ is the energy-momentum tensor defined as

$T^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g_{\mu\nu}}.$ (5)

Using that $\displaystyle
A^{\alpha}_{\sigma}A^{\nu\sigma}=A^{\alpha\tau}g_{\tau\sigma}A^{\sigma\nu}=A^{\alpha\tau}A^{\nu}_{\tau}=A^{\alpha\sigma}A^{\nu}_{\sigma}=A^{\nu}_{\sigma}A^{\alpha\sigma}$,
the field equations become

$\displaystyle\displaystyle R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}+\Lambda
g^{\mu\nu}-\kappa
A^{\mu\nu}-\frac{\kappa}{2}Ag^{\mu\nu}+\frac{\kappa}{2}\Bigl{[}2g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}(A_{\sigma}^{\alpha}A^{\nu\sigma})$
$\displaystyle\displaystyle-\nabla^{2}(A^{\mu}_{\sigma}A^{\nu\sigma})-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}(A_{\sigma}^{\alpha}A^{\rho\sigma})\Bigl{]}=T^{\mu\nu}.$
(6)

For simplicity, the Ricci-inverse gravity equations are written as

$\displaystyle\displaystyle R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}+\Lambda
g^{\mu\nu}+M^{\mu\nu}=T^{\mu\nu},$ (7)

where $\displaystyle M^{\mu\nu}$ contain the terms that modify the equations
of general relativity due to the anti-curvature tensor and is defined as

$\displaystyle\displaystyle
M^{\mu\nu}=\kappa\left[A^{\mu\nu}+\frac{A}{2}g^{\mu\nu}-\frac{1}{2}\left(2g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}(A_{\sigma}^{\alpha}A^{\nu\sigma})-\nabla^{2}(A^{\mu}_{\sigma}A^{\nu\sigma})-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}(A_{\sigma}^{\alpha}A^{\rho\sigma})\right)\right].$
(8)

Our aim is to investigate cosmological solutions associated with causality
violation. However, it is important to note that the Ricci-inverse gravity is
constructed assuming that there is an anti-curvature tensor, i.e.,
$\displaystyle A^{\mu\nu}=R_{\mu\nu}^{-1}$. Therefore, such a study is
possible only in spacetime where there exists this quantity which is defined
as

$A^{\mu\nu}=\frac{1}{det[R_{\mu\nu}]}adj[R_{\mu\nu}].$ (9)

Note that, if the determinant of the Ricci tensor is null, there is no way to
invert $\displaystyle R_{\mu\nu}$. Therefore, $\displaystyle
det(R_{\mu\nu})\neq 0$ is a necessary condition to investigate a metric as a
possible solution to Ricci-inverse gravity.

In the next section, an axially symmetric spacetime is studied in Ricci-
inverse gravity. The main objective is to investigate whether this
gravitational model allows solutions that leads to causality violation. In
addition to the axially symmetric metric, other solutions of general
relativity, such as Gödel and Gödel-type solutions, are discussed in the
approach described by Ricci-inverse gravity.

##  III Axially symmetric metric in Ricci-inverse gravity

Here, an axially symmetric metric is investigated in Ricci-inverse gravity.
The main feature of this solution is the presence of Closed Timelike Curves
(CTCs), as discussed in references [20, 21, 22]. The line element that
describes this spacetime at $\displaystyle(t,r,\phi,z)$ coordinates is given
as

$\mathrm{d}s^{2}=\frac{\mathrm{d}r^{2}}{\alpha^{2}r^{2}}+r^{2}\mathrm{d}z^{2}+\left(-2r^{2}\mathrm{d}t+\frac{\beta
z\mathrm{d}r}{r^{2}}-tr^{2}\mathrm{d}\phi\right)\mathrm{d}\phi,$ (10)

where $\displaystyle\alpha$ and $\displaystyle\beta$ are non-zero constants,
with $\displaystyle\beta>0$. It should be noted from Eq. (10) that this
spacetime has a coordinate singularity at $\displaystyle r=0$.

The most notable characteristic of this metric is its ability to display CTCs.
These curves can be obtained considering $\displaystyle r=r_{0}$,
$\displaystyle z=z_{0}$ and $\displaystyle t=t_{0}$, with $\displaystyle
r_{0},z_{0},t_{0}=\mathrm{const.}$ in Eq. (10). Then

$\displaystyle\displaystyle\mathrm{d}s^{2}=-tr^{2}\mathrm{d}\phi^{2}.$ (11)

This leads to three different curves: (i) null-like curve for $\displaystyle
t=0$; (ii) space-like curve for $\displaystyle t_{0}<0$ and (iii) time-like
curve for $\displaystyle t_{0}>0$. Therefore, in this spacetime, CTCs appear
in an instant of time $\displaystyle t=t_{0}>0$.

In order to study this axially symmetric metric in Ricci-inverse gravity and
compare this result with those obtained in general relativity, let’s first
review this solution in Einstein’s theory. For this proposal, some geometric
elements associated with the metric are necessary. Explicitly, the metric
components are given as

$\displaystyle\displaystyle g_{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle g_{20}=-r^{2},$ $\displaystyle\displaystyle
g_{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\alpha^{2}r^{2}},$
$\displaystyle\displaystyle g_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle g_{21}=\frac{\beta z}{2r^{2}},$
$\displaystyle\displaystyle g_{22}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-r^{2}t,$ $\displaystyle\displaystyle g_{33}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle r^{2},$ (12)

and their inverses are

$\displaystyle\displaystyle g^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\beta^{2}z^{2}}{4r^{6}}+\frac{t}{r^{2}},$
$\displaystyle\displaystyle g^{01}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle g^{10}=\frac{\alpha^{2}\beta z}{2r^{2}},$
$\displaystyle\displaystyle g^{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle g^{20}=-\frac{1}{r^{2}},$
$\displaystyle\displaystyle g^{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha^{2}r^{2},$ $\displaystyle\displaystyle
g^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{r^{2}}.$ (13)

These metric components lead to the non-zero Christoffel symbols

$\displaystyle\displaystyle\Gamma^{0}_{01}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{r},\quad\quad\Gamma^{0}_{02}=\frac{\alpha^{2}\beta
z+r}{2r},\quad\quad\Gamma^{0}_{11}=\frac{\beta z}{2r^{5}},$
$\displaystyle\displaystyle\Gamma^{0}_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\alpha^{2}\beta^{2}z^{2}}{4r^{5}},\quad\quad\Gamma^{0}_{13}=-\frac{\beta}{4r^{4}},$
$\displaystyle\displaystyle\Gamma^{0}_{22}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\beta^{2}z^{2}+4\alpha^{2}\beta
r^{3}tz+4r^{4}t}{8r^{4}},\quad\quad\Gamma^{0}_{23}=\frac{\alpha^{2}\beta^{2}z}{8r^{4}},$
$\displaystyle\displaystyle\Gamma^{0}_{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\alpha^{2}\beta
z}{2r},\quad\quad\Gamma^{1}_{02}=\alpha^{2}r^{3},\quad\quad\Gamma^{1}_{11}=-\frac{1}{r},$
$\displaystyle\displaystyle\Gamma^{1}_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\alpha^{2}\beta
z}{2r},\quad\quad\Gamma^{1}_{22}=\frac{1}{4}\alpha^{2}\left(\beta
z+4r^{3}t\right),$ $\displaystyle\displaystyle\Gamma^{1}_{23}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\beta}{4},\quad\quad\Gamma^{1}_{33}=-\alpha^{2}r^{3},\quad\quad\Gamma^{2}_{12}=\frac{1}{r},$
$\displaystyle\displaystyle\Gamma^{2}_{22}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2},\quad\quad\Gamma^{3}_{12}=-\frac{\beta}{4r^{4}},\quad\quad\Gamma^{3}_{13}=\frac{1}{r}.$

Thus, the non-zero components of the Ricci tensor are

$\displaystyle\displaystyle R_{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 3\alpha^{2}r^{2},$ $\displaystyle\displaystyle
R_{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3}{r^{2}},$ $\displaystyle\displaystyle
R_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{2}\beta z}{2r^{2}},$
$\displaystyle\displaystyle R_{22}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\left(\beta^{2}+24r^{6}t\right)}{8r^{4}},$
$\displaystyle\displaystyle R_{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-3\alpha^{2}r^{2}.$ (14)

and in the contravariant form become

$\displaystyle\displaystyle R^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\left[\beta^{2}\left(1-6\alpha^{2}r^{2}z^{2}\right)-24r^{6}t\right]}{8r^{8}},$
$\displaystyle\displaystyle R^{01}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{4}\beta z}{2r^{2}},$
$\displaystyle\displaystyle R^{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{3\alpha^{2}}{r^{2}},$
$\displaystyle\displaystyle R^{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-3\alpha^{4}r^{2}$ $\displaystyle\displaystyle
R^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{2}}{r^{2}}.$ (15)

The Ricci scalar is given as

$\displaystyle\displaystyle R=g_{\mu\nu}R^{\mu\nu}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-12\alpha^{2}.$

With these ingredients, the field equations of general relativity,

$R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}+\Lambda g^{\mu\nu}=T^{\mu\nu},$ (16)

can be solved. For such an analysis, pure radiation is chosen as the matter
content whose energy-momentum tensor is defined as

$T^{\mu\nu}=\rho\zeta^{\mu}\zeta^{\nu},$ (17)

with $\displaystyle\zeta^{\mu}=(1,0,0,0)$ being a null vector. Then the field
equations of general relativity are given as

$\displaystyle\displaystyle\rho$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{6\alpha^{4}\beta^{2}r^{2}z^{2}+\alpha^{2}\left[\beta^{2}\left(2\Lambda
r^{2}z^{2}+1\right)+24r^{6}t\right]+8\Lambda r^{6}t}{8r^{8}},$ (18)
$\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\beta
z\left(3\alpha^{2}+\Lambda\right)}{2r^{2}},$ (19) $\displaystyle\displaystyle
0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{2}+\Lambda}{r^{2}},$ (20)
$\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha^{2}r^{2}\left(3\alpha^{2}+\Lambda\right),$
(21) $\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{3\alpha^{2}+\Lambda}{r^{2}}.$ (22)

This set of equations leads to the solution

$\displaystyle\displaystyle\rho$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{2}\beta^{2}}{8r^{8}},$
$\displaystyle\displaystyle\Lambda$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-3\alpha^{2}.$ (23)

Therefore, the axially symmetric metric is a solution of general relativity
with a negative cosmological constant and an energy density that decreases
with $\displaystyle r$ [20, 21].

Now the goal is to study this cosmological solution in Ricci-inverse gravity
and discuss whether CTCs are present in this gravitational theory. To
calculate the anti-curvature tensor using Eq. (9), it is first necessary to
analyze the determinant of the Ricci tensor. For the axially symmetric metric
one finds that

$det[R_{\mu\nu}]=-81\alpha^{6}r^{4}.$ (24)

Then, as the determinant of the Ricci tensor is not zero, it is possible to
determine all components of the anti-curvature tensor. The non-zero components
are

$\displaystyle\displaystyle A^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{6\alpha^{2}\beta^{2}r^{2}z^{2}+\beta^{2}+24r^{6}t}{72\alpha^{2}r^{8}},$
$\displaystyle\displaystyle A^{01}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\beta z}{6r^{2}},$
$\displaystyle\displaystyle A^{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{3\alpha^{2}r^{2}},$
$\displaystyle\displaystyle A^{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{r^{2}}{3},$ $\displaystyle\displaystyle
A^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{3\alpha^{2}r^{2}},$ (25)

and in the covariant form become

$\displaystyle\displaystyle A_{02}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{r^{2}}{3\alpha^{2}},$
$\displaystyle\displaystyle A_{11}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{3\alpha^{4}r^{2}},$
$\displaystyle\displaystyle A_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\beta z}{6\alpha^{2}r^{2}},$
$\displaystyle\displaystyle A_{22}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\beta^{2}-24r^{6}t}{72\alpha^{2}r^{4}},$
$\displaystyle\displaystyle A_{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{r^{2}}{3\alpha^{2}}.$ (26)

And the anti-curvature scalar is given as

$\displaystyle\displaystyle A=g_{\mu\nu}A^{\mu\nu}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{4}{3\alpha^{2}}.$ (27)

With these quantities and considering pure radiation given in the energy-
momentum tensor (17) as the source of matter, the Ricci-inverse gravity field
equations, Eq. (7), become

$\displaystyle\displaystyle\rho$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{216\alpha^{2}r^{8}}\left\\{162\alpha^{6}\beta^{2}r^{2}z^{2}+27\alpha^{4}\left[\beta^{2}\left(2\Lambda
r^{2}z^{2}+1\right)+24r^{6}t\right]+\right\\}$
$\displaystyle\displaystyle+\frac{1}{216\alpha^{2}r^{8}}\left\\{54\alpha^{2}\left(\beta^{2}\kappa
r^{2}z^{2}+4\Lambda r^{6}t\right)-35b^{2}\kappa+216\kappa r^{6}t\right\\},$
$\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\beta
z\left(3\alpha^{4}+\alpha^{2}\Lambda+\kappa\right)}{2r^{2}},$ (29)
$\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{4}+\alpha^{2}\Lambda+\kappa}{\alpha^{2}r^{2}},$
(30) $\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
r^{2}\left(3\alpha^{4}+\alpha^{2}\Lambda+\kappa\right),$ (31)
$\displaystyle\displaystyle 0$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{3\alpha^{4}+\alpha^{2}\Lambda+\kappa}{\alpha^{2}r^{2}}.$
(32)

Solving this set of equations for the energy density $\displaystyle\rho$ and
the cosmological constant $\displaystyle\Lambda$, we get

$\displaystyle\displaystyle\rho$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\beta^{2}\left(27\alpha^{4}-35\kappa\right)}{216\alpha^{2}r^{8}},$
$\displaystyle\displaystyle\Lambda$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{3\alpha^{4}+\kappa}{\alpha^{2}}.$ (33)

With the above results, it can be stated that the axially symmetric metric is
a solution to the Ricci-inverse gravity field equations. Therefore, this
gravitational theory presents CTCs as defined in Eq. (11), and as a
consequence, causality is violated. Comparing with the results of general
relativity, Eq. (III), it is noted that in both cases the cosmological
constant assumes a negative value. With regard to energy density, some
observations must be made. In general relativity, the energy density satisfies
the energy condition and is always positive [20, 21]. In Ricci-inverse
gravity, to obtain a positive energy density, a relation arises between the
constant $\displaystyle\alpha$ and the coupling constant
$\displaystyle\kappa$, i.e., $\displaystyle\kappa<\frac{27\alpha^{4}}{35}$.
Also, it is important to note that the results of general relativity are
recovered when the coupling constant $\displaystyle\kappa$ becomes zero.

In addition to these results that lead to a non-causal universe in Ricci-
inverse gravity, other analyzes have been developed considering different
sources of matter. Two matter contents have been chosen: (i) a scalar field
whose energy-momentum tensor is given as

$T^{\mu\nu(S)}=\partial^{\mu}\phi\partial^{\nu}\phi-\frac{1}{2}g^{\mu\nu}g_{\rho\lambda}\partial^{\rho}\phi\partial^{\lambda}\phi,$
(34)

and (ii) an electromagnetic field described by energy-momentum tensor

$T^{\mu\nu(EM)}=-F^{\mu\alpha}F^{\nu}_{\alpha}+\frac{1}{4}g^{\mu\nu}F_{\beta\alpha}F^{\alpha\beta}.$
(35)

Using Eqs. (34) and (35) in Eq. (7), two sets of field equations are obtained.
However, there are no solutions to these sets of equations. Therefore, these
matter sources prevent the axially symmetric metric given in Eq. (10) from
being a solution of Ricci-inverse gravity. This implies that the causality
violation generated by this metric is avoided for proper matter content.

Although the main objective has been to study the axially symmetric metric
with causality violation, other metrics that present CTCs have also been
investigated in the context of Ricci-inverse gravity. The Gödel metric [23] is
an exact solution of general relativity, but it is not a solution in Ricci-
inverse gravity. The same result occurs for the Gödel-type metric [24]. In
fact, these metrics have that the determinant of the Ricci tensor is zero,
i.e., the inverse of the Ricci tensor is not determined. This means that the
anti-curvature tensor for both metrics can not be calculated. Therefore, these
metrics fail the first test to be studied in Ricci-inverse gravity.

##  IV Conclusion

An alternative theory of gravity has been considered. An anti-curvature tensor
that is defined as the inverse of the Ricci tensor is used to construct the
Ricci-inverse gravity. In this theory an axially symmetric spacetime is
studied. This metric allows the existence of CTCs, closed curves in time that
permit travel to the past. As a consequence, the violation of causality
arises. Considering pure radiation as matter content, our results show that
the axially symmetric metric is a solution in Ricci-inverse gravity. Then this
gravitational model allows for the violation of causality. It is found that
the cosmological constant has a negative value as obtained in general
relativity. But, a positive energy density, which satisfies the energy
conditions, requires a condition involving the Ricci-inverse gravity coupling
constant and the constant $\displaystyle\alpha$, a free parameter of the
metric. These results are consistent with the class I of the model which is
defined as a function of the form $\displaystyle f(R,A)$. Class II, which is
characterized by a function of Ricci scalar and square of the anti-curvature
tensor, is also analyzed. But the axially symmetric metric is not a solution
for this set of equations. For different sources of matter, such as scalar
field and electromagnetic field, it is shown that this spacetime is not a
solution for the gravitational model. Therefore, our results indicate that
there is a relation between the matter content of the universe and the
existence of CTCs or causality violation. Furthermore, the Gödel and Gödel-
type universes were also investigated in Ricci-inverse gravity. Our analysis
shows that these metrics lead to a $\displaystyle det(R_{\mu\nu})=0$, then
there is no Ricci tensor inverse. Thus, it is impossible to construct an anti-
curvature tensor associated with the Gödel-types universes. Therefore, these
exact solutions of general relativity do not satisfy the mandatory condition
to be investigated in Ricci-inverse gravity. Here a natural question arises:
how to investigate metrics that display $\displaystyle det(R_{\mu\nu})=0$ in
Ricci-inverse gravity? Are they simply excluded? or are they a problem of this
alternative theory of gravity? These are open questions and are under
investigation.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2. J.
C. R. S. thanks CAPES for financial support.

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# 6GAN: IPv6 Multi-Pattern Target Generation via Generative Adversarial Nets
with Reinforcement Learning

Tianyu Cui1,2, Gaopeng Gou(✉)1,2, Gang Xiong1,2, Chang Liu1,2, Peipei Fu1,2
and Zhen Li1,2 1.Institute of Information Engineering, Chinese Academy of
Sciences
2.School of Cyber Security, University of Chinese Academy of Sciences
{cuitianyu, gougaopeng, xionggang, liuchang, fupeipei<EMAIL_ADDRESS>
###### Abstract

Global IPv6 scanning has always been a challenge for researchers because of
the limited network speed and computational power. Target generation
algorithms are recently proposed to overcome the problem for Internet
assessments by predicting a candidate set to scan. However, IPv6 custom
address configuration emerges diverse addressing patterns discouraging
algorithmic inference. Widespread IPv6 alias could also mislead the algorithm
to discover aliased regions rather than valid host targets. In this paper, we
introduce 6GAN, a novel architecture built with Generative Adversarial Net
(GAN) and reinforcement learning for multi-pattern target generation. 6GAN
forces multiple generators to train with a multi-class discriminator and an
alias detector to generate non-aliased active targets with different
addressing pattern types. The rewards from the discriminator and the alias
detector help supervise the address sequence decision-making process. After
adversarial training, 6GAN’s generators could keep a strong imitating ability
for each pattern and 6GAN’s discriminator obtains outstanding pattern
discrimination ability with a 0.966 accuracy. Experiments indicate that our
work outperformed the state-of-the-art target generation algorithms by
reaching a higher-quality candidate set.

###### Index Terms:

IPv6, Internet-wide Scanning, Network Measurement, Deep Learning,
Reinforcement Learning

##  I Introduction

The exploration of the global IPv6 address space [1] could enhance the ability
of researchers to conduct wide-ranging assessments of the next-generation
Internet. Except for the particular case [2], the concrete instantiation of
prior exploration work mainly includes passive measurement [3, 4, 5] and
active scanning [6, 7, 8, 9, 10]. Because the former approach is limited by
vantage points to monitor the traffic, fast IPv6 scanning has become a
critical means required by the community.

However, IPv6 active hosts distribute in an extensive address space. A brute-
force scanning approach [11] with current network speed and computational
power will expanse considerable time, rendering exhaustive probing completely
infeasible. The recently proposed target generation algorithms [12, 13, 14,
15, 16, 17] seem the state-of-the-art breakthrough to overcome the problem.
Algorithms are required to learn features of the seed set and predict the
active individuals or regions in the real network space to provide the
candidate set waiting for scanning. The algorithmic design becomes essential
to guarantee the quality of the candidate set.

While prior work has developed sophisticated techniques, IPv6 Internet nature
begets these algorithms still facing challenges: (1) IPv6 addressing pattern.
Network administrators are allowed to freely select IPv6 address configuration
schemes [18], which enables multiple allocation pattern for interface
identifier (IID) in the address. Clients may use Stateless Address
Autoconfiguration and thus produce pseudorandom [19] or EUI-64 IIDs [20],
while servers and routers are usually assigned according to administrator
customs or employing DHCPv6 [21]. The algorithmic inference will be stuck in
trouble as these patterns are required to be opaque according to RFC 7136
[22]. (2) IPv6 aliasing. Prior experience [14] has proposed that the large-
scale aliased regions are the challenge that must be resolved in future IPv6
scanning because these addresses unconditionally respond to queries and seem
not bound to unique devices. While the related alias detection work [8, 15]
has gradually matured, these approaches are all scanning strategies for
generating candidate sets or regions rather than algorithmic optimization.
Therefore, the algorithm is still deceived to learn the aliased regions and
consume a large budget to generate low-quality candidate sets.

To address these problems, we raise a novel consideration to help push the
candidate set from quantity to quality. Firstly, with the application of
unsupervised learning to address sequences [8, 17], addressing patterns could
be clustered into limited categories. We followed the idea to perform a deeper
analysis under each pattern rather than the entire seed set to avoid multiple
pattern interference. Secondly, in the field of natural language processing,
the policy gradient of reinforcement learning [23, 24] helps Generative
Adversarial Nets (GANs) [25] solve the non-differentiable problem of the
discrete text. The generator is guided by the discriminator rewards and keeps
improving with sampling policy just like an agent. We leverage this design to
generate address sequences and increase a new reward mechanism with more
intelligence to discourage aliased address generation.

TABLE I: Comparison of Target Generation Algorithms Approach | Target Generation | Alias Detection | Goal
---|---|---|---
Entropy/IP [13] | Analyzing addressing structures through information entropy | - | Visual address distribution
6Gen [14] | Searching the densest address clusters to provide active regions | Sampling scanning | Remarkable performance
6Tree [15] | Dynamic adjusting search directions with a space tree | Dynamic scanning | Faster time complexity
6GCVAE [16] | Reconstructing addresses through variational autoencoder | - | Deep learning attempts
6VecLM [17] | Predicting address sequences through language modeling | - | IPv6 semantics exploration
6GAN | Multi-pattern target generation through adversarial training | Reinforcement learning | Higher-quality candidates

In this paper, we introduce a deep learning architecture 6GAN to replace
traditional target generation algorithms. 6GAN uses multiple generators to
learn IPv6 addressing patterns determined by known seed classification
methods. These generators employ policy gradient [23] and Monte Carlo search
[26] for the address sequence decision-making process, which is motivated by
two environmental rewards. One reward comes from a multi-class discriminator’s
estimation of whether the generated address conforms to the real pattern
address, the other reward is the evaluation of the aliased prefix detection to
judge whether to update the address prefix. The approach contributes to
uncovering active targets under multiple addressing patterns and avoiding
learning from aliased addresses.

Contributions. Our contributions can be summarized as follows:

  * •

We propose a novel architecture 6GAN to generate diversified non-aliased
active addresses of different addressing pattern types through using multiple
generators guided by rewards from a discriminator and an alias detector.

  * •

We employ a multi-class objective of 6GAN’s discriminator, which can identify
IPv6 addressing pattern categories.

  * •

We implement an alias detection approach embedded in the algorithm by
optimizing the generator, which saves algorithmic budget to generate high-
quality candidates.

  * •

We push the quality of candidate sets to a higher level. Experiments show that
6GAN outperforms state-of-the-art target generation algorithms on multiple
metrics.

Roadmap. Section II summarizes the prior researches related to our work.
Section III introduces the basic knowledge about IPv6 target generation.
Section IV highlights the overall design of 6GAN. Section V shows the
evaluation results and Section VI concludes the paper.

##  II Related Work

Prior work on IPv6 target generation falls into two broad categories: (1)
analyzing seed addresses to understand allocation patterns and (2) designing
algorithms that generate candidate targets to scan. In addition, we will
introduce (3) the related applications using GANs and reinforcement learning.

###  II-A Addressing Pattern Learning

RFC 7707 [27] points out known address configuration schemes and possible
administrator configuration customs to facilitate the exploration of IPv6
address space. The measurement work in the document indicates that most
addresses follow specific patterns. Gasser et al. [8] employed entropy
clustering to classify the known addresses into 6 addressing pattern
categories. The classification results show a great correlation with the
configuration schemes. Cui et al. [17] used IPv62Vec to explore IPv6 semantics
to learn address similarity. Addresses with similar semantic structures are
gathered in the same cluster in the vector space. These prior works determined
the possible addressing patterns in the IPv6 space and laid the foundation for
the target generation in addressing patterns.

###  II-B Target Generation Algorithms

1) Traditional Design Algorithms. Traditional design algorithms refer to
target generation algorithms designed based on human experience and
assumptions.

  * •

Ullrich et al. [12] used a recursive algorithm for the first attempt to target
generation. The algorithm greedily determines the bit by reaching the most
addresses in the seed set in each recursion until remaining reserved bits.

  * •

Foremski et al. [13] proposed Entropy/IP, which employs a Bayesian network to
model the statistical dependence between the values of different entropy
segments. The statistical model could generate addresses for scanning.

  * •

Murdock et al. [14] introduced 6Gen, which generates the densest address range
clusters by combining the closest Hamming distance addresses in each
iteration.

  * •

Zhizhu et al. [15] implemented 6Tree, which performs information entropy to
construct a space tree to discover active regions and dynamically adjust the
exploration direction according to the scan results. The work achieves a
faster linear time complexity compared to 6Gen.

2) Deep Learning Approaches. The exploitation of deep learning approaches to
achieve the algorithm has recently been developed because of the adaptability
to different datasets.

  * •

Cui et al. [16] proposed 6GCVAE to complete the first attempt to employ deep
learning architecture. They used the Variational Autoencoder [28] with gated
convolutional layers to discover active addresses and proved that seed
classification can improve the experimental performance.

  * •

Cui et al. [17] introduced 6VecLM, which utilizes a Transformer network [29]
to build an IPv6 language model, i.e., learning sequence relationships and
using softmax temperature [30] to generate creative addresses.

Table I compares the instantiations of these works with our approach. 6GAN
achieves address discovery in multiple patterns and detecting aliased regions
during model training to generate high-quality candidate sets without wasting
budget.

###  II-C Generative Adversarial Nets and Reinforcement Learning

Generative Adversarial Nets (GANs) [25] are a novel class of deep generative
models with great success in computer vision. However, discrete sequences are
not differentiable, causing little progress in text generation. Yu et al. [31]
leveraged reinforcement learning which treats the process of discriminator
guiding generator as a reinforcement learning policy [23, 24] to tackle the
problem and led more GANs to follow this scheme in natural language
generation. Wang et al. [32] proposed SentiGAN to generate text with multiple
sentiment labels, which is similar to our purpose. While we aim to perform
IPv6 multi-pattern target generation and employ two rewards to co-instruct
generators to complete active address generation and alias detection.

##  III Preliminaries

In this section, we provide the target generation problem definition and a
brief IPv6 background related to our work to help readers understand this
paper.

###  III-A Problem Definition

The IPv6 target generation problem requires an algorithm $\tau$ fed with a
seed set $S$ to predict a possible active candidate set $C$ with a limited
budget $|C|$. Considering the influence of aliased addresses, we assume that
$T$ is the set including all active addresses in the IPv6 space and $T_{a}$ is
a subset of $T$ including all aliased addresses. The algorithm $\tau$ finally
discovers a valid target set $\widehat{C}$:

$\widehat{C}=C\cap T-C\cap T_{a}-C\cap S$ (1)

Where $C\cap T$ could be verified through a scanner like Zmapv6 [11]. $C\cap
T_{a}$ could be obtained by alias detection. $C\cap S$ is collected through
algorithmic search.

To ensure a high-quality candidate set, the algorithm $\tau$ requires
minimizing the objective function $L_{\tau}$ to reach optimal:

$L_{\tau}=|C|-|\widehat{C}|$ (2)

###  III-B IPv6 Background

1) IPv6 Addressing. An IPv6 address consists of a global routing prefix, a
local subnet identifier, and an interface identifier (IID) [18]. While the
global routing prefix is determined to route traffic destined to a Local Area
Network (LAN), the configuration of IID is allowed more freedom to ensure the
uniqueness of the host interface in the local network segment. According to
RFC 7707 [27], these IIDs could be categorized into the following patterns:

  * •

Embedded-IPv4. In which the lowest-order 32 bits or each byte of IID embeds
IPv4 address of the network interface. For instance, 0:0:c0a8:20a or c0:a8:2:a
could embed an IPv4 address 192.168.2.10.

  * •

Embedded-port. In which the lowest-order byte of the IID embeds the service
port and the rest bytes are all zero. For instance, 0:0:0:80 or 0:0:0:50 embed
decimal or hexadecimal port 80 for HTTP.

  * •

IEEE-derived. In which the word “fffe” is inserted between the
Organizationally Unique Identifier (OUI) [19] and the rest of the Ethernet
address. For instance, 0250:56ff:fe89:49be.

  * •

Low-byte. In which all the bytes of the IID are set to zero except the least
significant bytes in one or two lowest-order. For instance, 0:0:0:a or
0:0:1:a.

  * •

Pattern-bytes. In which more than two bytes of the IID have identical values
and belong to the same prefix. These IIDs contain a specific addressing
pattern different from the above.

  * •

Randomized. In which the IID keeps a pseudorandom representation because these
addresses might be privacy addresses by using Stateless Address
Autoconfiguration (SLAAC) [20].

The complex address composition raises the significance of classification.
Target generation algorithms are urgently required deep eyes on each
addressing pattern rather than bearing the pressure of the whole IPv6 address
space.

2) Aliased Address. Aliased addresses refer to all addresses under aliased
prefixes, which unconditionally respond to scan queries. For instance, the
address 2001:db8::20:1a could be recognized as an aliased address because of
the known aliased prefix 2001:db8::/32. These aliased regions could be formed
through configuration like directed mapping to the same server. Prior work
[14] indicates more appearance in large CDN networks. Aliased addresses
seriously affect the accuracy of host discovery approaches because there is no
mapping relationship between these addresses and unique devices. Performing
alias detection has been a consensus in IPv6 scanning.

However, previous work [14, 8] detected the aliased address mainly on the
candidate set and ignored that algorithms still consume the budget to generate
aliased addresses. These aliased addresses mislead the algorithmic inference
during algorithm learning. Even though alias detection is pre-processed on the
seed set, algorithms may reconstruct the alias address during prediction
through sampling strategy. Therefore, an advanced algorithm must consider how
to discourage learning aliased prefixes during algorithmic execution.

Figure 1: The overall architecture of 6GAN. Real-world seed addresses are
classified into $k$ addressing patterns in seed classification. $k$ generators
imitate these pattern types through adversarial training with a multi-class
discriminator. The address sequence decision-making process of the generators
is indicated by the rewards from the discriminator and an alias detector
through reinforcement learning during training.

##  IV 6GAN

6GAN is a target generation architecture that integrates seed classification
and adversarial training with reinforcement learning, which is shown in Figure
1. The architecture could be divided into four objectives: seed
classification, generator learning, discriminator learning, and alias
detection.

The real-world seed addresses will first be performed pattern discovery
through known seed classification methods. Then, $k$ generators will be
respectively fed with the addresses of the $k$ classified addressing patterns
and complete the adversarial training with a discriminator. The goal of the
$i$-th generator $G_{i}$ is to generate addresses with the $i$-th pattern type
to deceive the discriminator $D$, while the goal of the discriminator $D$ is
to distinguish between fake addresses generated by generators and real
addresses with $k$ pattern labels. The learning of generator $G_{i}$ is guided
by the evaluation rewards of generated addresses, which is from the
discriminator $D$ and an alias detector $A$ through reinforcement learning.
The alias detector $A$ is fed with known aliased prefixes to estimate rewards.

###  IV-A Seed Classification

To determine the hidden IPv6 addressing pattern in the real world, we classify
the seed set and determine the number of generators $k$ according to the
classification result. Based on previous experience, we provide the following
three seed classification methods to promote pattern discovery:

  * •

RFC Based. According to possible IPv6 addressing patterns proposed in RFC 7707
[27], the addr6 tool in ipv6toolkit [33] can match the patterns mentioned in
Section III-B. We use the tool to perform the match on the seed set to obtain
RFC-based classification results.

  * •

Entropy Clustering. Gasser et al. [8] proposed entropy clustering, which uses
information entropy of the nybble value under the same prefix in the seed set
as a prefix fingerprint and utilizes k-means [34] to perform unsupervised
clustering to discover the prefix-level pattern set. The hyperparameter $k$
controls the number of generated pattern categories. We label an address
according to the pattern category to which the prefix corresponding to the
address belongs.

  * •

IPv62Vec. Cui et al. [17] proposed IPv62Vec, which implements the mapping from
address space to vector space by learning the context of words in the address.
IPv6 addresses with similar semantic structures are distributed in the same
cluster in the vector space. The method finally employs DBSCAN [35] to find
these clusters to identify possible addressing patterns. We restore the
category of the address corresponding to each vector by using the mapping
relationship between addresses and vectors.

Where the first method is the address division based on measurement
experience. The other two methods employ machine learning approaches and could
discover more unknown addressing patterns. 6GAN could freely choose one of the
classification methods to determine the addressing patterns to be learned by
generators.

###  IV-B Generator Learning

In our model architecture, the target generation problem could be considered
as an address sequence decision-making problem. A hexadecimal IPv6 address can
be denoted as an address sequence
$X_{0:T}=(x_{0},...,x_{t},...,x_{T}),x_{t}\in V=\\{0,1,...,f\\}$, where $V$ is
the vocabulary including all address nybble values to be selected. When
training a $\theta_{i}$-parameterized generator $G_{\theta_{i}}$ at timestep
$t$, the state $s$ is the current produced address nybbles
$X_{0:t-1}=(x_{0},...,x_{t-1})$ and the action $a$ is the next nybble value
$x_{t}$ to be select. Following Sutton et al. [23], the policy
$G_{\theta_{i}}(a|s)$ is the probability $G_{\theta_{i}}(a=x_{t}|s=X_{0:t-1})$
that action selects $x_{t}$ as the next nybble according to the previous
sequence of the current state $X_{0:t-1}$. The action-value function
$Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)$ is the assessment
$Q_{AD_{\phi}}^{G_{\theta_{i}}}(s=X_{0:t-1},a=x_{t})$ of the sequence
$X_{0:t}$ led by the action policy based on the rewards of a
$\phi$-parameterized discriminator $D_{\phi}$ and an alias detector $A$.
Therefore, the objective function $J(\theta_{i})$ of the $i$-th generator
$G_{\theta_{i}}$ is:

$J(\theta_{i})=\sum_{t=1}^{T}G_{\theta_{i}}(x_{t}|X_{0:t-1})Q_{AD_{\phi}}^{G_{\theta_{i}}}(X_{0:t-1},x_{t})$
(3)

Where $Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)$ is a penalty based action-value
function that we follow Wang et al. [32]. A high action value
$Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)$ could be considered that the generated
address is recognized as a fake address by the discriminator or an aliased
address marked by the alias detector. This reminds the generator to update the
current policy to exploit other nybble values $x_{t}$ until the objective
function $J(\theta_{i})$ is minimized.

The action value $Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)$ in 6GAN is composed of
discriminator reward $Q_{D_{\phi}}^{G_{\theta_{i}}}(s,a)$ and alias detection
reward $Q_{A}^{G_{\theta_{i}}}(s,a)$:

$Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)=Q_{D_{\phi}}^{G_{\theta_{i}}}(s,a)+\alpha
Q_{A}^{G_{\theta_{i}}}(s,a)$ (4)

Where $\alpha$ is the hyperparameter that controls the ratio of the two
rewards.

At each timestep $t$, the discriminator and alias detector are required to
evaluate the generator policy to obtain $Q_{AD_{\phi}}^{G_{\theta_{i}}}(s,a)$,
while the generator only keeps the incomplete address sequence $X_{0:t}$
composed of the current state $X_{0:t-1}$ and the action $x_{t}$. To produce a
complete sequence $X_{0:T}$ for judgment, the generator applies a $N$-time
Monte Carlo search [26] with a roll-out policy $G_{\theta_{i}}$ to sample the
future sequence $X_{t+1:T}$:

$\text{MC}^{G_{\theta_{i}}}(X_{0:t};N)=\\{X_{0:T}^{1},...,X_{0:T}^{N}\\}$ (5)

Where $N$ is the sampling times to produce a batch of address samples to more
accurately calculate an assessment of the action value.

All generators of 6GAN use Long Short-Term Memory (LSTM) [36] cells to model
$G_{\theta_{i}}(a|s)$. LSTM is a recurrent neural network outputs the $t$-th
address nybble $x_{t}$ based on the hidden $h_{t}$ retaining the previous
states:

$\begin{split}p(x_{t}|X_{0:t-1})=\text{softmax}(c+wh_{t})\\\ \text{where}\quad
h_{t}=\text{LSTM}(h_{t-1},x_{t-1})\end{split}$ (6)

Where the parameters are a bias matrix $c$ and a weight matrix $w$. Softmax
function achieves the selection probability of $x_{t}$.

It is worth noting that each generator of 6GAN applies the above framework
separately without parameter sharing. Each generator independently learns the
addressing pattern to generate diversified addresses.

###  IV-C Discriminator Learning

Our discriminator uses a multi-class classification objective to help
distinguish between the real active addresses with addressing pattern types
and the generated addresses. The objective is trained by providing the real-
world seed addresses and the synthetic addresses generated from generators.
When 6GAN sets $k$ generators for training, the discriminator classifies the
input addresses into $k+1$ categories by producing a softmax probability
distribution to obtain the class score
$D_{\phi}(X)=(D_{\phi}^{1}(X),...,D_{\phi}^{k+1}(X))$. The $i$-th of the first
$k$ scores represents the probability of a sample being judged as the $i$-th
pattern type address. The $(k+1)$-th score denotes the probability of a sample
being classified into the fake address class. The objective function $J(\phi)$
of the discriminator is:

$J(\phi)=-\sum_{i=1}^{k}\mathbb{E}_{X\sim
p_{i}}[\text{log}D_{\phi}^{i}(X)]-\mathbb{E}_{X\sim
G_{\theta}}[\text{log}D_{\phi}^{k+1}(X)]$ (7)

Where $p_{i}$ is real $i$-th type pattern addresses. The discriminator is
required to minimize the objective function $J(\phi)$ to optimize
discrimination ability.

At each timestep $t$, the discriminator provides the reward
$Q_{D_{\phi}}^{G_{\theta_{i}}}(s,a)$ through evaluating the batches of address
samples generated by generators with $N$-time Monte Carlo search:

$\begin{split}&Q_{D_{\phi}}^{G_{\theta_{i}}}(s=X_{0:t-1},a=x_{t})=\\\
&\left\\{\begin{aligned}
&\textstyle\frac{1}{N}\sum\nolimits_{n=1}^{N}(1-D_{\phi}^{i}(X_{0:T}^{n})),X_{0:T}^{n}\in\text{MC}^{G_{\theta_{i}}}(X_{0:t};N)&t<T\\\
&1-D_{\phi}^{i}(X_{0:t})&t=T\end{aligned}\right.\end{split}$ (8)

Algorithm 1 The adversarial training process in 6GAN

0: $k$ Generators $G_{\theta_{i}}$, $i$ = $(1,...,k)$; Discriminator
$D_{\phi}$; Alias detector $A$; Real-world address seed set with $k$ types of
addressing pattern $S=\\{S_{1},...,S_{k}\\}$

0: Optimal generators $G_{\theta_{i}}$ and discriminator $D_{\phi}$

1: Initialize $G_{\theta_{i}}$, $D_{\phi}$ with random wights $\theta_{i}$,
$\phi$

2: Pre-train $G_{\theta_{i}}$ on $S_{i}$

3: Generate fake addresses $F$ using $G_{\theta_{i}}$ for training $D_{\phi}$

4: Pre-train $D_{\phi}$ on $S\cup F=\\{S_{1},...,S_{k},F\\}$

5: repeat

6: for g-steps do

7: Generate $k$-type fake sequences $X_{0:T}$ using $G_{\theta_{i}}$

8: for $t$ in $0\sim T$ do

9: Calculate $Q_{AD_{\phi}}^{G_{\theta_{i}}}(s=X_{0:t-1},a=x_{t})$ by Eq. 4

10: end for

11: Update generators $G_{\theta_{i}}$ by minimizing Eq. 3

12: end for

13: for d-steps do

14: Generate $k$-type fake addresses $F$ using $G_{\theta_{i}}$

15: Update $D_{\phi}$ using $S\cup F$ by minimizing Eq. 7

16: end for

17: until 6GAN converges

18: return

The discriminator of 6GAN is implemented using Convolutional Neural Networks
(CNN) [37] with multiple filters. Filters of different sizes could contribute
to discovering multiple address sequence structures. This framework enhances
the ability of the discriminator to discriminate different addressing
patterns, which helps to reach optimal in adversarial training.

In 6GAN, $k$ generators and one discriminator will be trained alternately to
achieve their respective goals. Algorithm 1 shows the full details of the
adversarial training on the generators and the discriminator.

###  IV-D Alias Detection

According to prior studies, the discovery work of aliased prefixes has
gradually matured. In the 6GAN model, we consider applying these known aliased
prefixes at the algorithmic level to help prevent the generation of aliased
addresses, which form the alias detector $A$ in the architecture.

Consider an aliased prefix $P_{0:L}=(p_{0},...,p_{t},...,p_{L})$, where $L\leq
T$, a generator with Monte Carlo search generates an address sequence
$X_{0:T}$. The alias detector identifies an aliased address when
$P_{0:L}=X_{0:L}$. Therefore, the alias detector calculates an alias score
$A(X)$ according to the judgment:

$A(X)=\left\\{\begin{aligned} \lambda&&P_{0:L}=X_{0:L}\\\ 0&&P_{0:L}\neq
X_{0:L}\end{aligned}\right.$ (9)

Where $\lambda$ is a hyperparameter to control the reward strength when an
aliased address is detected.

At each timestep $t$, the alias detector evaluates generated the address
sequence $X_{0:T}$ to provide the reward $Q_{A}^{G_{\theta_{i}}}(s,a)$:

$\begin{split}&Q_{A}^{G_{\theta_{i}}}(s=X_{0:t-1},a=x_{t})=\\\
&\left\\{\begin{aligned}
&\textstyle\frac{t}{NL}\sum_{n=1}^{N}A(X_{0:T}^{n}),X_{0:T}^{n}\in\text{MC}^{G_{\theta_{i}}}(X_{0:t};N)&t\leq
L\\\ &0&T\geq t>L\end{aligned}\right.\end{split}$ (10)

Where a positive reward $Q_{A}^{G_{\theta_{i}}}(s,a)$ is only provided on the
prefix part. The coefficient $\frac{t}{L}$ performs hierarchical rewards for
the address sequence $X_{0:L}=(x_{0},..,x_{t},...,x_{L})$. The high indexes of
the prefix (like $x_{L}$) will obtain high rewards when meeting an aliased
prefix. This design forces the generator to more likely update the high index
and helps reduce the wide range changes of the prefix region.

##  V Evaluation

In this section, we present our experimental setup and all experimental
results to indicate 6GAN’s performance.

###  V-A Experimental Setup

1) Dataset. The dataset used in the paper is collected from a public dataset
IPv6 Hitlist [8] and a measurement dataset CERN IPv6 2018. Table II shows the
detail of these datasets.

  * •

IPv6 Hitlist. Gasser et al. [8] provided the public dataset IPv6 Hitlist to
help future IPv6 study. The dataset is formed by the results daily probing
multiple public address sets. They also used scanning strategies to provide
daily updated aliased prefix data. To ensure the reproducibility of the 6GAN’s
performance, we use the address data as the real-world seed set and utilize
the aliased prefix data as the benchmark for 6GAN’s alias detector.

  * •

CERN IPv6 2018. To explore the performance difference of the model on
different datasets, we additionally employ the passive measurement dataset
CERN IPv6 2018 to indicate the 6GAN’s robustness. We passively collected
address sets on China Education and Research Network (CERNET) from March to
July 2018 and provided keeping-active addresses to build the dataset.

2) Verification Method. To ensure the accuracy of the results, we design the
scanning operation to verify the active addresses generated through target
generation algorithms. Firstly, we use Zmapv6 tool [11] to perform scanning
work through multiple protocols including ICMPv6, TCP/80, TCP/443, UDP/53, and
UDP/443. An address is considered an active target when any scanning protocol
obtains a response. Secondly, to reduce the impact of the difference in host
activity at different times, we maintain continuous scanning for three days.
An address responding to the scanning on any one day will be considered an
active target.

3) Evaluation Metric. To evaluate the quality of the generated candidate set,
we design multiple evaluation metrics to measure from various aspects.

  * •

Pattern quality. We propose the pattern quality metric to evaluate the
imitating ability of the generators to each addressing pattern. Given a real-
world address seed set with $k$ types of addressing pattern
$S=\\{S_{1},...,S_{t},...,S_{k}\\}$, a candidate set $C$ is generated by using
the $t$-th pattern generator. The pattern quality of the $t$-th pattern
candidate set $C$ is:

$Pattern(C)=\frac{1}{|C|}\sum_{i=1}^{|C|}\text{min}\\{\psi(C_{i},S_{t_{j}})\\}_{j=1}^{j=|S_{t}|}$
(11)

Where $\psi$ is the Cosine similarity function to evaluate the similarity
between a seed and a generated address.

TABLE II: The Detail of Two Datasets Used in the Paper. Dataset | Description | Period | #Seeds
---|---|---|---
IPv6 Hitlist | Active addresses | June 27, 2020 | 610.9k
| Source addresses |  | 100.0k
| Aliased prefixes |  | 516.1k
CERN IPv6 2018 | Active addresses | March - July 2018 | 90.1k
TABLE III: Results of 6GAN’s Three Seed Classification Methods with 50k Budget on Each Pattern and 50k Total Seeds. RFC Based |  | Entropy Clustering |  | IPv62Vec |
---|---|---|---|---|---
Pattern | #Seeds | #Targets |  | Pattern | #Seeds | #Targets |  | Pattern | #Seeds | #Targets | Common Seed Example
Embedded-IPv4 | 10.1k | 13.0k |  | $E_{1}$ | 9.1k | 1.7k |  | $I_{1}$ | 3.8k | 3.7k | 2001:db8:ff01:2::c8c3:8c07
Embedded-port | 8.9k | 3.7k |  | $E_{2}$ | 9.5k | 1.4k |  | $I_{2}$ | 8.0k | 6.9k | 2001:db8::80
IEEE-derived | 5.0k | 4.0k |  | $E_{3}$ | 5.2k | 2.9k |  | $I_{3}$ | 17.1k | 5.8k | 2001:db8:900::21e:67ff:fe31:4cdf
Low-byte | 4.8k | 1.5k |  | $E_{4}$ | 16.0k | 10.1k |  | $I_{4}$ | 20.5k | 13.5k | 2001:db8:100:100::1
Pattern-bytes | 15.0k | 21.5k |  | $E_{5}$ | 3.3k | 2.4k |  | $I_{5}$ | 0.3k | 0.1k | 2001:db8:ff01:4:face:b00c::a7
Randomized | 6.2k | 10.2k |  | $E_{6}$ | 6.9k | 25.5k |  | $I_{6}$ | 0.3k | 0.1k | 2001:db8:8:68d3:b791:8741:c127:a75
Total | 50.0k | 53.9k |  |  | 50.0k | 44.0k |  |  | 50.0k | 30.1k |
  * •

Novelty quality. The novelty quality metric can measure the difference between
the candidate set $C$ and the seed set $S$ to indicate the algorithmic ability
to generate new address sequences. Given the seed set $S$, the novelty quality
of a candidate set $C$ is:

$Novelty(C)=\frac{e}{|C|}\sum_{i=1}^{|C|}(1-\text{max}\\{\varphi(C_{i},S_{j})\\}_{j=1}^{j=|S|})$
(12)

Where $\varphi$ is the Jaccard similarity function to measure whether the
sampled nybble set is the same as the seed. $e=100$ is a constant to help
enlarge the metric value.

  * •

Diversity quality. The diversity quality metric assessment whether a candidate
set $C$ is a diverse address set, which contains a variety of address
sequences. The diversity quality of a candidate set $C$ is:

$Diversity(C)=\frac{e}{|C|}\sum_{i=1}^{|C|}(1-\text{max}\\{\varphi(C_{i},C_{j})\\}_{j=1}^{j=|C|,j\neq
i})$ (13)

  * •

Hit rate. The hit rate refers to the proportion of active addresses in the
candidate set, which could verify the model’s learning ability. Assume $T$ is
the real active target set in the IPv6 space and $T_{a}$ is the real aliased
addresses set, the hit rate of a candidate set $C$ is:

$Hit(C)=\frac{|C\cap T-C\cap T_{a}|}{|C|}$ (14)

Where $C\cap T$ is the scanning result using Zmapv6 [11]. We use known aliased
prefix set to approximate $T_{a}$.

  * •

Generation rate. The generation rate is the proportion of the active addresses
in the candidate set that are not in the seed set. The metric highlights the
model’s generation ability in the IPv6 space. Given the seed set $S$, assume
the real active target set $T$ and real aliased addresses set $T_{a}$, the
generation rate of a candidate set $C$ is:

$Generation(C)=\frac{|C\cap T-C\cap T_{a}-C\cap S|}{|C|}$ (15)

Figure 2: Three metric scores of all pattern candidates generated by
generators with 50k budget on each pattern and 50k total seeds of IPv6 Hitlist
dataset.

4) Model Setting. 6GAN is required to first transform nybble word in an
address to a word embedding with the dimension size of 200. Word embeddings
are randomly initialized while training. The generators are set as single-
layer LSTM with the hidden dimension size of 200 and the max sample length of
32 nybbles. The discriminator is set as CNN with multiple kernel sizes from 1
to 16 and is added a highway architecture [38] based on the pooled feature
maps.

In adversarial training, the $\alpha$ is 0.9 to control the ratio of the
discriminator reward and the alias detector reward. The reward strength of the
alias detector $\lambda$ is 10. The sampling times $N$ in Monte Carlo search
is set as 15. In the pre-training step, we pre-train generators for 60 steps
and pre-train the discriminator for 20 steps. When adversarial training, the
number of g-steps is 5 and the number of d-steps is 1. The optimization
algorithm is RMSProp.

###  V-B Pattern Target Generation

To explore the influence of addressing pattern on target generation tasks and
6GAN’s performance on each addressing pattern, we separately use three seed
classification methods to classify the seed set to determine the $k$ possible
patterns in the IPv6 space. RFC Based could divide the seeds into 6 addressing
patterns described in RFC 7707 [27]. To keep the number of generation patterns
consistent, we set the parameter $k$ to 6 in entropy clustering and adjust the
parameters of DBSCAN [35] to maintain 6-categories result in IPv62Vec. Table
III shows the classification results and target generation results under the
three classification methods of 6GAN on active addresses in the IPv6 Hitlist
dataset.

TABLE IV: Results of Classification Modes with 50k Budget Allocation and 50k Total Seeds. Seed Classification | Budget Allocation | #Targets | Generation(C)
---|---|---|---
None | 1 | 0.5k | 1.06%
RFC Based | 11:3:3:1:19:10 | 12.7k | 25.43%
Entropy Clustering | 2:1:3:8:1:26 | 16.9k | 33.82%
IPv62Vec | 13:70:40:141:1:1 | 9.1k | 18.19%

There are common addresses between the classified patterns of the different
seed classification methods, which indicates that these methods may find
similar addressing patterns. However, the classification boundaries might be
slightly different. Pattern-bytes pattern includes the most address in RFC
Based method, while more addresses are classified into $E_{4}$ and $I_{4}$
pattern in Entropy Clustering and IPv62Vec. To exploit the pattern generation
performance of 6GAN, we set a 50k budget for each pattern to uncover the
active targets hidden in the pattern. Pattern-bytes, $E_{6}$, and $I_{4}$
enable 6GAN to reach the most hit among the three seed classification methods.

6GAN forces generators to simulate the addressing patterns through adversarial
training to achieve the purpose of deceiving the discriminator. In Figure 2,
we use the pattern quality and hit rate of the candidate sets to illustrate
the learning ability of 6GAN for each addressing pattern. The experiment shows
that 6GAN has a strong ability to imitate most patterns. The low score of the
patterns like $I_{5}$ and $I_{6}$ may be caused by the small size of the
classified seed set. In addition, we calculate the generation rate of each
pattern candidate set to reveal the active user distribution in the addressing
patterns and help guide 6GAN’s budget allocation. Given the generation rates
of $k$ patterns $(r_{1},...,r_{i},...,r_{k})$ and the total budget $|C|$, the
allocated budget of $i$-th pattern is $|C_{i}|$ in optimized 6GAN:

$|C_{i}|=\frac{r_{i}}{\sum_{j=1}^{k}r_{j}}\times|C|$ (16)

Therefore, the budget allocation of 6GAN could be represented as
$(|C_{1}|:...:|C_{k}|)$.

Table IV shows the performance of 6GAN with budget allocation under the three
seed classification methods. We compared the none classification method case
which uses one single generator to learn the seed set. The 18.2-33.8 times
discovered targets compared to none seed classification highlight the
significance of pattern discovery. Under the three classification modes with
budget allocation, Entropy Clustering achieves the best performance on the
IPv6 Hitlist dataset contributed by the high generation rate of $E_{6}$. We
strongly recommend readers to perform the budget allocation work to help 6GAN
reach the best results on target generation.

###  V-C Pattern Discrimination

Through adversarial training with multiple generators, 6GAN’s discriminator
can be optimized to achieve pattern discrimination. To explore the utility of
the discriminator, for a case study, we use 50k seeds in the IPv6 Hitlist
dataset to train 6GAN with RFC Based classification mode and label other 50k
addresses in the dataset with RFC Based classification method to test the
trained discriminator. Table V indicates the pattern discrimination ability of
6GAN’s discriminator. In most of the addressing patterns, the discriminator
keeps an outstanding identification performance. Pattern-bytes obtains a lower
accuracy than other pattern types because of mixing multiple unobvious
patterns. The overall accuracy of the discriminator reaches 0.966 scores in
the discrimination evaluation of the 6 pattern types. We believe that the
trained 6GAN discriminator possess sufficient capacity to recognize addressing
patterns in the network space, which allows researchers to fast infer host
attributes to help wide-range network assessments.

###  V-D Performance of Alias Detection

In the architecture of 6GAN, the de-aliasing work is accomplished through the
supervision of an alias detector when model training. To indicate the alias
detection performance, we operate the alias detection condition and separately
train 6GAN with 50k active addresses and 50k source addresses in the IPv6
Hitlist dataset, where the active addresses are all non-aliased addresses
while the source addresses contain 7.9k aliased addresses and 42.1k non-
aliased addresses. Table VI shows the experimental results of alias detection.

Even if 6GAN is fully trained with the active address set, the sampling
process of the generative model still keeps the opportunity to recombine the
aliased prefix to generate the aliased address. By increasing the reward
guidance for alias detection, 6GAN’s generator could intelligently avoid
exploiting alias regions to achieve effective alias detecting performance.
When forced to learn in the dataset containing considerable aliased addresses,
6GAN’s alias detector can still greatly reduce the generation of aliased
addresses. This design effectively prevents wasting the budget of the
candidate set to generate aliased addresses, thus enabling 6GAN to generate
high-quality candidate sets.

TABLE V: Case Study of Pattern Discrimination Ability of 6GAN’s Discriminator on RFC Based Addressing Patterns. Category | #Labels | # Preds | #Hits | #Errors | Accuracy
---|---|---|---|---|---
Embedded-IPv4 | 4.38k | 4.54k | 4.17k | 0.37k | 0.954
Embedded-port | 0.57k | 0.52k | 0.50k | 0.02k | 0.898
IEEE-derived | 3.19k | 3.37k | 3.18k | 0.19k | 0.998
Low-byte | 12.82k | 12.04k | 11.93k | 0.11k | 0.931
Pattern-bytes | 0.73k | 1.49k | 0.51k | 0.98k | 0.701
Randomized | 28.31k | 28.04k | 28.02k | 0.02k | 0.990
Total | 50.00k | 50.00k | 48.31k | 1.69k | 0.966
TABLE VI: Ablation Study on the Alias Detection with 50k Budget and Active / Source Seeds in the IPv6 Hitlist Dataset. Seed set | Alias Detection | #Aliased Targets | Percentage
---|---|---|---
Active addresses | W/o | 0.01k | 0.02%
Active addresses | W/ | 0.00k | 0.00%
Source addresses | W/o | 6.91k | 13.82%
Source addresses | W/ | 0.01k | 0.02%

###  V-E Quality of Generated Addresses

While in this paper we have described the effectiveness of each component in
the 6GAN architecture in detail, as a target generation approach, we still
want to know the superiority of 6GAN compared to other work. Since a variety
of concrete instantiation has been successful on IPv6 target generation,
previously measuring only on the target quantity under a limited candidate set
could not provide a comprehensive evaluation of these works. We incorporate
the quality of the candidate set into the perspective of judgment and present
multiple metrics to evaluate the quality of the generated addresses, which
could indicate the excellent performance of 6GAN compared to prior studies.

(a) IPv6 Hitlist

(b) CERN IPv6 2018

Figure 3: Comparison of discovered addresses quantity $|\widehat{C}|$ and
generation rate Generation(C) under different sizes of the candidate budget
$|C|$ on the two datasets. 6GAN reaches the best performance through training
with Entropy Clustering mode and setting 6 generators.

1) Baselines. The baselines in our experiments for comparison mainly contain
traditional design algorithms and deep learning approaches.

  * •

Traditional Design Algorithms. Entropy/IP [13], 6Gen [14], and 6Tree [15] are
the prior paradigms of the traditional design algorithm that have shown
significant gains in target generation. Entropy/IP and 6Tree employ entropy
information to construct address segments or space trees used to explore
active targets, while 6Gen uses algorithmic analysis to uncover active
clusters. In this paper, we use Entropy/IP and 6Tree’s open-source code and
reconstruct 6Gen according to the author’s algorithm description to achieve
the baselines of these works.

  * •

Deep Learning Approaches. 6GCVAE [16] and 6VecLM [17] are the recent
explorations of the target generation through deep learning. 6GCVAE is the
generative model using gated convolutional Variational Autoencoder (VAE) [28]
architecture, while 6VecLM leverages Transformer [29] and softmax temperature
[30] to build the IPv6 language model for predicting address sequences. Since
6GAN is the same deep learning architecture, we use 6GCVAE and 6VecLM
implemented through open-source code as the baselines for comparison to
highlight the superiority of 6GAN’s neural network architecture.

TABLE VII: Candidates Quality of Target Generation Algorithms with 50k Budget and 50k Seeds. Approach | Novelty(C) | Diversity(C) | Hit(C) | Generation(C)
---|---|---|---|---
Entropy/IP | 12.37 | 6.80 | 12.03% | 7.88%
6Gen | 11.09 | 2.05 | 14.81% | 10.33%
6Tree | 11.16 | 2.06 | 24.40% | 24.39%
6GCVAE | 12.00 | 7.66 | 13.61% | 9.50%
6VecLM | 12.35 | 6.03 | 33.16% | 12.20%
6GAN | 12.75 | 4.73 | 36.05% | 33.21%

2) Candidates Quality. To comprehensively evaluate the performance of target
generation algorithms, we adopt four metrics to measure the quality of the
candidate sets generated by all approaches. Table VII shows the evaluation
results of all target generation algorithms on the IPv6 Hitlist dataset. As
traditional design algorithms, 6Gen and 6Tree achieve a lower novelty and
diversity quality because they enumerate the addresses in the active regions,
thus less changing the nybble set. Entropy/IP and 6GCVAE keep well exploiting
the new addresses while they are rarely active targets. As deep learning
architecture with generative adversarial nets, 6GAN could generate creative
addresses with high novelty quality. However, 6GAN forces generators to learn
the addressing patterns, the generated addresses keep similar sequences in
each pattern type, thus obtaining a not high diversity quality score. We think
the problem could be improved by setting more pattern types $k$ in seed
classification for generator learning. Through a deep analysis in each
addressing pattern, 6GAN outperforms all the baseline on the hit rate and
generation rate in our experiments.

To explore the robustness of 6GAN to perform target generation under different
datasets, we measure target quantity and generation rate under the different
sizes of the budget with 50k total seeds in the IPv6 Hitlist and CERN IPv6
2018 datasets in Figure 3. Experimental results show that 6GAN could achieve a
stable performance on both the two datasets. While comparing with the state-
of-the-art target generation algorithm 6Tree, 6GAN could discover 1.03-1.33
times more active addresses than 6Tree in the limited budget. In addition,
6GAN’s candidates keep different types of addressing patterns. The generators
are controllable for researchers to generate specific pattern type addresses
as long as training with the corresponding pattern seed data. We believe that
this architecture implements a meaningful target generation work and could
help more researchers chasing IPv6 addressing patterns.

##  VI Conclusion

In this work, we explore the implementation of target generation algorithms to
help address discovery in the global IPv6 address space. We present 6GAN, a
generative adversarial nets architecture with reinforcement learning to
achieve multi-pattern target generation.

6GAN uses multiple generators to learn the addressing patterns existing in
IPv6 networks and optimizes them through the rewards of the discriminator and
alias detector. 6GAN’s generator can generate diversified addresses of
different addressing pattern types. The discriminator employs a multi-class
objective to recognize multiple pattern categories. 6GAN’s alias detector
provides rewards to avoid exploiting aliased prefixes and help 6GAN generate
non-aliased candidate sets. Experiments indicate that 6GAN outperformed the
state-of-the-art target generation algorithms in multiple metrics.

## Acknowledgment

This work is supported by The National Key Research and Development Program of
China (No. 2020YFE0200500, No. 2018YFB1800200 and No.2020YFB1006100) and Key
research and Development Program for Guangdong Province under grant No.
2019B010137003.

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