# Invertibility of generalized g-frame multipliers
in Hilbert spaces

M. Abolghasemi, Y. Tolooei and Z. Moosavianfard  Department of Mathematics,
Faculty of Sciences, Razi University, Kermanshah, Iran.
<EMAIL_ADDRESS>, Y<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the invertibility of generalized g-Bessel
multipliers. We show that for semi-normalized symbols, the inverse of any
invertible generalized g-frame multiplier can be represented as a generalized
g-frame multiplier. Also we give several approaches for constructing
invertible generalized g-frame multipliers from the given one. It is worth
mentioning that some of our results are quite different from those studied in
the previous literatures on this topic.

__2010_ MSC_: 42C15, 47A05, 41A58.
_Keywords:_g-Bessel sequences, g-frames, generalized multipliers,
perturbation.__

## Introduction

In recent years there has been shown considerable interest by functional
analysts in the study of Bessel multipliers as a generalization of the frame
operators, approximately dual frames [7], generalized dual frames [7, Remark
2.8(ii)] and atomic systems for subspaces [10]. In fact, the study of this
class of operators leads us to new results concerning dual frames and local
atoms, two concepts at the core of frame theory.

The notions Fourier and Gabor multipliers were extended to ordinary Bessel
multipliers in Hilbert spaces by Balazs [2], $p$-Bessel sequences in Banach
spaces by Balazs and Rahimi in [13], von Neumann-Schatten setting [9] and
continuous setting in [3]. In [14], sufficient and/or necessary conditions for
invertibility of ordinary Bessel multipliers have determined depending on the
properties of the analysis and synthesis sequences, as well as the symbol.
Later on, in [4], Stoeva and Balazs have considered the representation of the
inverse of an ordinary frame multiplier. Moreover, the invertibility of Bessel
multipliers in a much more general setting has been considered by Javanshiri
and his coauthor in [8, 9].

On the other hand, Rahimi [12] introduced and studied the concept of Bessel
multipliers for $g$-Bessel sequences in Hilbert spaces. Recall that,
$g$-Bessel sequences as an interesting generalization of ordinary Bessel
sequence were first considered by Sun [15, 16]. It seems to the author that
the invertibility of $g$-Bessel multipliers has not been touched so far. The
reader will remark that, $g$-frames are quite different from ordinary frames;
For example, an exact $g$-frame in a Hilbert space is not equivalent to a
$g$-Riesz basis, whereas an exact frame is equivalent to a Riesz basis. This
guarantees that the study of $g$-frames and other related concepts is more
complicated than that of ordinary frames in Hilbert spaces.

Our purpose here is to consider the representation of the inverse of an
invertible $g$-frame multiplier. For this purpose, we discuss a new result
about the dual of $g$-frames. Moreover, in the case where the symbols is semi-
normalized, we show that the inverse of any invertible $g$-frame multiplier
can always be represented as a $g$-frame multiplier with the reciprocal symbol
and dual $g$-frames of the given ones. Finally, we investigate the matrix
representation as well as the diagonalization of operators on a Hilbert space
$\mathcal{U}$ with $g$-frames.

##  1\. preliminaries

In this section we have collected some notations and results which are needed
for the subsequent sections. Throughout the paper $\mathcal{H}$ and
$\mathcal{K}$ are separable Hilbert spaces; $\mathit{I}$ is a subset of
$\mathbb{Z}$ and $\\{\mathcal{K}_{i}\\}_{i\in I}$ is a sequence of closed
subspaces of $\mathcal{K}$. The notation $B(\mathcal{H},\mathcal{K}_{i})$
denotes the Banach space of all bounded linear operators from $\mathcal{H}$
into $\mathcal{K}_{i}$; $Id_{\mathcal{H}}$ denotes the identity operator on
$\mathcal{H}$; $\Lambda$ and $\Gamma$ are used to denote the sequences
$\\{\Lambda_{i}\\}_{i\in I}$ and $\\{\Gamma_{i}\\}_{i\in I}$ with elements
from $B({\mathcal{H}},{\mathcal{K}}_{i})$, respectively. Moreover, we assume
that $\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$ is the Hilbert space

$\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})={\Big{\\{}}\\{x_{i}\\}_{i\in
I}:~{}\forall i~{}x_{i}\in{\mathcal{K}}_{i}~{}{\hbox{and}}~{}\sum_{i\in
I}\|x_{i}\|^{2}<\infty{\Big{\\}}},$

with the inner product given by $\big{<}\\{x_{i}\\}_{i\in I},\\{y_{i}\\}_{i\in
I}\big{>}=\sum_{i\in I}\big{<}x_{i},y_{i}\big{>}.$

Now, let us recall from the definition of g-frame which includes the ordinary
frames and many recent generalizations of ordinary frames.

######  Definition 1.1.

A sequence $\Lambda$ is called a generalized frame or simply a g-frame for
$\mathcal{H}$ with respect to $\\{{\mathcal{K}}_{i}\\}_{i\in I}$ if there are
two positive constants $A_{\Lambda}$ and $B_{\Lambda}$ such that

$A_{\Lambda}\|x\|^{2}\leq\sum_{i\in I}\|\Lambda_{i}x\|^{2}\leq
B_{\Lambda}\|x\|^{2},\quad\quad\quad(x\in{\mathcal{H}}).$ (1.1)

We call $A_{\Lambda}$ and $B_{\Lambda}$ the lower and upper frame bounds,
respectively. In particular, the sequence $\Lambda$ is called a $g$-Bessel
sequence, if only in (1.1) the second inequality holds.

If $\Lambda$ is a $g$-Bessel sequence for $\mathcal{H}$ with respect to
$\\{{\mathcal{K}}_{i}\\}_{i\in I}$, then

$T_{\Lambda}:\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i})\rightarrow{\mathcal{H}};\quad\\{x_{i}\\}_{i\in
I}\mapsto\sum_{i\in I}\Lambda^{*}_{i}(x_{i})$

denote the associated synthesis operator. Its adjoint $T_{\Lambda}^{*}$ is
called the analysis operator of $\Lambda$ which can be obtained as follows

$T_{\Lambda}^{*}:{\mathcal{H}}\rightarrow\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i});\quad x\mapsto\\{\Lambda_{i}(x)\\}_{i\in I}.$

It is proved that $\Lambda$ is a g-frame if and only if $T_{\Lambda}$ is a
bounded operator which maps $\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$
surjectively onto $\mathcal{H}$. In particular, if $\Lambda$ is a g-frame, its
frame operator given by

$S_{\Lambda}(x):=T_{\Lambda}T_{\Lambda}^{*}(x)=\sum_{i\in
I}\Lambda^{*}_{i}\Lambda_{i}(x)\quad\quad\quad(x\in{\mathcal{H}}),$

is a bounded and positive self-adjoint operator in $B(\mathcal{H})$, the
Banach space of all bounded operators from $\mathcal{H}$ into $\mathcal{H}$.
This leads to the following reconstruction formula

$x=\sum_{i\in I}\Lambda^{*}_{i}\Lambda_{i}S_{\Lambda}^{-1}(x)=\sum_{i\in
I}(\Lambda_{i}S_{\Lambda}^{-1})^{*}\Lambda_{i}(x),$

for all $x\in{\mathcal{H}}$. As usual, the sequence
$\widetilde{\Lambda}:=\\{\Lambda_{i}S_{\Lambda}^{-1}\\}_{i\in I}$ is called
the canonical dual g-frame of $\Lambda$ which is a g-frame for $\mathcal{H}$
with frame operator $S_{\Lambda}^{-1}$ and frame bounds $B_{\Lambda}^{-1}$ and
$A_{\Lambda}^{-1}$.

Finally, we recall from [6] that for two g-Bessel sequences $\Lambda$ and
$\Gamma$ and a bounded operator $U:\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i})\longrightarrow\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i})$, the operator
$M_{U,\Lambda,\Gamma}:\mathcal{H}\longrightarrow\mathcal{H}$ defined by

$M_{U,\Lambda,\Gamma}:=T_{\Lambda}UT^{*}_{\Gamma},$

is called generalized multiplier of g-Bessel sequences $\Lambda$ and $\Gamma$
with symbol $U$. Particularly, if for every $j\in I$, we define the operators
$\pi_{j}$ and $\iota_{j}$ as

$\pi_{j}:\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i})\rightarrow{\mathcal{H}}_{j};\quad\\{x_{i}\\}_{i\in
I}\mapsto x_{j},$

and

$\iota_{j}:{\mathcal{H}}_{j}\rightarrow\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i});\quad x_{j}\mapsto\\{\delta_{i,j}x_{j}\\}_{i\in I},$

where $\delta_{i,j}$ denotes the Kronecker delta, then the operator
$M_{U,\Lambda,\Gamma}$ enjoys the following representation

$M_{U,\Lambda,\Gamma}(x)=\sum_{i\in I}\sum_{j\in
I}{\Lambda}_{i}^{*}u_{ij}{\Gamma}_{j}(x),$ (1.2)

where $u_{ij}\in B(\mathcal{H}_{j},\mathcal{H}_{i})$ defined by
$u_{ij}(x_{j})=\pi_{i}U\iota_{j}(x_{j})$ and $U=[u_{ij}]$ is its matrix
description. We observe that, if one restricts the set of diagonal operators
$U=diag\\{u_{i}\\}_{i\in I}$ with $u_{i}\in
B(\mathcal{H}_{i},\mathcal{H}_{i})$, then formula (1.2) becomes considerably
simpler

$M_{U,\Lambda,\Gamma}(x)=\sum_{i\in I}{\Lambda}_{i}^{*}u_{i}{\Gamma}_{i}(x).$
(1.3)

Moreover, if for a weight $m=\\{m_{i}\\}_{i\in}\in\ell^{\infty}(I)$ we
consider $u_{i}:{\mathcal{H}}_{i}\rightarrow{\mathcal{H}}_{i}$ by
$u_{i}(x_{i})=m_{i}x_{i}$, then (1.3) reduces to

$M_{m,\Lambda,\Gamma}(x)=\sum_{i\in I}m_{i}{\Lambda}_{i}^{*}{\Gamma}_{i}(x),$

which has been studied by Rahimi [12] and in a much more general setting by
Javanshiri and Choubin in [9], where, here, and in the sequel
$\ell^{\infty}(I)$ has its usual meanings.

##  2\. Some basic results on invertibility

We commence this section by a discussion of why the invertibility of
multipliers with the form Eq. (1.3) is the main object of study of this paper.
To this end, first let us to note that, on the one hand, it is not hard to
check that the satisfying of g-Bessel sequences $\Lambda$ and $\Gamma$ in the
lower g-frame condition are necessary for the invertibility of a generalized
multiplier $M_{U,\Lambda,\Gamma}$ of the form Eq. (1.2). On the other hand,
for given g-frames $\Lambda$ and $\Gamma$ there exists always infinitely many
non-injective operators $U\in B(\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i}))$
such that the generalized multiplier
$M_{U,\Lambda,\Gamma}=T_{\Lambda}UT^{*}_{\Gamma}$ is invertible whereas the
injectivity of the operator

${\mathcal{M}}_{m}:\ell^{2}(I)\rightarrow\ell^{2}(I);\quad\\{c_{i}\\}_{i\in
I}\mapsto\\{m_{i}c_{i}\\}_{i\in I},$

was a very useful tool in the study of invertible ordinary Bessel multipliers,
see for example . Indeed, it suffices to set

$U:=T_{\Lambda}^{*}S_{\Lambda}^{-1}S_{\Gamma}^{-1}T_{\Gamma}+P_{\ker(T_{\Lambda})}R,$

where $R$ is an arbitrary operator in $B(\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i}))$. This shows that there is too much freedom in the
choice of the operator $U$ in Eq. (1.2) and it seems reasonable to work with
particular classes of multipliers of the form Eq. (1.3). Hence, in what
follows $U$ refers to an operator in $B(\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i}))$ which has the matrix description defined by
$diag\\{u_{i}\\}_{i\in I}$ and $u_{i}\in B({\mathcal{K}}_{i})$ for each $i\in
I$. Moreover, the letter semi-normalized is used for $U$ whenever in addition
to the invertibility of each $u_{i}$ ($i\in I$) the operator $D_{U}$ is also
boundedly invertible, that is, the operator

$D_{U^{-1}}:\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i})\rightarrow\ell^{2}(\oplus_{i\in
I}{\mathcal{K}}_{i});\quad\\{x_{i}\\}_{i\in
I}\mapsto\\{u_{i}^{-1}x_{i}\\}_{i\in I}$

is in $B(\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i}))$. It is worth mentioning
that if $M_{U,\Lambda,\Gamma}$ is invertible for some g-Bessel sequence
$\Gamma$, then routine calculations show that the g-Bessel sequences
$\Lambda$, $\Gamma$, $U\Gamma:=\\{u_{i}\Gamma_{i}\\}_{i\in I}$ and
$U\Lambda:=\\{u_{i}^{*}\Lambda_{i}\\}_{i\in I}$ must satisfy in the lower
g-frame condition.

Our starting point is the following result which for fixed g-Bessel sequence
$\Lambda$ and symbol $U$ characterizes all possible g-Bessel sequence $\Gamma$
that participate to construct invertible generalized multiplier
$M_{U,\Lambda,\Gamma}$.

######  Proposition 2.1.

Let $\Lambda$ be a g-Bessel sequence for $\mathcal{H}$ with respect to
$(\mathcal{K}_{i})_{i\in I}$ and let $U$ be a bounded operator on
$\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$. The following assertions hold.

  1. (1)

The g-Bessel sequences that participate to construct invertible generalized
multipliers with g-Bessel sequence $\Lambda$ and symbol $U$ are precisely the
sequence $\Gamma$ satisfying

$u_{i}\Gamma_{i}=\Lambda_{i}S^{-1}_{\Lambda}T+\pi_{i}\Phi\quad\quad\quad\quad(i\in
I)$ (2.1)

where $\Phi:\mathcal{H}\longrightarrow\ell^{2}(\oplus_{i}\mathcal{K}_{i})$ is
an operator such that $T_{U}\Phi=0$ and $T$ is an invertible operator in
$B(\mathcal{H})$.

  2. (2)

The g-Bessel sequence $\Gamma$ participates to construct invertible
generalized multipliers with g-frame $\Lambda$ and symbol $U$ for which the
analysis operator of $U\Gamma$ obtains the minimal norm if and only if
$u_{i}\Gamma_{i}=\Lambda_{i}S^{-1}_{\Lambda}T$ ($i\in I$) for some invertible
operator $T$ in $B(\mathcal{H})$.

###### Proof.

The backward implication of (1) being trivial, we give the proof of the direct
implication only. To this end, suppose that $\Gamma$ is a g-Bessel sequence
such that $M_{U,\Lambda,\Gamma}$ is invertible. Put $T=M_{U,\Lambda,\Gamma}$
and define the operator
$\Phi:\mathcal{H}\longrightarrow\ell^{2}(\oplus_{i}\mathcal{K}_{i})$ by

$\Phi=UT^{*}_{\Gamma}-T^{*}_{\Lambda}S^{-1}_{\Lambda}T$

Then we observe that

$T_{\Lambda}\Phi=T_{\Lambda}UT^{*}_{\Gamma}-T_{\Lambda}T^{*}_{\Lambda}S^{-1}_{\Lambda}T=0,$

and this completes the proof of (1).

In order to prove (2) it suffices to show that for any g-frame $\Gamma$ which
satisfies in 2.1 we have

$\|T_{U\Gamma}^{*}\|^{2}\geq\Big{(}\widetilde{A}_{\Lambda}\|{T}^{-1}\|^{2}\Big{)}^{-1},$

and $U\Gamma_{o}:=\\{\Lambda_{i}S^{-1}_{\Lambda}T\\}_{i\in I}$ is the unique
g-frame for which

$\|T_{U\Gamma_{0}}^{*}\|^{2}=\Big{(}\widetilde{A}_{\Lambda}\|T^{-1}\|^{2}\Big{)}^{-1},$

where $T=M_{U,\Lambda,\Gamma}$. To this end, by definition, we observe that

$\|x\|^{2}\leq\frac{1}{A_{\Lambda}}\|T_{\Lambda}^{*}x\|^{2}\quad\quad\quad(x\in{\mathcal{H}}).$

It follows that

$\frac{1}{\widetilde{A}_{\Gamma}}=\inf\Big{\\{}A:~{}\|x\|^{2}\leq
A\|T_{\Lambda}^{*}x\|^{2}\quad\forall~{}x\in{\mathcal{H}}\Big{\\}}.$ (2.2)

On the other hand, we have

$\displaystyle\|T^{*}x\|^{2}$ $\displaystyle=\big{<}TT^{*}x,x\big{>}$
$\displaystyle=\big{<}T_{\Lambda}T_{U\Gamma}^{*}T_{U\Gamma}T_{\Lambda}^{*}x,x\big{>}$
$\displaystyle\leq\|T_{U\Gamma}^{*}\|^{2}\big{<}T_{\Lambda}T_{\Lambda}^{*}x,x\big{>}$
$\displaystyle=\|T_{U\Gamma}^{*}\|^{2}\|T_{\Lambda}^{*}x\|^{2}.$

From this, by equality $\|x\|\leq\|T^{-1}\|\|T^{*}x\|$, we deduce that

$\|x\|^{2}\leq\|T^{-1}\|^{2}\|T_{U\Gamma}^{*}\|^{2}\|T_{\Lambda}(x)\|^{2}.$

This together with (2.2) implies that

$\|T^{-1}\|^{2}\|T_{U\Gamma}^{*}\|^{2}\geq\frac{1}{\widetilde{A}_{\Lambda}}.$

In order to prove that $\Gamma_{0}$ is the unique g-frame for which

$\|T_{U\Gamma_{0}}^{*}\|^{2}=\Big{(}\widetilde{A}_{\Lambda}\|T^{-1}\|^{2}\Big{)}^{-1},$

we first make use of Douglas’ Theorem for surjective operators $T_{\Lambda}$
and $T$ and find that there exists a unique operator
$R:{\mathcal{H}}\rightarrow\ell^{2}(\oplus_{i}\mathcal{K}_{i})$ of minimal
norm for which $T=T_{\Lambda}R$, particularly, we have

$\|R\|^{2}=\inf\Big{\\{}B:~{}\|T^{*}x\|^{2}\leq
B\|Rx\|^{2}\quad\forall~{}x\in{\mathcal{H}}\Big{\\}}.$

On the other hand, an argument similar to the proof of [16, Lemma 2.1] shows
that if $x$ has a representation $Tx=\sum_{i\in I}\Lambda^{*}_{i}x_{i}$ for
some sequence $\\{x_{i}\\}_{i\in I}\in\ell^{2}(\oplus_{i}\mathcal{K}_{i})$,
then

$\sum_{i\in I}\|x_{i}\|^{2}=\sum_{i\in
I}\|\Lambda_{i}S_{\Lambda}^{-1}T\|^{2}+\sum_{i\in
I}\|x_{i}-\Lambda_{i}S_{\Lambda}^{-1}T\|^{2}.$

It follows that $R=T_{U\Gamma_{0}}^{*}$ and thus

$\displaystyle\|T_{U\Gamma_{0}}^{*}\|^{2}$
$\displaystyle=\inf\Big{\\{}B:~{}\|T^{*}x\|^{2}\leq
B\|T_{\Lambda}^{*}x\|^{2}\quad\forall~{}x\in{\mathcal{H}}\Big{\\}}$
$\displaystyle=\inf\Big{\\{}B:~{}\|x\|^{2}\leq
B\|T^{-1}\|^{2}\|T_{\Lambda}^{*}x\|^{2}\quad\forall~{}x\in{\mathcal{H}}\Big{\\}}$
$\displaystyle=\frac{1}{\|T^{-1}\|^{2}}\inf\Big{\\{}A:~{}\|x\|^{2}\leq
A\|T_{\Lambda}^{*}x\|^{2}\quad\forall~{}x\in{\mathcal{H}}\Big{\\}}$
$\displaystyle=\frac{1}{\widetilde{A}_{\Lambda}\|T^{-1}\|^{2}}.$

We have now completed the proof of the proposition. ∎

Next we turn our attention to the characterization of g-frames $\Lambda$ that
participate to construct invertible generalized multiplier
$M_{U,\Lambda,\Gamma}$ for given g-frame $\Gamma$ and certain symbol $U$. Here
it should be noted that the class of symbol $U$ satisfying the property of the
next result is quite rich. It contains for instance all positive semi-
normalized sequence $\\{u_{i}\\}_{i\in I}\subset B({\mathcal{K}}_{i})$ and
positive semi-normalized scaler sequence $\\{m_{i}\\}_{i\in
I}\subset(0,\infty)$ as well.

######  Proposition 2.2.

Let $\Gamma$ be a g-frame for $\mathcal{H}$ with respect to
$\\{\mathcal{K}_{i}\\}_{i\in I}$ and let $U$ be a bounded operator on
$\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$. Assume also that the sequence
$V=\\{v_{i}\\}_{i\in I}\subset B({\mathcal{K}}_{i})$ is such that
$u_{i}=v_{i}^{*}v_{i}$ ($i\in I$) and the sequence
$V\Gamma:=\\{v_{i}\Gamma_{i}\\}_{i\in I}$ is a g-frame. Then
$M_{U,\Gamma,\Gamma}$ is invertible and particularly, the g-frame $\Lambda$
participates to construct invertible g-Bessel multiplier
$M_{U,\Lambda,\Gamma}$ if and only if there is an operator
$\Psi:\ell^{2}(\oplus_{i}\mathcal{K}_{i})\longrightarrow\mathcal{H}$ and
invertible operators $T_{1},T_{2}\in B(\mathcal{H})$ such that

$T_{\Lambda}=T_{1}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+T_{2}^{-1}\Psi(Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma})\Big{)}.$

###### Proof.

That $M_{U,\Gamma,\Gamma}$ is invertible follows from the fact that it equal
to the frame operator of g-frame $V\Gamma$. Now, suppose that
$M_{U,\Lambda,\Gamma}$ is invertible and take $\Psi=T_{\Lambda}$ and
$T_{1}=T_{2}=M_{U,\Lambda,\Gamma}$, then we observe that

$\displaystyle T_{\Lambda}$
$\displaystyle=M_{U,\Lambda,\Gamma}M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}$
$\displaystyle=M_{U,\Lambda,\Gamma}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}-M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}\Big{)}$
$\displaystyle=M_{U,\Lambda,\Gamma}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}-M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma})$
$\displaystyle=T_{1}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+T_{2}^{-1}\Psi(Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma})\Big{)}$

Conversely, suppose that $\Lambda$ is a g-frame for which

$T_{\Lambda}=T_{1}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+T_{2}^{-1}\Psi(Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma})\Big{)},$

where $\Psi$ is an operator in
$B(\ell^{2}(\oplus_{i}\mathcal{K}_{i}),\mathcal{H})$ and $T_{1},T_{2}\in
B(\mathcal{H})$ are invertible operators. Then we have

$\displaystyle M_{U,\Lambda,\Gamma}=T_{\Lambda}UT^{*}_{\Gamma}$
$\displaystyle=T_{1}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}+T_{2}^{-1}\Psi(Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma})\Big{)}UT^{*}_{\Gamma}$
$\displaystyle=T_{1}\Big{(}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}UT^{*}_{\Gamma}+T_{2}^{-1}\Psi(UT^{*}_{\Gamma}-UT^{*}_{\Gamma}M^{-1}_{U,\Gamma,\Gamma}T_{\Gamma}UT^{*}_{\Gamma})\Big{)}$
$\displaystyle=T_{1}$

It follows that $M_{U,\Lambda,\Gamma}$ is invertible. ∎

The proof of Theorem 2.4 below which characterizes the invertibility of
generalized g-Riesz multipliers relies on the following proposition.

######  Proposition 2.3.

Let $\Lambda$ be a g-Bessel sequence for $\mathcal{H}$ with respect to
$(\mathcal{K}_{i})_{i\in I}$ and let $U$ be a bounded operator on
$\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$ which is also semi-normalized.
Then the equality of the excess of g-frame $\Gamma$ with the excess of
$\Lambda$, that is, $\dim(\ker(T_{\Gamma}))=\dim(\ker(T_{\Lambda}))$ is
necessary for $\Gamma$ to participate to construct invertible g-Bessel
multipliers with g-frame $\Lambda$ and symbol $U$.

###### Proof.

If we define
$V:\ell^{2}(\oplus\mathcal{K}_{i})\longrightarrow\ell^{2}(\oplus\mathcal{K}_{i})$
by

$\displaystyle V\\{x_{i}\\}_{i\in
I}:=(Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}U)\\{x_{i}\\}_{i\in
I},$

then it is not hard to check that

$\displaystyle ran(V)=\ker(T_{\Lambda}U)$

On the other hand, using the equality

$\displaystyle
ran(T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1})\oplus\ker((T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1})^{*})=\ell^{2}(\oplus\mathcal{K}_{i})$

and the equality
$T_{\Lambda}UT^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1}=Id_{\mathcal{H}}$ we see
that

$\displaystyle ran(V)=V(\ker((T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1})^{*}))$

Hence,we have

$\displaystyle\dim(\ker(T_{\Lambda}))$
$\displaystyle=\dim(U^{-1}(\ker(T_{\Lambda}))$
$\displaystyle=\dim(\ker(T_{\Lambda}U))$
$\displaystyle=\dim(V(\ker((T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1})^{*}))$
$\displaystyle\leq\dim(\ker((T^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1})^{*}))$
$\displaystyle=\dim(\ker((M_{U,\Lambda,\Gamma}^{*})^{-1}T_{\Gamma})$
$\displaystyle=\dim(\ker(T_{\Gamma})).$

Similarly, if we define
$V^{\prime}:=Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}-U^{*}T^{*}_{\Lambda}(M_{U,\Lambda,\Gamma}^{*})^{-1}T_{\Gamma}$,
then, using the equality

$(M_{U,\Lambda,\Gamma}^{*})^{-1}T_{\Gamma}U^{*}T^{*}_{\Lambda}=Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})},$

one can show that

$\displaystyle\dim(\ker(T_{\Gamma}))\leq\dim(\ker(T_{\Lambda}))$

We have now completed the proof of proposition. ∎

The following result completely characterizes the invertibility of generalized
multiplier $M_{U,\Lambda,\Gamma}$ when one of the sequences is a g-Riesz
basis.

######  Theorem 2.4.

Let $\Lambda$ be a g-Riesz basis for $\mathcal{H}$ with respect to
$\\{\mathcal{K}_{i}\\}_{i\in I}$ and let $U$ be a bounded operator on
$\ell^{2}(\oplus_{i\in I}{\mathcal{K}}_{i})$. Then the following assertion
hold.

  1. (1)

If $U$ is semi-normalized, then $M_{U,\Lambda,\Gamma}$ is invertible if and
only if $\Gamma$ is a g-Riesz basis.

  2. (2)

If $\Gamma$ is a g-Riesz basis, then

    1. (a)

the mapping $U\mapsto M_{U,\Lambda,\Gamma}$ from
$B(\ell^{2}(\oplus_{i}\mathcal{K}_{i}))$ into $B({\mathcal{H}})$ is injective;

    2. (b)

$M_{U,\Lambda,\Gamma}$ is invertible if and only if $U$ is semi-normalized;

    3. (c)

if $U$ is semi-normalized, then
$M_{U,\Lambda,\Gamma}^{-1}=M_{U^{-1},\widetilde{\Gamma},\widetilde{\Lambda}}$;

    4. (d)

if $\\{\xi_{j}\\}_{j}$ refers to the orthonormal basis of
$\ell^{2}(\oplus_{i}\mathcal{K}_{i})$ and $K=\sup_{j}\|U(\xi_{j})\|$, then we
have

$K\sqrt{A_{\Lambda}A_{\Gamma}}\leq\|M_{U,\Lambda,\Gamma}\|\leq\sqrt{B_{\Lambda}B_{\Gamma}}\|U\|.$

###### Proof.

(1) First note that by Proposition 2.3 the invertibility of
$M_{U,\Lambda,\Gamma}$ together with the fact that $U$ is semi-normalized, we
have

$\dim(\ker(T_{\Gamma}))=\dim(\ker(T_{\Lambda})).$

From this, we can deduce that $\ker(T_{\Gamma})$ is isomorphic to
$\ker(T_{\Lambda})=\\{0\\}$. It follows that the operator $T_{\Gamma}$ is
injective and thus it is invertible. This means that $\Gamma$ is a g-Riesz
basis. By biorthogonality of the sequences $\widetilde{\Lambda}$, $\Lambda$
and $\widetilde{\Gamma}$, $\Gamma$ the backward implication is trivial. In
fact, $M_{U^{-1},\widetilde{\Gamma},\widetilde{\Lambda}}$ is the inverse of
$M_{U,\Lambda,\Gamma}$.

To prove part (a) of (2), suppose that
$M_{U_{1},\Lambda,\Gamma}=M_{U_{2},\Lambda,\Gamma}$. If $U:=U_{1}-U_{2}\neq
0$, then there exists $\\{x_{i}\\}_{i\in
I}\in{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}$ such that $U\\{x_{i}\\}_{i\in
I}\neq 0$. We now invoke the surjectivity of the operator $T_{\Gamma}^{*}$ to
conclude that there exists $x\in{\mathcal{H}}$ for which
$T_{\Gamma}^{*}(x)=\\{x_{i}\\}_{i\in I}$. It follows that
$UT_{\Gamma}^{*}(x)\neq 0$ whereas $T_{\Lambda}UT_{\Gamma}^{*}(x)=0$. Hence,
we have

$\big{<}T_{\Lambda}UT_{\Gamma}^{*}(x),y\big{>}=0\quad\quad\quad(y\in{\mathcal{H}}).$

This means that $0\neq UT_{\Gamma}^{*}(x)\perp
ran(T_{\Lambda}^{*})={\mathcal{H}}$ which is a contradiction.

Now suppose that $M_{U,\Lambda,\Gamma}$ is invertible, then we observe that

$\displaystyle UT^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}$
$\displaystyle=T^{*}_{\widetilde{\Lambda}}T_{\Lambda}UT^{*}_{\Gamma}M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}$
$\displaystyle=T^{*}_{\widetilde{\Lambda}}M_{U,\Lambda,\Gamma}M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}$
$\displaystyle=T^{*}_{\widetilde{\Lambda}}T_{\Lambda}$
$\displaystyle=Id_{\ell^{2}(\oplus_{i}\mathcal{K}_{i})}.$

This together with part (1) and its proof proves parts (b) and (c).

Finally, in order to prove part (d) of (2), suppose that $\xi_{i_{0}}$ is an
arbitrary element of $\\{\xi_{j}\\}_{j}$. The surjectivity of the operator
$T^{*}_{\Gamma}$ implies that there exists $x_{0}\in\mathcal{H}$ such that
$T^{*}_{\Gamma}(x_{0})=\xi_{i_{0}}$. Hence, we have

$\displaystyle\sqrt{A_{\Gamma}}\|x_{0}\|\leq\|T^{*}_{\Gamma}(x_{0})\|=\|\xi_{i_{0}}\|=1,$

and thus

$\displaystyle\|M_{U,\Lambda,\Gamma}\|=\sup_{x\in{\mathcal{H}},x\neq
0}\dfrac{\|T_{\Lambda}UT^{*}_{\Gamma}(x)\|}{\|x\|}\geq\dfrac{\|T_{\Lambda}UT^{*}_{\Gamma}(x_{0})\|}{\|x_{0}\|}=\sqrt{A_{\Lambda}A_{\Gamma}}\|U(\xi_{i_{0}})\|.$

We have now completed the proof of proposition. ∎

We conclude this section by the following two results on the representation of
the inverse of a generalized g-frame multiplier. The first one looks for a
unique dual g-frame $\Gamma^{\dagger}$ of $\Gamma$ such that for any dual
g-frame $\Lambda^{d}$ of $\Lambda$ the inverse of $M_{U,\Lambda,\Gamma}$ can
be represented using the diagonal operator $U^{-1}$, $\Gamma^{\dagger}$ and
$\Lambda^{d}$, that is, $M_{U,\Lambda,\Gamma}^{-1}$ is again a generalized
g-Bessel multiplier. The second one investigates invertible generalized
g-frame multipliers $M_{U,\Lambda,\Gamma}$ whose inverses can be written as
$M_{U^{-1},\widetilde{\Gamma},\widetilde{\Lambda}}$.

######  Theorem 2.5.

Let $\Lambda$ and $\Gamma$ be g-frames for $\mathcal{H}$ with respect to
$\\{\mathcal{K}_{i}\\}_{i\in I}$ and let $U$ be semi-normalized. If
$M_{U,\Lambda,\Gamma}$ is invertible, then there exists a unique dual g-frame
$\Gamma^{\dagger}=\\{\Gamma^{\dagger}_{i}\\}_{i\in I}$ of $\Gamma$ such that

$M_{U,\Lambda,\Gamma}^{-1}=M_{U^{-1},\Gamma^{\dagger},\Lambda^{d}}.$

for all dual g-frame $\Lambda^{d}$ of $\Lambda$.

###### Proof.

The existence of $\Gamma^{\dagger}$ follows from the fact that

$x=M_{U,\Lambda,\Gamma}^{*}(M_{U,\Lambda,\Gamma}^{-1})^{*}=\sum_{i\in
I}\Gamma_{i}^{*}u_{i}^{*}\Lambda_{i}(M_{U,\Lambda,\Gamma}^{-1})^{*}.$

In detail,
$\Gamma^{\dagger}:=\\{u_{i}^{*}\Lambda_{i}(M_{U,\Lambda,\Gamma}^{-1})^{*}\\}_{i\in
I}$ is a dual g-frame of $\Gamma$ for which
$M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}=T_{\Gamma^{\dagger}}U^{-1}$ and we get

$M_{U,\Lambda,\Gamma}^{-1}=M_{U,\Lambda,\Gamma}^{-1}T_{\Lambda}T^{*}_{\Lambda^{d}}=T_{\Gamma^{\dagger}}U^{-1}T^{*}_{\Lambda^{d}},$

for all dual g-frame $\Lambda^{d}$ of $\Lambda$ as a consequence. Let us now
prove the uniqueness of $\Gamma^{\dagger}$, which is the essential part of the
theorem. To this end, suppose that $\Gamma^{\ddagger}$ is another dual g-frame
of $\Gamma$ for which
$M_{U,\Lambda,\Gamma}^{-1}=M_{U^{-1},\Gamma^{\ddagger},\Lambda^{d}}$, then we
would have the following equality

$\displaystyle
T_{\Lambda^{d}}(U^{-1})^{*}T^{*}_{\Gamma^{\dagger}}=T_{\Lambda^{d}}(U^{-1})^{*}T^{*}_{\Gamma^{\ddagger}},$
(2.3)

for all dual g-frame $\Lambda^{d}$ of $\Lambda$. From this we deduce that
$(U^{-1})^{*}T^{*}_{\Gamma^{\dagger}}=(U^{-1})^{*}T^{*}_{\Gamma^{\ddagger}}$
and thus the invertibility of $U$ implies that
$\Gamma^{\ddagger}=\Gamma^{\dagger}$. Indeed, if for arbitrary $x$ in
$\mathcal{H}$ we set

$\\{x_{i}\\}_{i\in
I}:=(U^{-1})^{*}T^{*}_{\Gamma^{\dagger}}(x)\quad\quad\quad{\hbox{and}}\quad\quad\quad(U^{-1})^{*}T^{*}_{\Gamma^{\ddagger}}(x):=\\{y_{i}\\}_{i\in
I},$

then equality 2.3 implies that $T_{\Lambda^{d}}\\{x_{i}-y_{i}\\}_{i\in I}=0$
for all dual g-frame $\Lambda^{d}$ of $\Lambda$. Specially, for canonical dual
$\widetilde{\Lambda}$ of $\Lambda$ we have

$S_{\Lambda}^{-1}T_{\Lambda}\\{x_{i}-y_{i}\\}_{i\in
I}=T_{\widetilde{\Lambda}}\\{x_{i}-y_{i}\\}_{i\in I}=0$

and thus $T_{\Lambda}\\{x_{i}-y_{i}\\}_{i\in I}=0$. If now for arbitrary
$\\{z_{i}\\}_{i\in I}\in\ell^{2}(\oplus_{i}{\mathcal{K}}_{i})$ and a fixed
$e\in{\mathcal{H}}$ with $\|e\|=1$ we define

$\Psi:{\mathcal{H}}\rightarrow\ell^{2}(\oplus_{i}{\mathcal{K}}_{i});\quad
x\mapsto\big{<}x,e\big{>}\\{z_{i}\\}_{i\in I},$

then we observe that

$\displaystyle\big{<}\\{x_{i}-y_{i}\\}_{i\in I},\\{z_{i}\\}_{i\in I}\big{>}$
$\displaystyle=\big{<}\\{x_{i}-y_{i}\\}_{i\in I},\Psi(e)\big{>}$
$\displaystyle=\big{<}\Psi^{*}P_{\ker(T_{\Lambda})}\\{x_{i}-y_{i}\\}_{i\in
I},e\big{>}$
$\displaystyle=\big{<}(T_{\widetilde{\Lambda}}+\Psi^{*}P_{\ker(T_{\Lambda})})\\{x_{i}-y_{i}\\}_{i\in
I},e\big{>}$ $\displaystyle=\big{<}(T_{\Lambda_{e}^{d}}\\{x_{i}-y_{i}\\}_{i\in
I},e\big{>}$ $\displaystyle=0$

where
$\Lambda_{e}^{d}:=\widetilde{\Lambda}_{i}+\pi_{i}\Psi^{*}P_{\ker(T_{\Lambda})}$
is a dual g-frame of $\Lambda$. This means that

$\\{x_{i}-y_{i}\\}_{i\in I}\perp\ell^{2}(\oplus_{i}{\mathcal{K}}_{i})$

and therefore we should have $\\{x_{i}\\}_{i\in I}=\\{y_{i}\\}_{i\in I}$. ∎

Having reached this state it remains to find conditions guaranteeing the
equality of $\Gamma^{\dagger}$ with $\widetilde{\Gamma}$. This is the subject
matter of the next result.

######  Theorem 2.6.

Suppose that $\Lambda$ and $\Gamma$ are g-frames for $\mathcal{H}$ and that
$U$ is semi-normalized. The following statements are equivalent.

  1. (1)

$M_{U,\Lambda,\Gamma}$ is invertible and the unique g-frame $\Gamma^{\dagger}$
in Theorem 2.5 is $\widetilde{\Gamma}$.

  2. (2)

The optimal upper frame bound of the g-frame $\Gamma^{\dagger}$ is
$\Big{(}\widetilde{A}_{\Gamma}\Big{)}^{-1}$.

  3. (3)

$U\Gamma$ and $\Lambda$ are $Q$-equivalent g-frames.

  4. (4)

The analysis operator of $U\Gamma$ obtains the minimal norm, that is

$\|T^{*}_{U\Gamma}\|^{2}=\Big{(}\widetilde{A}_{\Gamma}\|Q^{-1}\|^{2}\Big{)}^{-1}.$

###### Proof.

The implication (3)$\Leftrightarrow$(4) is proved in Proposition 2.1. Let us
first prove that (1)$\Rightarrow$(2). To this end, suppose that
$\Gamma^{\dagger}$ is the unique dual g-frames of $\Gamma$ for which

$\displaystyle
M^{-1}_{U,\Lambda,\Gamma}=M_{U^{-1},\Gamma^{\dagger},{\Lambda}^{d}},$

for all dual g-frame $\Lambda^{d}$ of $\Lambda$. In light of [1, Theorem 3.4]
we have

$\displaystyle\Gamma^{\dagger}_{i}=\widetilde{\Gamma}_{i}+\pi_{i}\Psi$ (2.4)

for some $\Psi\in B({\mathcal{H}},\ell^{2}(\oplus_{i}{\mathcal{K}}_{i}))$ such
that $T_{\Gamma}\Psi=0$, where, here and in the sequel, $\pi_{i}$ is the
standard projection on the $i$-th component. Observe that
$T_{\Gamma^{\dagger}}=T_{\widetilde{\Gamma}}+\Psi^{*}$ and we get
$S_{\Gamma^{\dagger}}=S_{\widetilde{\Gamma}}+\Psi^{*}\Psi$. We now invoke the
positivity of operators $S_{\Gamma^{\dagger}}$, $S_{\widetilde{\Gamma}}$ and
$\Psi^{*}\Psi$ as well as the equality
$S_{\widetilde{\Gamma}}=S_{\Gamma}^{-1}$ to conclude that

$\displaystyle\|S_{\Gamma^{\dagger}}\|_{\rm op}$
$\displaystyle=\sup_{\|x\|=1}\langle\Gamma^{\dagger}x,x\rangle$
$\displaystyle=\|S_{\widetilde{\Gamma}}\|_{\rm op}+\sup_{\|x\|=1}\|\Psi
x\|^{2}$ $\displaystyle=\|S_{\Gamma}^{-1}\|_{\rm op}+\sup_{\|x\|=1}\|\Psi
x\|^{2}.$ (2.5)

From this, by equality $\Gamma^{\dagger}=\widetilde{\Gamma}$, we deduce that
$\Psi=0$. On the other hand, an argument similar to the proof of Proposition
5.4.4 of [5] with the aid of [16, Lemma 2.1] shows that
$\|S_{\Gamma}^{-1}\|_{\rm op}=\Big{(}\widetilde{A}_{\Gamma}\Big{)}^{-1}$ and
particularly $\|S_{\Gamma^{\dagger}}\|_{\rm
op}=\widetilde{B}_{\Gamma^{\dagger}}$. It follows that
$\widetilde{B}_{\Gamma^{\dagger}}=\Big{(}\widetilde{A}_{\Gamma}\Big{)}^{-1}$.

In order to prove that (2)$\Rightarrow$(3), first note that Eq. (2) together
with the equality of the optimal upper frame bound of the g-frame
$\Gamma^{\dagger}$ with $\Big{(}\widetilde{A}_{\Gamma}\Big{)}^{-1}$ imply that
$\Psi=0$. Hence, the equivalency of the g-frames $U\Gamma$ and $\Lambda$
follows from Eq. (2.4) and the definition of $\Gamma^{\dagger}$.

Finally, the proof will be completed by showing that (3)$\Rightarrow$(1). To
do that, just noting that this is nothing more than routine calculations. ∎

##  3\. Some approaches for constructing invertible generalized multipliers

In this section, we present some approaches for constructing of invertible
generalized multipliers from a given one. In this respect, we first recall the
following perturbation condition from [11].

######  Definition 3.1.

Let $\Lambda=\\{\Lambda_{i}\\}_{i}$ be a sequence in
$B(\mathcal{H},\mathcal{H}_{i})$ we say that a sequence
$\Lambda^{\prime}=\\{\Lambda^{\prime}_{i}\\}_{i}$ in
$B(\mathcal{H},\mathcal{H}_{i})$ is a $\mu$-perturbation of $\Lambda$ if
$\parallel T_{\Lambda}-T_{\Lambda^{\prime}}\parallel\leq\mu$.

######  Theorem 3.2.

Suppose that $\Lambda$ and $\Gamma$ are g-frames for $\mathcal{H}$ and suppose
that $U$ is semi-normalized. For any $\Lambda^{\prime}$ which is a
$\mu$-perturbation of $\Lambda$ with $\mu<\sqrt{A_{\Lambda}}$, there exists a
Bessel sequence $\Gamma^{\prime}$ which is a $\lambda\mu$-perturbation of
$\Gamma$ for some $\lambda>0$ and
$M_{U,\Lambda^{\prime},\Gamma^{\prime}}=M_{U,\Lambda,\Gamma}$.

###### Proof.

First note that Theorem 3.5 of [11] implies that $\Lambda^{\prime}$ is a
g-frame for $\mathcal{H}$ with lower bound
$A_{\Lambda^{\prime}}=(\sqrt{A_{\Lambda}}-\mu)^{2}$. Moreover, it is not hard
to check that the sequence $U\Lambda$ is a g-frame for $\mathcal{H}$. Hence,
we have

$\displaystyle\ell^{2}(\oplus_{i}\mathcal{K}_{i})=\mathcal{R}(T^{*}_{U\Lambda})\oplus\ker(T_{U\Lambda})\hskip
28.45274ptand\hskip
28.45274ptT^{*}_{U\Lambda}T_{\widetilde{U\Lambda}}=T^{*}_{U\Lambda}S_{U\Lambda}^{-1}T_{U\Lambda}=P_{\mathcal{R}(T^{*}_{U\Lambda})}.$

If we define

$\displaystyle\Gamma^{\prime}:=\\{u_{j}^{-1}\Lambda^{\prime}_{i}S^{-1}_{U\Lambda}M_{U,\Lambda,\Gamma}+\pi_{i}P_{\ker(T_{U\Lambda})}T^{*}_{\Gamma}\\}_{i\in
I},$

then we have

$\displaystyle
T_{\Gamma^{\prime}}=T_{\Gamma}U^{*}T^{*}_{\Lambda}T_{\widetilde{\Lambda^{\prime}}}(U^{*})^{-1}+T_{\Gamma}P_{\ker(T_{U\Lambda})}.$

Hence, we observe that

$\displaystyle T_{\Gamma^{\prime}}-T_{\Gamma}$
$\displaystyle=T_{\Gamma}U^{*}T^{*}_{\Lambda}T_{\widetilde{\Lambda^{\prime}}}(U^{*})^{-1}+T_{\Gamma}P_{\ker(T_{U\Lambda})}$
$\displaystyle-
T_{\Gamma}(P_{\mathcal{R}(T^{*}_{U\Lambda})}+P_{\ker(T_{U\Lambda})})$
$\displaystyle=T_{\Gamma}U^{*}T^{*}_{\Lambda}T_{\widetilde{\Lambda^{\prime}}}(U^{*})^{-1}-T_{\Gamma}P_{\mathcal{R}(T^{*}_{U\Lambda})}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{\Lambda^{\prime}}}(U^{*})^{-1}.$
(3.1)

On the other hand, by the Open Mapping Theorem, one can conclude that there
are constants $a,b>0$ such that $a<\|U\|<b$ and thus

$\displaystyle\|T_{\Gamma^{\prime}}-T_{\Gamma}\|_{\rm op}$
$\displaystyle\leq\|T_{\Gamma}\|_{\rm op}\|U^{*}\|_{\rm
op}\|(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})\|_{\rm
op}\|T_{\widetilde{\Lambda^{\prime}}}\|_{\rm op}\|(U^{*})^{-1}\|_{\rm op}$
$\displaystyle\leq\dfrac{b\sqrt{B_{\Gamma}}\mu}{a(\sqrt{A_{\Lambda}}-\mu)}.$

If we set $\lambda=\dfrac{b\sqrt{B_{\Gamma}}}{a(\sqrt{A_{\Lambda}}-\mu)}$,
then $\Gamma^{\prime}$ is a $\lambda\mu$-perturbation of $\Gamma$ and
particularly

$\displaystyle M_{U,\Lambda^{\prime},\Gamma^{\prime}}$
$\displaystyle=T_{\Lambda^{\prime}}UT^{*}_{\Gamma^{\prime}}$
$\displaystyle=T_{\Lambda^{\prime}}U(U^{-1}T^{*}_{\widetilde{\Lambda^{\prime}}}T_{\Lambda}UT^{*}_{\Gamma})+T_{\Lambda^{\prime}}U(P_{\ker
T_{\Lambda^{\prime}}U}T_{\Gamma})$ $\displaystyle=T_{\Lambda}UT^{*}_{\Gamma}$
$\displaystyle=M_{U,\Lambda,\Gamma}.$

This completes the proof. ∎

The next result shows that in Theorem 3.2 above our choice of
$\Gamma^{\prime}$ turns out to be perfect in terms of best approximations with
respect to the norm $\|{\mathcal{X}}\|_{\mathcal{F}}:=\|T_{\mathcal{X}}\|_{\rm
op}$, where ${\mathcal{X}}=\\{{\mathcal{X}}_{i}\\}_{i\in I}$ is in
$g{\mathcal{F}}({\mathcal{H}})$, the set of all g-frames in $\mathcal{H}$,
whenever $M_{U,\Lambda,\Gamma}$ is invertibe g-frame multiplier.

######  Theorem 3.3.

Let $\Lambda$ and $\Gamma$ be g-frames for $\mathcal{H}$ and let $U$ be semi-
normalized symbol such that $M_{U,\Lambda,\Gamma}$ is invertible. If
$\Lambda^{\prime}$ is a $\mu$-perturbation of $\Lambda$ with
$\mu<\sqrt{A_{\Lambda}}$, Then

$\displaystyle\Gamma^{\prime}:=\\{u_{i}^{-1}\Lambda^{\prime}_{i}S^{-1}_{U\Lambda}M_{U,\Lambda,\Gamma}+\pi_{i}P_{\ker(T_{U\Lambda})}T^{*}_{\Gamma}\\}_{i\in
I}$

is a best approximation of $\Gamma$ in $g{\mathcal{F}}({\mathcal{H}})$ and a
$\lambda\mu$-perturbation of $\Gamma$ such that
$M_{U,\Lambda^{\prime},\Gamma^{\prime}}=M_{U,\Lambda,\Gamma}$, where
$\lambda=\frac{b\sqrt{B_{\Gamma}}}{a\sqrt{A_{\Lambda}}-\mu}$.

###### Proof.

According to the theorem 3.2 and its proof, it suffices to show that
$\Gamma^{\prime}$ is the best approximation of $\Gamma$. To this end, suppose
that $\Gamma^{{}^{\prime\prime}}$ is another g-frame for $\mathcal{H}$ such
that $M_{U,\Lambda^{\prime},\Gamma^{{}^{\prime\prime}}}=M_{U,\Lambda,\Gamma}$.
For each $x\in\mathcal{H}$, we have

$\displaystyle
x=M_{U,\Lambda^{\prime},\Gamma^{{}^{\prime\prime}}}(M_{U,\Lambda,\Gamma}^{-1}x)=T_{U\Lambda}T^{*}_{\Gamma^{{}^{\prime\prime}}M_{U,\Lambda,\Gamma}^{-1}}x.$

Hence, $\Gamma^{{}^{\prime\prime}}M_{U,\Lambda,\Gamma}^{-1}$ is a dual g-frame
for $U\Lambda$ and thus there exists a bounded operator
$\Psi:\mathcal{H}\longrightarrow\ell^{2}(\oplus_{i}\mathcal{K}_{i})$ such that

$\displaystyle\Gamma^{{}^{\prime\prime}}_{i}M_{U,\Lambda,\Gamma}^{-1}=u_{i}^{*}\Lambda_{i}S^{-1}_{U\Lambda}+\pi_{i}\Psi$

and $T_{U\Lambda}\Psi=0$. Hence, the equality
$\ell^{2}(\oplus_{i}\mathcal{K}_{i})=\mathcal{R}(T^{*}_{U\Lambda})\oplus\ker(T_{U\Lambda})$,
implies that

$\displaystyle\Psi^{*}\\{x_{i}\\}_{i\in
I}=\Psi^{*}(P_{\mathcal{R}(T^{*}_{U\Lambda})}\\{x_{i}\\}_{i\in
I}+P_{\ker(T_{U\Lambda})}\\{x_{i}\\}_{i\in
I})=\Psi^{*}P_{\ker(T_{U\Lambda})}\\{x_{i}\\}_{i\in I},$

for all $\\{x_{i}\\}_{i\in I}\in\ell^{2}(\oplus_{i}\mathcal{K}_{i})$.
Moreover, we see that

$\displaystyle
T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{U\Lambda}}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})S^{-1}T_{U\Lambda}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})S^{-1}T_{U\Lambda}(P_{\mathcal{R}(T^{*}_{U\Lambda})}+P_{\ker(T_{\theta})})$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})S^{-1}T_{U\Lambda}P_{\mathcal{R}(T^{*}_{U\Lambda})}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{U\Lambda}}P_{\mathcal{R}(T^{*}_{U\Lambda})}.$
(3.2)

Now by equality

$\displaystyle\Gamma^{{}^{\prime\prime}}_{i}=(u_{i}^{*}\Lambda_{i}S^{-1}_{U\Lambda}+\pi_{i}\Psi)M_{U,\Lambda,\Gamma}\hskip
28.45274pt(i\in I)$

we have

$\displaystyle T_{\Gamma^{{}^{\prime\prime}}}\\{x_{i}\\}$
$\displaystyle=\sum_{i\in I}(\Gamma^{{}^{\prime\prime}}_{i})^{*}(x_{i})$
$\displaystyle=\sum_{i\in
I}M^{*}_{U,\Lambda,\Gamma}(S^{-1}_{U\Lambda}\Lambda^{*}_{i}u_{i}+\Psi^{*}\pi_{i}^{*})\\{x_{i}\\}$
$\displaystyle=M_{U^{*},\Gamma,\Lambda}(S^{-1}_{U\Lambda}T_{U\Lambda}\\{x_{i}\\}+\Psi^{*}\\{x_{i}\\})$
$\displaystyle=M_{U^{*},\Gamma,\Lambda}(T_{\widetilde{U\Lambda}}\\{x_{i}\\}+\Psi^{*}\\{x_{i}\\})$

for each $\\{x_{i}\\}_{i\in I}\in\ell^{2}(\oplus_{i}\mathcal{K}_{i})$. Hence,
the equation
$T^{*}_{U\Lambda}T_{\widetilde{U\Lambda}}=P_{\mathcal{R}(T^{*}_{\theta})}$
conclude that

$\displaystyle T_{\Gamma^{{}^{\prime\prime}}}-T_{\Gamma}$
$\displaystyle=M_{U^{*},\Gamma,\Lambda}(T_{\widetilde{U\Lambda}}+\Psi^{*})-T_{\Gamma}(P_{\mathcal{R}(T^{*}_{\widetilde{U\Lambda}})}+P_{\ker(T_{\widetilde{U\Lambda}})})$
$\displaystyle=M_{U^{*},\Gamma,\Lambda}T_{\widetilde{U\Lambda}}-T_{\Gamma}P_{\mathcal{R}(T^{*}_{\widetilde{U\Lambda}})}+M_{U^{*},\Gamma,\Lambda}\Psi^{*}-T_{\Gamma}P_{\ker(T_{\widetilde{U\Lambda}})}$
$\displaystyle=T_{\Gamma}U^{*}T^{*}_{\Lambda}T_{\widetilde{U\Lambda}}-T_{\Gamma}T^{*}_{U\Lambda}T_{\widetilde{U\Lambda}}+M_{U^{*},\Gamma,\Lambda}\Psi^{*}-T_{\Gamma}P_{\ker(T_{\widetilde{U\Lambda}})}$
$\displaystyle=T_{\Gamma}U^{*}T^{*}_{\Lambda}T_{\widetilde{U\Lambda}}-T_{\Gamma}U^{*}T^{*}_{\Lambda^{\prime}}T_{\widetilde{U\Lambda}}+M_{U^{*},\Gamma,\Lambda}\Psi^{*}-T_{\Gamma}P_{\ker(T_{\widetilde{U\Lambda}})}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{U\Lambda}}+M_{U^{*},\Gamma,\Lambda}\Psi^{*}P_{\ker(T_{\widetilde{U\Lambda}})}-T_{\Gamma}P_{\ker(T_{\widetilde{U\Lambda}})}$
$\displaystyle=T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{U\Lambda}}P_{\mathcal{R}(T^{*}_{\widetilde{U\Lambda}})}+(M_{U^{*},\Gamma,\Lambda}V^{*}-T_{\Gamma})P_{\ker(T_{\widetilde{U\Lambda}})}.$

We now invoke the equality
$\mathcal{R}(T^{*}_{\widetilde{U\Lambda}})=\ker(T_{\widetilde{U\Lambda}})^{\perp}$
to deduce that

$\displaystyle\|(T_{\Gamma^{{}^{\prime\prime}}}-T_{\Gamma})\\{x_{i}\\}\|^{2}\geq\|T_{\Gamma}U^{*}(T^{*}_{\Lambda}-T^{*}_{\Lambda^{\prime}})T_{\widetilde{U\Lambda}}P_{\mathcal{R}(T^{*}_{\widetilde{U\Lambda}})}\\{x_{i}\\}\|^{2}$

for all $\\{x_{i}\\}_{i}\in\ell^{2}(\oplus_{i}\mathcal{K}_{i})$. By using
equations (3) and (3), we have

$\displaystyle\|(T_{\Gamma^{{}^{\prime\prime}}}-T_{\Gamma})\|_{\rm
op}^{2}\geq\|(T_{\Gamma}-T_{\Gamma^{\prime}})\|^{2}_{\rm op}.$

Therefore
$\|(T_{\Gamma^{{}^{\prime\prime}}}-T_{\Gamma})\|_{\mathcal{F}}\geq\|(T_{\Gamma}-T_{\Gamma^{\prime}})\|_{\mathcal{F}}$
and the proof is completed. ∎

We conclude this paper with the following necessary condition for
invertibility of generalized g-Bessel multiplier which is quite different from
those studied in the previous literatures on this topic.

######  Theorem 3.4.

Let $\Lambda$ be a g-frame for $\mathcal{H}$ and $\Lambda^{d}$ be the dual
g-frame of $\Lambda$. Assume that $\Gamma$ is a sequence of bounded operators
from $\mathcal{H}$ into ${\mathcal{K}}_{i}$ ($i\in I$) which satisfies the
following two conditions:

  1. (1)

$\lambda:=\sum_{i\in I}\parallel\Lambda_{i}-\Gamma_{i}\parallel^{2}<\infty$;

  2. (2)

$\mu:=\sum_{i\in
I}\parallel\Lambda_{i}-u_{i}^{*}\Gamma_{i}\parallel\parallel\Lambda_{i}^{d}\parallel<1$.

If $M$ denote the operator $M_{U,\Gamma,\Lambda^{d}}$, then $\Gamma$ is a
g-frame for $\mathcal{H}$ with bounds
$\frac{1}{\|U\|^{2}B_{\Lambda^{d}}}(1-\mu)^{2}$ and
$B_{\Lambda}(1+\frac{\sqrt{\lambda}}{\sqrt{B_{\Lambda}}})^{2}$ and $M$ is
invertible on $\mathcal{H}$ and for all $x\in\mathcal{H}$

$\displaystyle\frac{1}{1+\mu}\parallel x\parallel\leq\parallel
M^{-1}x\parallel\leq\frac{1}{1-\mu}\parallel x\parallel.$

###### Proof.

We first show that $\Gamma$ is a g-Bessel sequence. To do that, we define

$\displaystyle
T_{\Gamma}:\ell^{2}(\oplus_{i}\mathcal{K}_{i})\longrightarrow{\mathcal{H}},\hskip
14.22636ptT_{\Gamma}(\\{x_{i}\\}_{i\in I})=\sum_{i\in
I}\Gamma^{*}_{i}(x_{i}).$

The condition (1) implies that $T_{\Gamma}$ is well defined and
$\|T_{\Gamma}\|_{\rm op}\leq\sqrt{\lambda}+\sqrt{B}$. Moreover, by condition
(2), for each $x\in\mathcal{H}$, we have

$\displaystyle\|x-M_{U,\Gamma,\Lambda^{d}}x\|$ $\displaystyle=\|\sum_{i\in
I}(\Lambda^{*}_{i}-\Gamma_{i}^{*}u_{i})(\Lambda_{i}^{d}(x))\|$
$\displaystyle\leq\sum_{i\in
I}\|\Lambda^{*}_{i}-\Gamma_{i}^{*}u_{i}\|\|\Lambda_{i}^{d}(x)\|\|x\|$
$\displaystyle=\mu\|x\|.$

This means that $M_{U,\Gamma,\Lambda^{d}}$ is invertible and
$\frac{1}{1+\mu}\leq\|M^{-1}\|\leq\frac{1}{1-\mu}$. Particularly, the
invertibility of $M$ implies that every $x\in\mathcal{H}$ can be written as

$\displaystyle x=MM^{-1}x=\sum_{i\in
I}\Gamma_{i}^{*}u_{i}(\Lambda_{i}^{d}(M^{-1}x)).$

This leads to

$\displaystyle\|x\|^{2}=\langle x,x\rangle$ $\displaystyle=\langle\sum_{i\in
I}\Gamma_{i}^{*}u_{i}(\Lambda_{i}^{d}(M^{-1}x)),x\rangle$
$\displaystyle\leq\sum_{i\in
I}\langle\Lambda^{d}_{i}(M^{-1}x),u_{i}^{*}\Gamma_{i}x\rangle$
$\displaystyle\leq\left(\sum_{i\in
I}\|\Lambda^{d}_{i}(M^{-1}x)\|^{2}\right)^{{\frac{1}{2}}}\left(\sum_{i\in
I}\|u_{i}^{*}\Gamma_{i}x\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{D}\|M^{-1}x\|\left(\|\sum_{i\in
I}u_{i}^{*}\Gamma_{i}x\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{D}\frac{1}{1-\mu}\|x\|\|U^{*}T^{*}_{\Gamma}x\|$
$\displaystyle\leq\sqrt{D}\frac{1}{1-\mu}\|x\|\|U^{*}\|\|T^{*}_{\Gamma}x\|.$

Therefore,

$\displaystyle\frac{1}{\parallel U\parallel D}(1-\mu)^{2}\parallel
f\parallel^{2}\leq\parallel T^{*}_{\Gamma}f\parallel$

This completes the proof. ∎

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# A Trusted, Verifiable and Differential Cyber Threat Intelligence Sharing
Framework using Blockchain

Kealan Dunnett∗, Shantanu Pal∗∗, Guntur Dharma Putra§, Zahra Jadidi§§, Raja
Jurdak∗ ∗School of Computer Science, Queensland University of Technology,
Brisbane, QLD 4000, Australia
∗∗School of Information Technology, Deakin University, Melbourne, VIC 3125,
Australia
§School of Computer Science and Engineering, University of New South Wales,
Sydney, NSW 2052, Australia
§§School of Information and Communication Technology, Griffith University,
Gold Coast Campus, QLD 4222, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Cyber Threat Intelligence (CTI) is the knowledge of cyber and physical threats
that help mitigate potential cyber attacks. The rapid evolution of the current
threat landscape has seen many organisations share CTI to strengthen their
security posture for mutual benefit. However, in many cases, CTI data contains
attributes (e.g., software versions) that have the potential to leak sensitive
information or cause reputational damage to the sharing organisation. While
current approaches allow restricting CTI sharing to trusted organisations,
they lack solutions where the shared data can be verified and disseminated
‘differentially’ (i.e., selective information sharing) with policies and
metrics flexibly defined by an organisation. In this paper, we propose a
blockchain-based CTI sharing framework that allows organisations to share
sensitive CTI data in a trusted, verifiable and differential manner. We
discuss the limitations associated with existing approaches and highlight the
advantages of the proposed CTI sharing framework. We further present a
detailed proof of concept using the Ethereum blockchain network. Our
experimental results show that the proposed framework can facilitate the
exchange of CTI without creating significant additional overheads.

###### Index Terms:

Cyber Threat Intelligence, Sharing Information, Privacy, Trust, Verifiability,
Accountability, Blockchain.

##  I Introduction

The rapid evolution of the threat landscape in recent years has caused many
organisations to reassess the risks associated with their digital
infrastructure [1]. For example, during the 2020-2021 financial year, almost
500 ransomware cybercrimes were reported in Australia, an increase of 15%
compared to the previous year [2]. As a result, many organisations have
started integrating Cyber Threat Intelligence (CTI) into traditional Cyber
Security Risks Management (CSRM) frameworks to try and achieve a more
proactive security posture [3]. CTI can be defined as the collection and
synthesis of evidence-based knowledge about actors, threats, and
vulnerabilities of harmful events occurring in cyberspace. Subsequently, the
contents of CTI are highly variable in nature and can range from Indicators of
Compromise (IoC) to sophisticated information detailing an attacker’s
Techniques, Tactics and Procedures (TTPs) [4].

Given that integration of CTI data into traditional CSRM frameworks can be
utilised to provide organisations with a more proactive defense posture, inter
organisational CTI sharing holds additional benefits [3]. In the case where
one organisation (e.g., a CTI producer) has been attacked, sharing CTI has the
potential to allow other organisations (e.g., a CTI consumer) to proactively
implement mitigation strategies [4]. Note, we refer to CTI producers as the
entities who share CTI, while CTI consumers are the entities that use shared
CTI. As a result, in recent years several Open-source/Vendor-created CTI
sharing platforms have been developed.

###  I-A Motivation and Problem Statement

In many cases, sharing CTI data in an uncontrolled way can result in CTI
producers sustaining reputational damage. Subsequently, for cases where CTI
contains sensitive data, trust-based data sharing has been highlighted as a
possible solution [4]. Trust-based CTI sharing ensures that potentially
sensitive CTI data is shared with consumers with which pre-established or
developed trust exists. Moreover, trust issues are significant when sharing
takes place in large-scale industrial systems, e.g., Industry 4.0 [5].

While trust-based sharing has the potential to reduce sharing risks, CTI
producers require more fine-grained control over what data they share. For
instance, in [4], it is highlighted that organisations are likely to differ in
their perceptions of what data is considered sensitive. As a result, CTI
sharing platforms need to provide organisations with the flexibility to share
CTI in a way that caters to this variation. Organisations are also likely to
differ in their perception of how trust-based sharing should be facilitated.
For instance, an organisation A may determine their trust in another
organisation B based on B’s attribute or using identity-based policies
specific to B. Subsequently, constraining organisations to share in a fixed
way, either through share or deny or policies, or group-based policies that
share or deny CTI with specific groups, does not provide the granularity
required for versatile CTI sharing. We propose a framework that allows CTI
producers to share sensitive CTI with consumers in a differential way. We
refer to differential as the ability of CTI producers to control the amount of
information exchanged with CTI consumers at a fine-grained level (more
discussion in Section III-A).

Furthermore, CTI consumers should have the ability to verify requested
information to ensure that the supplied intelligence has not be tampered with
or altered by the CTI producer. Subsequently, an efficient verification
mechanism which simultaneously preserves the privacy in data sharing is
required. Thus, we argue that CTI sharing must be trusted, performed in a
verifiable way, and without compromising the privacy of sensitive CTI data.

Several blockchain-based CTI sharing proposals have been proposed [6] in
recent years. Blockchain has a number of salient properties that can be
leveraged to facilitate CTI sharing. For example, among others, blockchain’s
consensus, auditability, and immutability properties allow organisations to
interact with one another in a distributed way without the need of a trusted
third party. Another advantage of using blockchain to facilitate CTI sharing
is verifiability. Considering that the information stored on the blockchain is
immutable, CTI consumers are able to validate the integrity of received data
and, therefore, can be sure it has not been tampered with or changed [7].

###  I-B Contributions

Existing proposals only allow CTI producers to share intelligence in fixed way
using a platform defined trust-based policy scheme. While these approaches do
provide verifiable and trust-based sharing, constraining CTI producers to
share in a fixed way, e.g. binary or group-based sharing, has the potential to
limit the effectiveness of sharing between organisations. Subsequently, in our
work we define a blockchain-based framework which provides CTI producers with
the ability to share CTI differentially using policies/metrics which they
define. Moreover, we further demonstrate how CTI producers can share in this
way without compromising the verifiability of the intelligence from the CTI
consumers perspective. To the best of our knowledge, our work is the first to
use blockchain for trusted, verifiable, and differential CTI sharing. The
major contributions of this paper are:

  * •

We propose the concept of differential sharing within the CTI sharing context.

  * •

We detail how differential sharing can provide more granuality compared to
existing solutions by allowing data segmentation and pluggable trust-based
policy/metric schemes, without compromising verifiability through the use of
two integrity hash schemes.

  * •

We discuss a detailed design for the proposed CTI sharing framework, its
system functionality, and communications among the various components.

  * •

We demonstrate and evaluate a detailed proof of concept implementation using
Ethereum private blockchain. The experimental results indicate that the
proposed framework is feasible and creates insignificant overheads compared to
a baseline sharing scenario.

  * •

We present a privacy and trust analysis of the proposed framework.

Next, in Section II, we discuss the related work. In Section III, we introduce
the proposed CTI sharing framework. Section IV describes and evaluates the
proof of concept implementation of the proposed framework and present the key
results. Moreover, this section also analysis the privacy and trust of the
proposed framework. Finally, Section V concludes the paper and highlights the
future work.

TABLE I: Available blockchain-based CTI sharing proposals and their comparison
with our work.

Ref. | Differential | Sensitive | Producer-Defined | Verifiable | Blockchain
---|---|---|---|---|---
| Sharing | Data Grouping | Trust Scheme | Sharing | Platform
[8] | ✗ | ✗ | ✗ | ✓ | CITA
[9] | ✗ | ✗ | ✗ | ✓ | Ethereum
[6] | ✗ | ✗ | ✗ | ✓ | Ethereum
[10] | ✗ | ✗ | ✗ | ✓ | Hyperledger
[11] | ✗ | ✗ | ✗ | ✓ | Hyperledger
[12] | ✗ | ✗ | ✗ | ✓ | Hyperledger
[13] | ✗ | ✗ | ✗ | ✓ | Hyperledger
[Our Work] | ✓ | ✓ | ✓ | ✓ | Ethereum

##  II Related Work

In Table I, we provide a list of the available related works on blockchain-
based CTI sharing and their comparison with our work. For example, The authors
in [8] developed a blockchain-based CTI sharing platform using the CITA model.
This model provides CTI producers with the ability to store sensitive data
off-chain. When sharing this CTI data with authorised CTI consumers, the
authors propose using point-to-point communication to ensure fine-grained
access control over which CTI consumers receive the sensitive data. To ensure
verifiability, a hash of the sensitive data is stored on-chain such that CTI
consumers can validate the integrity of data received off-chain. While [8]
provides trust-based verifiable sharing, CTI producers are constrained to
share CTI data with consumers in a coarse-grained and binary way.

‘BloCyNfo-Share’ is a blockchain-based CTI sharing framework proposed in [9].
It utilises proxy re-encryption and attribute-based encryption to achieve
fine-grained access control. When sharing, CTI producers provide an encrypted
copy of the data to a cloud server. When a CTI consumer requests access to
this CTI, the CTI producer generates a re-encryption key and adds it to a
smart contract. This re-encryption key is subsequently used by the cloud
server to generate a new cipher text. The resulting cipher text is
subsequently sent to the CTI consumer, who can decrypt it given they satisfy
the policy specified by the CTI producer. Similar to [8], in this framework,
CTI producers are limited to making binary sharing decisions. Thus, unlike our
approach, proposals [9] and [8] do not provide differential sharing.

Figure 1: Illustration of our differential CTI sharing compared to traditional
binary and group-based CTI sharing approaches.

In [6], the blockchain-based CTI sharing platform TRusted Anonymous Data
Exchange (TRADE) is proposed. To facilitate the exchange of CTI data, the
TRADE platform proposes the use of off-chain data storage, policy-based access
control, on-chain policy validation and off-chain data exchange using Trusted
Automated Exchange of Intelligence Information (TAXII) [14]. When a CTI
producer wants to share, they add an integrity hash and access policy (e.g.,
eXtensible Access Control Markup Language (XACML) policy) to the blockchain.
However, TRADE again limits CTI producers to binary sharing decisions.

In contrast to [9], proposal [10] utilises the existing CTI sharing platform
MISP [15] to facilitate the exchange of CTI between organisations. To ensure
that the privacy of sensitive data is maintained, a Trustworthy API for Threat
Intelligence Sharing (TATIS) Reverse Proxy is used. This proxy uses ciphertext
attribute-based encryption to restrict the access of sensitive data to trusted
organisations. To provide verifiability, CTI producers store a hash of the
original data on-chain. However, like the above proposals, [10] also limits
producers to binary sharing decisions.

In a few other proposals, e.g., [11], [12], and [13], a group-based approach
for sensitive CTI sharing is utilised. Instead of limiting CTI producers to
binary sharing decisions, a group-based approach allows CTI producers and
consumers to exchange sensitive CTI as part of a trusted group. Moreover,
these proposals utilise the Traffic Light Protocol (TLP). Instead of limiting
CTI producers to share in a binary way, more or less sensitive data can be
shared using a layered approach. In this case, layered refers to a grouping of
organisations into trust groups (e.g., red, amber, green, and white), in which
CTI of varying sensitivity can be shared. However, the major limitation of a
group-based approach is that it assumes organisations trust each other in a
pre-defined and fixed way. While these approaches extend sharing beyond binary
decisions, we argue they still lack the granularity required for more
effective CTI sharing. Subsequently, our proposal allows CTI producers to
share differentially, and therefore supports dynamic and time-varying trust
relationships.

In summary, the existing blockchain-based CTI sharing proposals limit a CTI
producer’s ability to share sensitive CTI in a coarse-grained manner, either
based on binary share/deny policies or as part of a pre-defined trust group.
In most cases, the existing solutions provide trust-based policies/metrics
that limit the dissemination of sensitive data to trusted organisations, but
they fail to deliver more granular sharing. Given that a more fine-grained
sharing mechanism is required to facilitate effective CTI sharing, we propose
a framework with the concept of differential sharing discussed in the next
section.

##  III Proposed Framework

In this section, we present the proposed blockchain-based CTI sharing
framework. First, we discuss the concept of differential CTI sharing in
detail. Then we present the various components of the architecture. We also
discuss the system functionality and communication among these components.

###  III-A Differential Sharing

In Section I, we highlight that organisations are likely to differ in their
perceptions of data sensitivity and what trust-based policy/metrics should be
used to determine a CTI consumers access. We propose the concept of
differential CTI sharing, which aims to address this challenge. Figure 1
compares differential CTI sharing with binary and trusted group-based CTI
sharing. From this comparison, we highlight that differential sharing can
tailor the CTI data to be shared with each CTI consumer. As a result, the
proposed differential sharing approach does not restrict sharing to only
highly trusted CTI consumers, and therefore, allows other CTI consumers to
benefit from CTI data they otherwise would not have received.

Given that CTI data contains many attributes, e.g., chained objects in
Structured Threat Information Expression (STIX), their individual sensitivity
can be evaluated by the CTI producer. Furthermore, as these attributes are
likely to vary in their sensitivity, attributes of similar sensitivities can
be grouped together. For example, chain-objects of type vulnerability often
contain specific software versions that could be valuable information to
potential attacks, while chain-objects of type indicator often only contain
IoC. As a result, based on the information contained within specific STIX
chain-objects, Fig. 2 demonstrates how CTI attributes can be broken down into
a variable number of sensitive data groups.

Figure 2: Sensitive data groups based on CTI attributes.

Once the CTI has been segmented using the above method, a subset of these
sensitive data groups can be given to CTI consumers based on a single or set
of trust-based policies/metrics (e.g., XACML policy). By allowing CTI
producers to segment CTI data into groups based on a sensitivity assessment, a
variable proportion of these groups can be shared with each CTI consumer. To
ensure verifiability of the received data, we propose the use of two group-
based hashing approaches: single hash and multi-hash. Given that CTI consumers
receive a distinct number of the sensitive data groups, a set of integrity
hashes constructed using either of these two approaches is required. In Fig. 3
we illustrate how these group-based hashes can be constructed using a single
and multi hash approach.

Figure 3: Proposed single hash (a) and multi hash (b) approaches. Note, dotted
boxes represent integrity hashes. ($N_{i}$ = One-time random nonce).

Sharing CTI using the proposed differential approach gives CTI producers the
ability to share with more granularity compared to traditional solutions,
e.g., binary and group-based. We further note that greater granularity is
achieved in two ways: (i) CTI producers are given the ability to define what
subset of the CTI data is shared with an individual or group of CTI consumers,
and (ii) CTI producers are also able to change how they segment CTI data and
the policies/metrics they use to evaluate trust each time they share.

###  III-B Architectural Components

In Fig. 4, we illustrate the proposed blockchain-based CTI sharing framework.
It contains the following components:

  * •

Organisation: An entity that participates in CTI sharing. Organisations can
assume the role of a producer and/or consumer. Organisations can participate
in both roles over time. However, we assume that an organisation performs only
one role at a time for a specific interaction.

  * •

Request Manager: A component within an organisation responsible for handling
and submitting access requests of CTI data. The request manager role is
primarily broken into two tasks, submitting an access request and processing
an access request. They are as follows:

    1. 1.

Submitting an access request occurs when new CTI is made available. Given that
another organisation has added a CTI, the request manager submits an access
request to the blockchain.

    2. 2.

When an organisation adds a new CTI, the organisation’s request manager is
responsible for processing the access requests added by other organisations.

  * •

Data Manager: A component inside of an organisation responsible for storing
sensitive CTI data and their corresponding access policies. This component is
also responsible for indicating to other organisations that new CTI is
available. Subsequently, it adds content on and off-chain when an organisation
acts as a CTI producer.

  * •

Access Manager: A component inside an organisation responsible for determining
what portion of requested CTI should be shared with a CTI consumer. Using the
access policies stored by the data manager and credentials (e.g., identity-
proof) provided by the CTI consumer, the access manager determines what
portion of the CTI data should be given to a particular CTI consumer.

Figure 4: Proposed blockchain-based CTI sharing framework.

  * •

Blockchain: A distributed digital ledger of information that is shared,
duplicated and distributed across the entire network of computer systems. It
can be seen as a shared database that stores a series of digital records using
cryptographic operations (called hash). Ethereum blockchain is used for our
proof-of-concept implementation.

  * •

Smart Contract: Self-executable programs stored on a blockchain. We introduce
a data storage smart contract in our framework. It is composed of three
functions: (i) share function provided by the smart contract is used by a CTI
producer to indicate the availability of CTI, (ii) request function provided
by the smart contract is used by a CTI consumer to request access to sensitive
CTI data, and (iii) response function provided by the smart contract is used
by a CTI producer to respond to a request submitted to the request function.

  * •

Interplanetary File System (IPFS): A peer-to-peer hypermedia protocol that is
designed to distribute the storage of data across a network of computers. IPFS
provides highly available and consistent data storage of large files that
otherwise would be expensive to store using a traditional blockchain network
(e.g., Ethereum). IPFS is utilised to store off-chain data associated with the
sharing process [16].

###  III-C System Functionality

This section provides an overview of the proposed CTI sharing framework’s
functionality. The framework follows five main phases to share and consume
CTI. They are:

Share CTI: The objective of this phase is for the CTI producer to create and
share all of the data required to facilitate the exchange of CTI. Given that
an organisation has produced CTI data, a sensitivity analysis is conducted by
the CTI producer. This analysis identifies sensitive CTI attributes and
segments them into a number of sensitivity groups (cf. Fig. 2).

Once the sensitive data groups have been defined, a set of integrity hashes
are created. To construct these hashes, the single or multi hash approach (cf.
Fig. 3) can be used by the CTI producer. For both of the proposed approaches,
the sensitive data groups and a set of one-time random nonce’s are used to
construct the integrity hashes. In this case, the set of one-time random
nonce’s are used to protect the privacy of sensitive data (more discussion in
Section IV-C). To generate these integrity hashes, a single or set of data
groups are hashed using these nonce’s in sequence.

As discussed in Section II, several trust-based metric/policy schemes have
been presented by existing works. Subsequently, our framework allows CTI
producers to plug-in a scheme dynamically. For example, an organisation might
assign an individual XACML policy for each of the sensitive data groups.

Once the above process has been completed, the sensitive data groups and
access policy are stored securely off-chain by the CTI producer. Moreover the
non-sensitive CTI data group, set of integrity hashes and access policy are
added to IPFS, while its associated content identifier (CID) (IPFS File
Pointer) and CTI metadata are added to the data storage smart contract.

Request CTI: During this phase, a CTI consumer requests access to a particular
CTI of interest. An access request is submitted via a data storage smart
contract. This smart contract is the same one used during the Share CTI phase.
The access request indicates which CTI data the CTI consumer wishes to access
and contains the credentials (e.g., identity-proof) required to meet the
associated access policy.

To ensure that these credentials are shared in a privacy-preserving way, they
are encrypted using the CTI producer’s public key before being added to IPFS.
By encrypting their credentials, a CTI consumer can ensure that only the
intended CTI producer can access these them. Once the access request has been
submitted, the CTI producer receives the request, accesses the encrypted
credentials from IPFS using the provided IPFS CID, and decrypts them using
their private key.

Policy Evaluation: During this phase, the CTI producer processes the access
request submitted by the CTI consumer. To determine the sensitive data groups
that the CTI consumer should have access to, the set of credentials provided
during the Request CTI phase and the access policy associated with the
requested CTI is used. Once the provided credentials are validated, the CTI
producer evaluates the access policy to determine which subset of the
sensitive data groups the requesting CTI consumer should receive.

Response CTI: During this phase, the CTI producer provides the CTI consumer
with the sensitive data groups they are able to access. Note, in the Policy
Evaluation phase above, the CTI producer determines which subset of the
sensitive data groups a CTI consumer is able to access. To ensure that the CTI
consumer can validate the integrity of the provided sensitive data groups, the
corresponding one-time random nonce values are also included in the response.
To ensure that the response data is exchanged in a privacy-preserving way, it
is encrypted using the CTI consumer’s public key. Once encrypted, the response
data is added to IPFS and the associated IPFS CID is added to the data storage
smart contract by the CTI producer. The CTI consumer can then access the
encrypted response data using the provided IPFS CID. To decrypt the response
data, the CTI consumers uses their private key.

Validate CTI: During this phase, the CTI consumer validates the integrity of
the received sensitive data. As part of the Share CTI phase, the CTI producer
added a set of integrity hashes to IPFS. Subsequently, the CTI consumer can
access these integrity hashes and use them to validate the sensitive CTI data
received during the Response CTI phase.

Figure 5: Communication between various components of the proposed CTI sharing
framework (on-chain interactionss).

###  III-D Communications

In Fig. 5, we illustrate the communication flow within our CTI framework. Note
that for simplicity, we contemplate the interactions of various components
with the blockchain. Therefore, only on-chain interactions are shown, and
interactions with IPFS are excluded. The steps are as follows:

Step 0: CTI producer completes the initial setup steps. Firstly, the CTI data
is segmented into a variable number of sensitivity data groups (cf. Fig.2).
The sensitive data groups and a set of one-time random nonces are used to
construct a set of integrity hashes using either of the proposed methods (cf.
Fig.3). The non sensitive data group, CTI producer-defined access policy, and
integrity hashes are added to IPFS. The access policy and sensitive data
groups are stored off-chain by the CTI producer’s access manager. Lastly, the
IPFS CID and CTI metadata (e.g., threat type) are added to the data storage
smart contract using the share function.

Step 1a: The CTI consumer’s request manager adds an access request to the data
storage smart contract using the request function. As discussed in Section
III-C, the request contains a IPFS CID that is associated with an encrypted
set of the CTI consumers’ credentials.

Step 1b: The CTI producer’s request manager receives the access request. The
request manager decrypts the credentials associated with the IPFS CID provided
by the CTI consumer using their private key.

Step 2: The CTI producer’s request manager forwards the decrypted credentials
and requested CTI to the access manager.

Step 3: The CTI producers access manager validates the received credentials
(e.g., certificate signature is checked). Using the access policy associated
with the requested CTI, and the verified credentials, the access manager
assess the subset of sensitive data that should be provided to the CTI
consumer.

Step 4: The CTI producers access manager returns the accessible sensitive data
groups and their associated one-time random nonce to the CTI producers request
manager.

Step 5a: The CTI producer’s request manager responds to the access request
submitted during Step 1a using the data storage smart contract’s response
function. As discussed in Section III-C, this response contains an IPFS CID
that is associated with the encrypted set of the accessible sensitive data
groups and one-time random nonce’s.

Step 5b: The CTI consumer’s request manager receives the response. The request
manager decrypts the accessible sensitive data groups and one-time random
nonce’s associated with the IPFS CID using their private key.

Step 6: The CTI consumer’s request manager forwards the received sensitive
data, one-time random nonce’s and original IPFS CID referencing the original
CTI data, to the CTI consumers data manager.

Step 7: Using the original IPFS CID, the CTI consumers data manager gets the
set of integrity hashes shared by the CTI producer during Step 0.

Step 8: Using the received sensitive data groups and one-time random nonce’s,
the CTI consumer’s data manager computes the integrity hashes. This hash(s) is
then compared to the set of integrity hashes received during Step 7.

##  IV System Design and Evaluation

This section discusses the implementation of the proposed CTI sharing
framework. We evaluate the developed proof-of-concept prototype using several
key on- and off-chain metrics, e.g. gas cost and computational performance. In
addition, we also provide a privacy and trust analysis.

Figure 6: Sample data integrity hash (a) generation, and (b) validation time
(ms) using the single (Single) and multi (Multi) hash approaches.

###  IV-A Implementation

We implemented a proof-of-concept of our proposed framework using Ethereum
blockchain, in which we deploy our data storage smart contract written in
Solidity v0.8.11. We select the Ethereum platform as it can be deployed as a
public or private blockchain, allowing higher flexibility when evaluating our
framework. We wrote a Python script using web3.py library [17] to evaluate the
blockchain implementation.

In addition, we developed the off-chain components of our framework in the
Python programming language (e.g., request, access, and data managers).
Specifically, we used hashlib library [18] to implement the proposed integrity
hashing mechanisms (cf. Fig. 3), where we use the SHA-256 hashing function and
a public CTI dataset available on GitHub [19] to evaluate its performance. In
addition, we also generate a number of synthetic data files larger and smaller
than the CTI sample data. For both the sample and synthetic files, the data is
segmented into various groups. We deployed a resource-constrained Amazon
Elastic Compute Cloud instance (i.e., t2.micro) to benchmark the off-chain
operations, as CTI sharing may also occur in settings with computational
limitations (e.g., in an IoT setting). As such, we consider these constrained
environments as a worst case scenario when measuring the effect of additional
hashing operations.

###  IV-B Results

To evaluate the on-chain components of the developed data storage smart
contract, we evaluated the transaction cost associated with the share, request
and response functions under two different market conditions, current (1 gas =
20 gwei, 1 Eth = $1,097.2) and all time high (1 gas = 979 gwei, 1 Eth =
$4891.70). In this context, gas cost represents the execution cost of the
smart contact that is payed for in Eth (i.e., a gas fee that is required for
an Ethereum blockchain network user to conduct a transaction on the network)
based on the cost per gas unit, measured in gwei (note, each gwei is equal to
$10^{-9}$ Eth).

In Table II, the gas cost associated with the data storage smart contract
functions are calculated using the two above-defined market conditions. These
results show that under current market conditions, the transaction cost
associated with sharing and requesting CTI is US$0.96 and US$2.57
respectively. As a result, the additional overhead created by our proposed
differential sharing mechanism is US$2.57. Given that this increased cost is
incurred by CTI consumer who in-exchange receive potentially valuable threat
information, we argue this additional cost is unlikely to prevent
participating.

Furthermore, Table II High (US$), also highlights that transaction costs
increase drastically (21762%) in the case where the market price for both gas
and Eth rise significantly. While extreme, this result highlights the
potential effect fluctuations in transaction costs may have on the
effectiveness of the proposed framework. Subsequently, we argue that a private
Ethereum network should be considered in a real world context, as this would
allow participating organisations to better moderate these costs.

Additionally, we also evaluated the off-chain computational overheads
associated with the proposed integrity hashing schemes (cf. Fig. 3) using the
sample and synthetic data. For each of these data types, the execution time
associated with the generation and validation of an integrity hash set using
both of the proposed hashing schemes was measured. In addition, the execution
time associated with hashing the sample and synthetic data was also measured
as a baseline. Given that the generation and validation execution times are
highly dependent on the number of additional hashing operations, the baseline
case indicates the additional overheads created by the two proposed hashing
schemes relative to the size of the input data. As discussed in Section IV-A,
these operations were performed on an Amazon Elastic Compute Cloud t2.micro
instance, with the average of 1000 iterations recorded.

Figure 7: Synthetic data integrity hash generation time using the single
(Single) and multi (Multi) hash approaches. Figure 8: Synthetic data integrity
hash validation time using the single (Single) and multi (Multi) hash
approaches.

In Table III, a summary of the key results presented in Figs. 6-8 is provided.
These results show that both the single hash and multi hash approaches result
in an increased execution time compared to the baseline. However, we argue
that these additional overheads are outweighed by the broader access to CTI
and the greater granularity achieved by the proposed differential approach.
Moreover, given that these results were obtained using a resource-constrained
test bed, they indicate that the proposed differential sharing mechanism can
operate efficiently in resource-constrained scenarios, e.g., with IoT.

When the performance of these two approaches (i.e, single and multi hash)
using the sample data are compared (cf. Fig. 6), two trends can be observed.
In terms of generating the integrity hashes, the multi hash approach can be
observed to have an increased execution time when compared to the single hash
approach. This result is due to the redundancy associated with hashing the
same data exponentially more times as the number of data groups get larger for
the multi hash case (cf. Fig. 3). In contrast, the opposite relationship can
be observed if the validation times of these two approaches are compared. As
the number of data groups increases, it can be observed that the single hash
approach increases in execution time compared to the multi hash approach. This
result is due to the redundancy associated with the number of comparisons
required to validate a set of groups being equal to number of groups for the
single hash case (cf. Fig. 3).

For the synthetic data, similar trends were also observed. In Figs. 7 and 8,
the single hash generation and multi hash validation efficiency differences
are mostly consistent. However, we observed that the multi hash validation
efficiency reduced proportional to the size of synthetic data. More
specifically, after a data size of 150(KB), we observed that the single hash
approach was either comparable or more efficient in terms of validation
performance.

Based on both the sample and synthetic data results, we argue that the multi-
hash approach is more efficient. Given that CTI data is shared with C number
of CTI consumers, integrity hash validation can be considered a C-time
operation, and therefore, additional validation overheads are exacerbated in a
real-world setting where C is denoted as a large number. In contrast,
integrity hash generation is a one-time operation completed by the CTI
producer each time they share. However, we note that in circumstances where
CTI data is significantly larger than the sample data, the single hash
approach may become more efficient, considering the validation time results
for the 150-500KB synthetic data in Fig. 8.

Using the discussed results, we make three major findings: (i) the proposed
differential sharing mechanism does not significantly increase transaction
costs, (ii) market conditions could affect the effectiveness of CTI sharing,
and (iii) the proposed integrity hashing mechanisms (cf. Fig. 3) do not create
large computational overheads, with the multi hash approach being more
efficient for CTI data size less than 150KB.

TABLE II: Gas cost and price in US$ for each data storage smart contract
function.

Function | Gas Cost | Current (US$) | High (US$)
---|---|---|---
Share | 43,897 | $0.96 | $210.22
Request | 66,628 | $1.46 | $319.07
Response | 50,625 | $1.11 | $242.44
Total | 161,150 | $3.53 | $771.73

TABLE III: Summary of computational times associated with Figs. 6-8. (Gen =
Generation Time, Val = Validation Time, SH = Single Hash, MH = Multi Hash).

Data | Size | Baseline | Number of | Gen (ms) | Val (ms)
---|---|---|---|---|---
Type | (KB) | (ms) | Group | SH | MH | SH | MH
Sample | 94 | 0.24 | 10 | 0.38 | 1.69 | 0.27 | 0.26
20 | 0.51 | 3.30 | 0.30 | 0.27
50 | 0.89 | 8.10 | 0.36 | 0.26
Synthetic | 50 | 0.14 | 5 | 0.22 | 0.54 | 0.16 | 0.16
20 | 0.41 | 1.9 | 0.2 | 0.17
50 | 0.78 | 4.6 | 0.26 | 0.17
200 | 0.55 | 5 | 0.67 | 2.1 | 0.61 | 0.83
20 | 0.88 | 7.2 | 0.66 | 0.83
50 | 1.2 | 17 | 0.72 | 0.83
500 | 1.4 | 5 | 1.6 | 5.1 | 2.5 | 1.9
20 | 1.8 | 18 | 1.5 | 1.9
50 | 2.2 | 45 | 1.6 | 2.1

###  IV-C Privacy and Trust Analysis

Now we examine a number of privacy and trust-based considerations with our
proposed framework. From a CTI producer’s perspective, it is important that
sensitive data is shared in a privacy-preserving way. Subsequently, the
proposed differential approach seeks to provide a more granular approach that
maintains privacy at rest and in transit. To ensure that privacy is maintained
at rest, sensitive data groups are stored securely off-chain by CTI producers,
and therefore cannot be accessed by untrusted entities. Moreover, when
sensitive data groups are exchanged with trusted CTI consumers, encryption is
used to ensure that man-in-the-middle and eavesdropping attacks (e.g.,
sniffing or snooping) are not possible. Finally, the use of one-time random
nonce’s as part of the integrity hash generation and validation process,
ensures that the privacy of sensitive data common across multiple sharing
instances or which contains low entropy (e.g., software versions), are
protected from dictionary and brute force attacks.

Further, two major trust-based factors must also be considered. Firstly, CTI
producers must be provided with a dynamic approach to trust management that
ensures that sensitive data is only shared with trusted entities.
Subsequently, our framework allows CTI producers to define a trust-based
policy/metric scheme that best suits their requirements. Finally, the
verifiability of CTI data from the perspective of CTI consumers must also be
considered. For say, if a CTI consumer receives a subset of the CTI data, it
could be possible for a malicious CTI producer to modify this data depending
on the identity of the requesting organisation (e.g., a competitor). In such a
case, to protect CTI consumers from this attack, our proposed integrity
hashing schemes ensure that CTI consumers can validate that the received
subset of the CTI data is contained within the original set of CTI data shared
by the CTI producer.

##  V Conclusion

We have proposed a blockchain-based CTI sharing framework that allows for the
trusted, verifiable, and differential exchange of CTI between producers and
consumers. We refer to differential as the ability of a CTI producer to
control the amount of information exchanged with a CTI consumer. The framework
thus allows CTI producers to segment CTI data into sensitive data groups,
which can be variably shared with CTI consumers in a differential manner. Our
results showed that the proposed framework could facilitate the exchange of
CTI between producers and consumers in a differential way without compromising
trust or variability. An interesting direction for future work is to conduct a
more comprehensive study to examine the impact of incentives on CTI sharing.

## Acknowledgment

The authors acknowledge the support of the Commonwealth of Australia and
Cybersecurity Research Centre Limited.

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