# Using Constraint Programming and Graph Representation Learning for
Generating Interpretable Cloud Security Policies

Mikhail Kazdagli1,2 Mohit Tiwari1,2 Akshat Kumar1,3
1Symmetry Systems
2The University of Texas at Austin
3Singapore Management University
{mikhail, mohit<EMAIL_ADDRESS>
###### Abstract

Modern software systems rely on mining insights from business sensitive data
stored in public clouds. A data breach usually incurs significant (monetary)
loss for a commercial organization. Conceptually, cloud security heavily
relies on Identity Access Management (IAM) policies that IT admins need to
properly configure and periodically update. Security negligence and human
errors often lead to misconfiguring IAM policies which may open a backdoor for
attackers. To address these challenges, first, we develop a novel framework
that encodes generating optimal IAM policies using constraint programming
(CP). We identify reducing dormant permissions of cloud users as an optimality
criterion, which intuitively implies minimizing unnecessary datastore access
permissions. Second, to make IAM policies interpretable, we use graph
representation learning applied to historical access patterns of users to
augment our CP model with similarity constraints: similar users should be
grouped together and share common IAM policies. Third, we describe multiple
attack models and show that our optimized IAM policies significantly reduce
the impact of security attacks using real data from 8 commercial
organizations, and synthetic instances.

##  1 Introduction

Cloud computing has recently become the dominant computing paradigm which
provides the flexibility of on-demand compute, and reduced cost due to
economies of scale. However, such benefits come at a cost—private and business
sensitive data is stored on public clouds. Managing identity access policies
(IAM), which intuitively means deciding which user should have access to which
datastore, for public clouds such as Amazon AWS IAM (2021), is complex even
for small and medium-sized companies with few hundred users and datastores
Kuenzli (2020). IAM policies are configured by IT admins of the organization
who often do not have access to automated decision support tools for IAM
policy optimization. As a result, due to the inherent complexity of IAM policy
optimization, security negligence, and human errors may often lead to
misconfigured IAM policies and result in data breaches Security (2021);
Scroxton (2020); Marks (2021b, a).

#### Related work.

There has been prior work in AI and ML on securing the sensitive data in a
cloud. Traditionally security researchers focus on developing dynamic
behavioral detectors that analyze system behavior using (un)supervised ML
methods at different levels - system and API calls Canali et al. (2012); Wu et
al. (2012), hardware signals Demme et al. (2013); Kazdagli et al. (2016),
network traffic Mirsky et al. (2018); Sommer and Paxson (2010); Handley et al.
(2001), and domain reputations Huang et al. (2017); Oprea et al. (2018). Such
methods often suffer from raising unacceptably high volume of false positives
Hassan et al. (2019) when analyzing large amount of system events even if the
false positive rate is very low Axelsson (1999). Moreover, ML-based detectors
are susceptible to adversarial attacks Lei et al. (2019); Wang et al. (2019);
Liu et al. (2021). Our proposed method, IAMAX, avoids such shortcomings by
securing an organization’s cloud infrastructure by hardening IAM policies,
which are the first line of defense, rather than detecting attackers inside
the cloud. Even after a user’s account is compromised, our optimized IAM
policies minimize the leak of sensitive data.

There has been a history of applying tools from AI planning for security in
the context of penetration testing (or pentesting) Sarraute et al. (2012);
Hoffmann (2015); Shmaryahu et al. (2018). In pentesting, automated tools based
on planning are used to identify vulnerabilities in the network by launching
controlled attacks under a variety of settings, such as fully or partially
observable network settings. Once vulnerabilities are identified, they can be
patched by network administrators. Our proposed work is different from such
pentesting methods as our target is to design the IAM policies (analogous to
network design) from grounds up so that opportunities for catastrophic attacks
(where compromising very few users gives hackers access to majority of
datastores) are minimized. Furthermore, our approach is customized for public
clouds and their security configurations, which is a different setting than
the standard pentesting Shmaryahu et al. (2018).

#### Contributions.

Our main contributions are as follows. First, we formalize the problem of IAM
policy optimization for public clouds using the constraint programming (CP)
framework Rossi et al. (2006), identfying core objectives and constraints. We
highlight the key role that dormant permissions play in defining secure IAM
policies. Second, given that organizations that we have worked with do not
provide the identities and job roles of its cloud users due to privacy
concerns, we use graph neural network Kipf and Welling (2017); Hamilton et al.
(2017) to learn embeddings for users and datastores based on the information
flow between them. These embeddings are used for defining constraints that
make IAM policies interpretable. Third, we describe multiple attack models,
based on randomly compromising $k$ users, and adversially compromising those
users that lead to worst case outcome. We test our optimized IAM policies on
real cloud infrastructures of 8 medium size commercial companies and several
realistic synthetic instances generated using the properties of real world
data sets, and show the significant potential of our approach to secure cloud
infrastructures by reducing dormant permissions significantly. We released the
code and the data sets used in our
experiments111https://github.com/mikhail247/IAMAX.

##  2 IAM Policies and Cloud Computing

An IAM policy is the fundamental security primitive in any cloud. Though our
real world data sets were collected across multiple cloud platforms, we use
Amazon AWS IAM policies as an example. An IAM policy is expressed using a
declarative policy language that defines what operations (“Action”) are
(dis)allowed (controlled by the “Effect” field) on a resource (“Resource”
field). Figure 1 shows an oversimplified IAM policy to be attached to an AWS
identity (e.g. a user, a role) or a group of identities. It stipulates that
the identity that the policy is attached to is allowed to perform the
operation “s3:ListObjects” (read the bucket content) on the S3 bucket
“arn:aws:s3:::bucket-name”.

#### Overpriviliged security policies.

To make the number of policies manageable, IT admins group identities together
into permission groups and/or roles (on AWS) and attach IAM policies to them.
We call such user aggregations data access groups. Typically, each data access
group carries a certain semantic meaning (e.g. web developers). A policy
assigned to a data access group grants access to all cloud resources that
individual identities in the group access to. Such an approach creates over-
privileged identities by granting each identity access to cloud resources that
are used by some identities in a group Security (2021); Verizon (2021). We
call permissions that identities are granted, but never use, dormant
permissions. Dormant permissions violate the fundamental security principle -
the principle of least privilege Saltzer and Schroeder (1975) that stipulates
that every user and an automated service should operate using the least set of
privileges necessary to complete the job. Dormant permissions allow attackers
after compromising an identity’s credentials to not only get access to cloud
resources that identity actively uses, but also get access to additional
resources that the compromised identity has access to because of dormant
permissions.

IAM policies are not necessarily poorly designed from the beginning - they are
likely to deteriorate over time as organization evolves (e.g. new automated
services get developed, employees move between teams, etc). Policies rarely
get updated to reduce the number of dormant permissions, thus causing the set
of dormant permissions to expand over time which gives an advantage to
attackers.

“Statement”: [{
“Effect”: “Allow”,
“Action”: “s3:ListObjects”,
“Resource”: “arn:aws:s3:::bucket-name” }]

Figure 1: An example of an AWS IAM policy Figure 2: Schematic representation
of the way users access datastores in AWS cloud. For simplicity we refer to
both Roles and Groups as existing data access groups.

#### Our approach.

We minimize the amount of dormant permissions to reduce damage caused by
potential attacks. We intentionally refrain from considering a trivial
solution - block a user from accessing unused datastores. Though such an
approach may work, but in practice it is not being used because it would be
hard for IT admins to maintain up-to-date per-user policies (specific accesses
that should be allowed/blocked per user) especially when users transition
between teams within an organization.

Figure 2 shows the schema denoting how users are partitioned into roles and
groups and how access to different datastores is determined based on
permissions granted to roles, groups and users themselves. Our approach to
minimizing dormant permissions is to create a set of additional datastore
access groups $\mathcal{G}$ (shown in figure 2). Solid edges represent
existing IAM policies, while the dashed edges represent generated IAM
policies. Conceptually, dashed edges are supposed to replace solid edges and
thus remove dormant permissions. Both types of edges depict mapping of users
to datastore access groups, which are named as roles and groups by cloud
platforms, and mapping of those groups to datastores. Using CP, we optimize:
(1) partitioning of users into different generated groups $g\in\mathcal{G}$,
and (2) what datastores are accessible by each group $g\in\mathcal{G}$. From
an implementation perspective, the policy generated by the CP solver is
concatenated with original over-permissive policies via logical AND operation
using cloud IAM policy management tools, thereby reducing dormant permissions.
We did not edit existing IAM policies to avoid introducing unintentional
errors due to high complexity of the policy language.

##  3 Optimizing IAM Policies

We start by introducing different objects and relationship among them (as
depicted in Figure 2). These objects and relationships will define the
variables, constraints and objective of the IAM policy optimization problem.
Set $U$ denotes the set of users; $D$ denotes the set of datastores; $G$
denotes the set of existing data access groups; set $\mathcal{G}$ denotes the
generated access groups as noted in Section 2. Set ${T}$ denotes data types in
a datastore according to the organization’s data classification scheme (e.g,
social security number, email, credit card number among others). Real world
data sets used in our experiments contain at most 10 data types. Such objects
within an organization are extracted from mining its cloud infrastructure. The
number of datastore access groups ($|\mathcal{G}|$) is a configurable
parameter that intuitively defines the flexibility we have in reconfiguring
the IAM policies (will be discussed later). We deliberately avoid making the
number of datastore access groups ($|\mathcal{G}|$) to be a variable to reduce
model complexity and thus increase scalability.

#### Users and existing groups.

As noted in Section 2, users are end users (either humans or services) that
need to access different datastores. Users may directly access a datastore,
however, usually they either assume a role or inherit permissions from a
permission group. Original IAM policies include 13–150 users, 90–1792 roles,
and 5–833 permission groups. Without loss of generality, we denote both roles
and permission groups as ‘existing data access groups’ $G$. Thus, users are
mapped to different groups $G$, which are mapped to datastores. We mine all
such links, and create following relationships.

#### Users and datastores.

Based on existing permissions (users-roles-groups-datastores), we can also
compute all the datastores $d\in D$ that a user $u\in U$ can potentially
access. This is stored in variable
$\operatorname{\widehat{UD}}(u,d)\in\\{0,1\\}$. These variables are constants
as they are based on pre-defined permissions in an organization’s cloud.

#### Users and accessed datastores.

Using an organization’s historical data, we can determine which datastores
have been accessed by different users. Let
$\operatorname{UD}(u,d)\in\\{0,1\\}$ denote if user $u\in U$ has accessed
datastore $d\in D$ in the past. We shall require any refactored IAM policy to
preserve access of different users to the datastores they have accessed in the
past.

#### Groups and datastores.

Let $\operatorname{GD}(g,d)\in\\{0,1\\}$ denote if group $g\in G$ provides
access to the datastore $d\in D$. This is a constant mapping.

#### Users and datastore access groups.

We create $|\mathcal{G}|$ datastore access profiles that allow us to further
control user access to datastores. Let variable
$\operatorname{U\mathcal{G}}(u,\operatorname{\tilde{g}})\in\\{0,1\\}$ denote
whether user $u\in U$ belongs to the access group
$\operatorname{\tilde{g}}\in\mathcal{G}$.

#### Datastore access groups and datastores.

Let variable
$\operatorname{\mathcal{G}D}(\operatorname{\tilde{g}},d)\in\\{0,1\\}$ denote
whether datastore access group $\operatorname{\tilde{g}}\in\mathcal{G}$
provides access to the datastore $d\in D$.

#### Datastores and data types.

Let $\operatorname{DT}(d,t)\in\\{0,1\\}$ denote whether datastore $d$ contains
data of type $t\in T$. This is a constant mapping.

#### Constraints.

We now describe the relationship between different variables that define the
IAM policy optimization problem. We next show for a given relationship among
users, groups and datastore access groups, how to compute how many total
datastores a user $u$ can access, and based on that compute redundant
datastores a user has access to. These redundant permissions constitute the
so-called dormant permissions that we want to minimize.

We create a variable $\widetilde{\operatorname{UD}}(u,d)\in\\{0,1\\}$ to
denote if user $u$ can access datastore $d$ after we incorporate additional
permissions and relations from the datastore access groups $\mathcal{G}$.
Constraints on it are the following:

#### Frequently accessed datastores must still be accessible.

$\widetilde{\operatorname{UD}}(u,d)\geq\operatorname{UD}(u,d)\;\forall u\in
U,\;d\in D$

#### Computing $\widetilde{\operatorname{UD}}(u,d)$.

A user $u$ can only access datastore $d$ iff the following conditions hold:

  * •

User $u$ is part of a datastore access group $\operatorname{\tilde{g}}$ and
$\operatorname{\tilde{g}}$ has permission to access the datastore $d$ AND

  * •

The current permissions based on user-role-groups-datastore also allow user
$u$ to access $d$

$\displaystyle\widetilde{\operatorname{UD}}(u,d)=\text{OR}\Big{(}\operatorname{U\mathcal{G}}(u,\operatorname{\tilde{g}})\bigwedge\operatorname{\mathcal{G}D}(\operatorname{\tilde{g}},d)\forall\operatorname{\tilde{g}}\in\mathcal{G}\Big{)}\bigwedge\operatorname{\widehat{UD}}(u,d)$

where OR stands for the logical or operator.

#### Data type constraints.

A data type constraint stipulates that a user should not get an access to a
completely new data type that they never worked before with. We make an
assumption that a user accesses all data types that are in the datastore $d$
(if the user is allowed to access $d$). More fine-grained formulation of this
constraint would require operating at an object level (e.g. directory/table
level in a datastore), but this would significantly increase the number of
variables in the problem and make it computationally intractable.

The data type constraint can be formulated as:

$\displaystyle\text{OR}\Big{(}\operatorname{UD}(u,d)\bigwedge\operatorname{DT}(d,t)\;\forall
d\in{D}\Big{)}\geq$
$\displaystyle\text{OR}\Big{(}\widetilde{\operatorname{UD}}(u,d)\bigwedge\operatorname{DT}(d,t)\;\forall
d\in{D}\Big{)}\forall u\in U,t\in T$ (1)

###  3.1 Constraint Program for Reducing Dormant Permissions

Based on the variables and relationships among them, we now describe the
constraint program (CP) that optimizes IAM policies by minimizing the dormant
permissions. Formally, the reduction in the number of user-datastore
permissions in the refactored IAM policy is:

$\displaystyle\sum_{u,d}\operatorname{\widehat{UD}}(u,d)-\sum_{u,d}\widetilde{\operatorname{UD}}(u,d))$
(2)

Maximizing the above objective would minimize the second term in the objective
(first term is constant based on the existing IAM policy). Therefore, users in
the refactored policy would have access to as few datastores as necessary
given the problem constraints, and number of access groups $|\mathcal{G}|$.
Thus, it will also reduce dormant permissions, and result in a least-privilege
IAM policy.

#### Penalty on dormant permissions per user.

The objective (2) minimizes the total number of dormant permissions in the
system. However, we also want to limit the maximum number of dormant
permissions per user for increased robustness to attacks. We formulate such
constrains as a soft constraints, violation of which incurs a penalty of
${\psi_{i}}(v)$ computed as below:

$\displaystyle
v_{u}=\sum_{d}\widetilde{\operatorname{UD}}(u,d)-(1.0+\epsilon)\operatorname{UD}(u,d)\forall
u\in U$ (3) $\displaystyle{\psi_{u}=\max(v_{u},\gamma v_{u})\forall u\in U}$
(4)

where $\epsilon$ is a hyper-parameter in our model, which should be tuned by a
domain expert according to the amount of risk they are willing to take. If the
parameter is set to 0, then our CP formulation will softly penalize any user
that has non-zero amount of dormant permissions. Hence, if $\epsilon>0$, then
the model does not penalize users with less than $\epsilon$ fraction of
dormant permissions. Based on the empirical analysis of IAM policies, we
believe that allowing a small fraction of dormant permissions (e.g. 15%)
increases policy interpretability and is unlikely to significantly affect
security aspect of the problem. The parameter $\gamma$ penalizes such soft
constraint violations more harshly if it is more than 1. We can also put per
user dormant permissions as constraints, however in practice it made the
program infeasible. The overall CP formulation is given in Table 1.

$\displaystyle\max\Big{(}$
$\displaystyle\sum_{u,d}\operatorname{\widehat{UD}}(u,d)-\sum_{u,d}\widetilde{\operatorname{UD}}(u,d)\Big{)}-\sum_{u}\psi_{u}$
(5)
$\displaystyle\widetilde{\operatorname{UD}}(u,d)\geq\operatorname{UD}(u,d)\;\forall
u\in U,\;d\in D$ (6)
$\displaystyle\widetilde{\operatorname{UD}}(u,d)=\text{OR}\Big{(}\operatorname{U\mathcal{G}}(u,\operatorname{\tilde{g}})\bigwedge\operatorname{\mathcal{G}D}(\operatorname{\tilde{g}},d)\;\forall\operatorname{\tilde{g}}\in\mathcal{G}\Big{)}$
$\displaystyle\hskip 70.0pt\bigwedge\operatorname{\widehat{UD}}(u,d)\;\forall
u\in U,\;d\in D$ (7)
$\displaystyle\text{OR}\Big{(}\operatorname{UD}(u,d)\bigwedge\operatorname{DT}(d,t)\;\forall
d\in{D}\Big{)}\geq$
$\displaystyle\text{OR}\Big{(}\widetilde{\operatorname{UD}}(u,d)\bigwedge\operatorname{DT}(d,t)\;\forall
d\in{D}\Big{)}\forall u\in U,t\in T$ (8) $\displaystyle
v_{u}=\sum_{d}\widetilde{\operatorname{UD}}(u,d)-(1.0+\epsilon)\operatorname{UD}(u,d)\forall
u\in U$ (9) $\displaystyle{\psi_{u}=\max(v_{u},\gamma v_{u})\forall u\in U}$
(10)
$\displaystyle\operatorname{\mathcal{G}D}(\operatorname{\tilde{g}},d),\operatorname{U\mathcal{G}}(u,\operatorname{\tilde{g}})\in\\{0,1\\}\;\forall
u,\operatorname{\tilde{g}},d$ (11)
---

Table 1: Constraint program for minimizing dormant permissions

###  3.2 Group Homogeneity Constraints

We further enhance the basic CP formulation (Table 1) by including group
homogeneity constraints. Intuitively, all users in a group
$\operatorname{\tilde{g}}\in\mathcal{G}$ should be similar w.r.t. a job role
in an organization to make the solution more explainable and interpretable to
IT admins. Moreover, heterogeneous groups may increase an impact of a
potential attack: similar users are usually susceptible to the same attack
vector (e.g. phishing, waterhole, etc). If they are spread across multiple
groups, then attackers may get additional advantage if the number of residual
dormant permissions across all those groups is larger than in the case when
similar users are grouped together.

We assume a precomputed pairwise function ${\alpha_{user}}$ defined on $U$x$U$
denoting dissimilarity between users. Section 4 shows how this function can be
computed using graph neural networks (GNNs) and the data that we mine from an
organization’s cloud infrastructure. We add the homogeneity constraints as
below. If a user $u$ is mapped to datastore access group
$\operatorname{\tilde{g}}$ (i.e.,
$\operatorname{U\mathcal{G}}(u,\operatorname{\tilde{g}})=1$), then we use the
shorthand $u\in\operatorname{\tilde{g}}$ for exposition clarity.

$\displaystyle\max\big{\\{}{\alpha_{user}}(u_{1},u_{2})\forall
u_{1},u_{2}\in\operatorname{\tilde{g}}\big{\\}}\leq\alpha,\forall\operatorname{\tilde{g}}\in\mathcal{G}$
(12)

where $\alpha$ is a diversity threshold that we show in Section 4 how to
compute from the data.

#### Constraint generation.

Constraints (12) create memory issues when incorporated in a single program as
quadratic number of terms are there in the number of users in a group
$\operatorname{\tilde{g}}$. Also, they often make the CP problem infeasible.
Therefore, we follow a constraint generation approach Ben-Ameur and Neto
(2006) where we first solve the program in Table 1 with no homogeneity
constraints. We then follow an iterative constraint generation procedure to
add back most violated homogeneity constraints and solve the program again.

#### Separation oracle.

At every iteration for each datastore access group
$\operatorname{\tilde{g}}\in\mathcal{G}$, we identify user pairs
$(u_{1},u_{2})\text{ s.t.
}\operatorname{U\mathcal{G}}(u_{1},\operatorname{\tilde{g}})=\operatorname{U\mathcal{G}}(u_{2},\operatorname{\tilde{g}})=1$,
and $\alpha_{user}(u_{1},u_{2})>\alpha$. We then add the constraint
$\small\operatorname{U\mathcal{G}}(u_{1},\operatorname{\tilde{g}})+\operatorname{U\mathcal{G}}(u_{2},\operatorname{\tilde{g}})\leq
1\;\;\forall\operatorname{\tilde{g}}\in\mathcal{G}$ to the CP and solve again.
This process continues until all the homogeneity constraints are satisfied or
we get an infeasible program, in which case we output the solution generated
in the last iteration. It may happen that no solution may exist that satisfies
all the homogeneity constraints (based on the threshold $\alpha$). In such a
case, the constraint generation approach provides the solution with most
violated constraints being satisfied.

##  4 Graph Representation Learning

#### Incorporating GNN into CP.

Interpretable policies require group homogeneity with respect to users’ job
roles. However, such information is not shared with us due to high business
sensitivity. Group homogeneity constraints are defined in terms of the user
dissimilarity function ${\alpha_{user}}$ and the diversity threshold $\alpha$
(Section 3.2). We use a graph neural network (GNN) to approximate both
parameters and share them with the CP solver.

#### User behavioral graph (UBG).

Intuitively, job roles can be inferred from users’ interaction with a cloud
environment (e.g. executed operations, accessed cloud resources). We represent
records of all executed cloud operations over the 6-12 months period as a
weighted heterogeneous graph.

  * •

Nodes, set $\mathcal{V}$: users, roles, groups, datastores. Each node has its
own handcrafted features. For example, a datastore includes distribution of
data types, each user node has an associated risk score.

  * •

Edges: a pair of nodes is connected with an undirected edge, if they have
participated in the execution of a cloud operation (e.g. reading data,
modifying resource configuration, etc) and its weight, $e_{v,u,\tau}$, is
proportional to the frequency of such an operation.

  * •

Edge types, set $\mathcal{R}$ carry semantic meaning of executed operations:
data flow edges (e.g. transferring data), configuration update edges (e.g.
updating configuration of cloud resources) and others.

#### Graph neural network.

We adapted Relational GCN Schlichtkrull et al. (2018) approach to
heterogeneous graphs, however, we replaced GCN Kipf and Welling (2017) modules
with GraphSage Hamilton et al. (2017) modules (one per each graph relation
type) to achieve inductive graph representation learning. Our GNN-based
embeddings are quite general, we successfully use them for multiple downstream
tasks such as anomaly detection and data visualization.

Conceptually, our GNN is a superposition of multiple weighted GraphSage neural
networks where each network operates on a specific relation type
$\tau\in\mathcal{R}$ (Eq. (13)–(15)). At each search depth (parameter $k$)
nodes aggregate information from their local neighbors,
$\mathcal{N_{\tau}}(v)$, using the weighted mean aggregator, into the vector
$\textbf{h}^{k}_{\mathcal{N_{\tau}}(v)}$, where weights are normalized edge
frequencies $e_{v,u,\tau}$ (Eq. (13)). After that, each node embedding
$\textbf{h}^{k}_{v,\tau}$ gets updated according to the rule (14), where
$\textbf{W}^{k}_{\tau}$ are trainable weight matrices and $\oplus$ stands for
concatenation of two vectors. The final node embedding $\textbf{h}^{k}_{v}$ at
the search depth $k$ (Eq. (16)) is a mean value of normalized relation-
specific embedding vectors $\textbf{h}^{k}_{v,\tau}$ (Eq. (15)).

The search depth is a design parameter and if it is too high, then a GNN may
suffer from over-smoothing. Thus, we set it to 2 because it is sufficient for
our experiments. We use 50-dimensional $\textbf{h}^{2}_{v}$ (Eq. (16)) vectors
as node embeddings in our experiments. Note that $\textbf{h}^{0}_{v}$ vectors
in the Eq. (13), (14) are initialized by corresponding node feature vectors.
We train our GNN on link prediction task with a cosine distance between node
embeddings for 500 epochs.

$\displaystyle\textbf{h}^{k}_{\mathcal{N_{\tau}}(v)}\xleftarrow{}mean_{e_{v,u,\tau}}(\textbf{h}^{k-1}_{u,\tau}\quad\forall
u\in\mathcal{N_{\tau}}(v))$ (13)
$\displaystyle\textbf{h}^{k}_{v,\tau}\xleftarrow{}\sigma(\textbf{W}^{k}_{\tau}\cdot(\textbf{h}^{k-1}_{v,\tau}\oplus\textbf{h}^{k}_{\mathcal{N_{\tau}}(v)}))\quad\forall
v\in\mathcal{V}\;\forall\tau\in\mathcal{R}$ (14)
$\displaystyle\textbf{h}^{k}_{v,\tau}\xleftarrow{}\textbf{h}^{k}_{v,\tau}/||\textbf{h}^{k}_{v,\tau}||_{2}\quad\forall
v\in\mathcal{V}$ (15)
$\displaystyle\textbf{h}^{k}_{v}\xleftarrow{}mean(\textbf{h}^{k}_{v,\tau}\quad\forall\tau\in\mathcal{R})$
(16)

#### User embeddings to CP parameters.

After training is complete we extract user embeddings and cluster them with
k-means algorithm. A grid search with k in range of [5, 25] gives us an
optimal value of k, $k_{opt}$, that corresponds to a point of maximum
curvature (‘knee’ point) of the function that maps k to k-means’ objective
value (sum of squared distances of samples to their closest cluster center)
Satopaa et al. (2011). The range of k encodes our domain-specific knowledge -
we expect to identify between 5 and 25 sufficiently different job roles at an
organization. Figure 3 visualizes 15 clusters detected in one of the data
sets. We shared clustering results with organizations and received back a
confirmation of good approximation quality of employees’ job roles.

GNN embeddings let us define user dissimilarity function
$\alpha_{user}(u_{1},u_{2})$ consistently with GNN design: it is the cosine
distance between embeddings of users $u_{1}$ and $u_{2}$. The threshold
$\alpha$ is the maximal cluster diameter computed as the cosine distance
between the most dissimilar users within any cluster when setting the
parameter k to $k_{opt}$. Significant entropy reduction (Section 7.2, Figure
4) of data access groups after iteratively adding homogeneity constraints
empirically validates the choice of the parameter $\alpha$. If $\alpha$ was
too high, then we would not observe entropy reduction.

##  5 Algorithm

1:$\textbf{Input:}\ org^{\prime}s\ cloud\ configuration$, $|\mathcal{G}|$
$\triangleright$ Section 3

2:$\textbf{Output:}\ Hardened\ IAM\ policies$

3:

4:$Train\;Graph\ Neural\ Network\ (GNN)$ $\triangleright$ Section 4

5:$\alpha_{user}(u_{1},u_{2}),user\ cluster\ assignment\leftarrow GNN$

6:$CP\leftarrow CP_{basic}$ $\triangleright$ Table 1

7:$solution\leftarrow solve(CP)$

8:

9:if $solution=\emptyset$ then

10: return Infeasible CP

11:end if

12:

13:repeat$\triangleright$ Section 3, ”Constraint generation”

14: for all $\operatorname{\tilde{g}}\in\mathcal{G}$ do

15: # Identify dissimilar user pair $(u_{1},u_{2})$

16: $(u_{1},u_{2})\ s.t.\ u_{1},u_{2}\in\operatorname{\tilde{g}}and\
\alpha_{user}(u_{1},u_{2})>\alpha$

17: # Add additional constraints

18:
$CP\cup\\{\small\operatorname{U\mathcal{G}}(u_{1},g\prime)+\operatorname{U\mathcal{G}}(u_{2},g\prime)\leq
1\ \forall g\prime\in\mathcal{G}\\}$

19: end for

20: $solution\leftarrow solve(CP)$

21:until $solution\neq\emptyset$

22:

23:Hardened IAM poclicies $\leftarrow$ extract(last feasible solution)

Algorithm 1 IAMAX: high-level description

(a)

(b)

(c)

Figure 3:  (a) T-SNE projection of 50-dimensional Graph-NN identity embeddings
(data set 7); 15 distinct clusters corresponding to different job roles are
highlighted. (b) Relative percentage of remaining dormant permissions after
IAM policy optimization as a function of the number of generated data access
groups. IAMAX significantly reduces the amount of dormant permissions and
often reaches 0% level with a small number of datastore access groups. For
group count more than 50, there are no dormant permissions, therefore not
shown in the plot. (c) To verify the generalizability of our approach, we
evaluate IAMAX on 280 synthetic data sets. In most cases, IAMAX achieves
significant reduction of dormant permissions.

In this section we turn concepts outlined in Sections 3, 4 into an imperative
Algorithm 1 to clarify IAMAX’s workflow. After IAMAX gets deployed in an
organization’s cloud environment, it starts mining cloud configuration to
convert it into a graph representation. Hence, those details lie outside the
scope of our paper. Having computed the graph representation, IAMAX trains a
GNN (Section 4), which is used to generate graph node embeddings and to
cluster users (Algorithm 1, lines 4,5). At this point, IAMAX can proceed with
solving the basic CP problem outlined in the Table 1 (Algorithm 1, lines 6,7).
The basic CP problem may turn out to be infeasible (Algorithm 1, lines 9–11)
if the specified number of datastore access groups is insufficient (input
parameter $|\mathcal{G}|$).

If IAMAX can find a feasible solution to the basic CP problem (Algorithm 1,
line 7), then it proceeds with iterative constraint generation (Algorithm 1,
lines 13–21). This process continues until all user dissimilarity constraints
get satisfied or the CP problem turns infeasible at some iteration. In
practice, constraint generation usually stops due to encountering an
infeasible CP problem. Also, at each iteration we add mini-batches of
constraints rather than individual constraints to amortize time needed for
running the CP solver. When the execution reaches the end (Algorithm 1, line
23), it is guaranteed that the feasible solution has been found at either the
final iteration or the one before that. Finally, the CP solution gets decoded
into new IAM policies that are concatenated with existing ones using logical
AND operation to reduce the amount of dormant permissions.

##  6 Attack Models

Besides reducing the dormant permissions, we now describe different attack
models that further test our optimized IAM policies. In random attack model,
we assume that an attacker does not have information about the internal IAM
configuration of an organization. Therefore, the attacker randomly tries to
compromise $k$ users. Once compromising $k$ users, the attacker gets access to
all the datastores that can be accessible by any of the $k$ users.

In the worst-case attack model, we assume that an attacker has full
observability of an organization’s IAM policies. Therefore, the attacker
carefully plans to attack $k$ users such that it can maximize the blast radius
(or the number of datastores it can access). We show that this problem is NP-
Hard; also, it is monotone and submodular, thus a simple greedy approach
provides a constant factor approximation.

###  6.1 Formalization of the Worst-case Attack Problem

The proof of NP-hardness relies on reduction from the set cover problem.
Suppose we have an instance of the Set Cover problem, where the sets are
$X_{1}$ to $X_{m}$. Each set $X_{i}$ is composed of elements $u_{i1}$ to
$u_{in_{i}}$ where $n_{i}$ denotes the number of elements in set $X_{i}$. Let
the total number of elements in the universe be $n$. We create an instance of
IAM problem as next.

  * •

We create $m$ users corresponding to each set $X_{1}$ to $X_{m}$.

  * •

We create $n$ datastores, one for each element of the universe.

  * •

If set $X_{i}$ contains element $j$, we allow user $X_{i}$ access to the
datastore $j$.

We now solve the worst-case attack problem on this datastore access graph by
choosing $k$ users to compromise. If the number of datastores accessible is
the same as the universe, then the set cover problem has a solution with
$k$-cover, otherwise not.

To properly simulate the worst-case attack, which is NP-hard, we have to
uncover its submodular nature and use an appropriate approximation.

#### Monotonicity.

Let $S$ be the set of users being compromised. Let $f(S)$ denotes the number
of datastores that become accessible as a result. The function $f$ is
monotone. Let $u\notin S$ be another user. $f(S\cup\\{u\\})\geq f(S)$ as the
number of accessible datastores cannot decrease if an additional user is
compromised.

#### Submodularity.

The function $f$ is also submodular. Let $S^{\prime}\subseteq S$, and let $u$
be a user not in $S$. Then, we must show:

$\displaystyle f(S\cup\\{u\\})-f(S)\leq
f(S^{\prime}\cup\\{u\\})-f(S^{\prime})$ (17)

Let us consider the expression $f(S\cup\\{u\\})-f(S)$. It will count those
datastores that can only be accessible by user $u$ and none of the users in
the set $S$. Let us denote this number as $n_{u|S}$; $n_{u|S^{\prime}}$ is
defined analogously. We must have $n_{u|S}\leq n_{u|S^{\prime}}$ because
$S^{\prime}\subseteq S$. This is becuase the additional access to datastores
granted by the user $u$ over datastores accessible by $S$ must be smaller than
the additional access granted by the user $u$ over datastores accessible by
$S^{\prime}$.

#### Constant factor approximation.

The standard greedy algorithm that iteratively selects the user $u$ that
provides the maximum marginal gain in terms of the function $f$ is guaranteed
to provide $(1-1/e)$-approximation Nemhauser et al. (1978). Using this
approach, we can approximately select $k$ users to compromise for simulating
the worst-case attack.

##  7 Experimental Setup

(a)

(b)

(c)

(d)

Figure 4: (a) Cumulative fraction of satisfied user embedding constraints at each iteration. (b) Entropy of generated solutions before and after adding user embedding constraints. (c, d) Attack impact (lower better) on an organization under an assumption of 1-3 compromised users. (c) The average-case impact considering an attacker possesses incomplete information about IAM policies. (d) The worst-case impact considering an attacker possesses perfect information about IAM policies.  Graph |  Users |  Data stores |  Dynamic edges |  Permission edges
---|---|---|---|---
1 |  13 |  56 |  513 |  1969
2 |  32 |  89 |  1192 |  16791
3 |  39 |  341 |  1602 |  28880
4 |  57 |  572 |  7153 |  37854
5 |  60 |  88 |  757 |  4115
6 |  64 |  258 |  2418 |  72272
7 |  112 |  163 |  2789 |  4025
8 |  150 |  600 |  3095 |  11517
Table 2: Properties of real world graph instances

We evaluate IAMAX on both synthetic and real world instances. Each real world
instance represents cloud infrastructure within one or more departments at a
commercial organization. In addition to 8 real world data sets, we generated
280 synthetic data sets using real data sets as baselines. Throughout the
paper we mostly focus on real world data sets and use synthetic data only for
evaluation of statistical effectiveness of the proposed approach. Independent
security experts verified correctness and interpretability of generated IAM
policies. Moreover, generated policies are correct by design (Section 3). The
section is organized as follows: first we show effectiveness of our basic CP
formulation, then we demonstrate the effect of adding constraints based on GNN
embeddings, and finally we conclude with attack simulations.

#### CP Versus MILP.

Even though our CP formulation can be solved with an MILP solver after being
linearized, we used IBM CP solver IBM (2021) because linearization of non-
linear functions (e.g. max, logical operators, etc) introduces a large number
of additional variables, thus making the problem unsolvable in a practical
amount of time. In the case of the largest instance (instance #8)
linearization increases the number of variables by $\sim$32 times and the
number of constraints by $\sim$9 times. Specifically, MILP formulation
contains 3,900,750 variables and 4,280,130 constraints vs 120,000 variables
and 472,801 constraints in the CP case (not counting homogeneity constraints).
As a consequence, IBM MILP solver failed to find any feasible solution within
2.5 hours. Across all 8 real data sets, the size of CP problem lies within the
range of 2,108–196,000 variables and 6,528–300,000 constraints (after
incorporating homogeneity constraints). IAMAX is designed to be used as a
decision support tool by system admins, thus we prioritize fast solving time
by setting the time limit to 15 minutes. All experiments were conducted on AWS
c5.24xlarge virtual machine equipped with 96 vCPU and 192 GB of RAM.

#### Real world graphs.

Real world graphs were shared by IT departments of 8 commercial organizations
(Table 2). Graphs are very sparse as we would expect in a security setting -
only certain accesses (permission edges) are allowed. However, most of them
remain unused: the number of dynamic edges that represent actual data accesses
is even smaller. Densities of graphs built on permission and dynamic edges
differ by 1.4 - 29.9 times. High coefficients correspond to poorly designed
(over-priviledged) security models. The number of users varies between 13 and
150, while the number of datastore nodes lies within the range of 56 - 600.

#### Synthetic graphs.

We use real graphs as baselines to generate 280 synthetic graphs. We vary the
number of nodes, but keep graph density as is, i.e. in the range of 0.259
$\pm$ 0.198 (avg $\pm$ std). To generate a synthetic graph, we first sample
the number of users and datastores from uniform distributions over the
following intervals [10, 150] and [50, 300] respectively that cover variations
of those parameters across real graphs. We deliberately set the maximum number
of data stores fewer than 600 (instance 8) to speed up computations. After
fixing node counts we sample with replacement the actual nodes from a real
world graph, which is chosen at random. Then we add Gaussian N(0, 0.01) noise
to node embeddings and renormalize them. To match the graph density with the
density of the underlying baseline we sample edges from a multinomial
distribution, where each component is proportional to the cosine distance
between a user and a datastore embeddings. Also we enforce the invariant that
dynamic edges are always a subset of all permission edges. A synthetic graph
generated in such a way is an ”upsampled” version of an underlying real world
graph.

###  7.1 Reduction of Dormant Permissions

#### Real world instances.

IAMAX significantly reduces dormant permissions (Figure 3) over the current
IAM policy of companies. For this purpose we vary the number of datastore
access groups $|\mathcal{G}|$ from 1 up to 100 with an increment of 5. The
solver’s time limit is fixed at 15 minutes. When the number of datastore
access group is too small, the problem often becomes infeasible. For most data
sets, the fraction of remaining dormant permissions quickly goes down to 0% as
we increase the number of datastore access groups. The only exception is the
instance 8 (the largest data set), which requires 50 data access groups.
According to these results we set the number of data access groups to 20 for
instances 1-7, and 40 for the instance 8 in other experiments. We also
highlight that it is important to have as few as possible datastore access
groups to keep IAM policy interpretable. Our results show that even with
additional 5 groups, dormant permissions reduce significantly over the
existing IAM policies.

#### Synthetic experiments.

To evaluate IAMAX across larger number of instances, we use 280 synthetic
graphs. Figure 3 shows unnormalized distribution of the remaining dormant
permissions in a graph while running the solver for at most 15 minutes, and
fixing the datastore access groups at 20. Each bin spans 5% interval, graphs
with no dormant permissions (0%) fall into the very first bin. The number
above a bar denotes how many instances fall in the corresponding interval. The
histogram is skewed towards the left-hand side: in 46.7% of cases IAMAX
reduces dormant permissions down to 0%. However, we observe some outliers
where the solver is unable to reach dormant permissions 0% level. This mostly
happens because of either exceeding the time limit or the need for a larger
number of data access groups due to instance complexity. These results can be
improved by setting a higher time limit for the CP solver or increasing the
number of data access groups.

###  7.2 User Embedding Constraints

To make generated IAM policies interpretable we follow constraint generation
approach outlined in Section 3.2, which leads to generating mostly homogeneous
data access groups.

Figure 4 shows the fraction of unsatisfied embedding constraints at each
iteration when setting the number of data access groups to 20. Depending on
the data set, it takes 10-20 iterations to satisfy more than 95% of embedding
constraints. To verify that such an approach produces mostly homogeneous
groups, we compare the entropy of generated data access groups with respect to
users’ cluster assignment (Figure 4) before and after adding embedding
constraints. In all cases except the data sets 4 and 6, embedding constraints
drastically reduce group entropy. In the case of the data sets 4 and 6, the
problem becomes infeasible at the very first constraint relaxation iteration.

###  7.3 Simulated Security Attacks

We consider two security attacks that fundamentally differ in terms of
knowledge that an attacker posses. In both cases an attacker tries to
compromise $k$ users, where $k\in\\{1,2,3\\}$. An attack’s impact is the
number of datastores that an attacker can get access to. Hence, we report the
relative attack impact, which is the the ratio between the number of
compromised datastores after applying IAMAX and the number of compromised
datastores in the existing cloud infrastructure. The lower ratio is, the more
noticeable effect IAMAX has on the IAM policy optimization. We notice that
IAMAX’s effect can be masked by high-degree user nodes, especially, in the
worst-case attack because the greedy algorithm keeps selecting such nodes.
High-degree user nodes impose a severe security risk on an organization and
this issue should be mitigated using traditional software engineering methods
- splitting nodes into multiple nodes of lower degrees. To evaluate IAMAX in
the worst-case attack scenario, we removed top-30% of high-degree nodes.

If an attacker has no information about the implemented IAM policies, then we
can estimate the average impact of compromising $k$ user identities (Figure
4). For example, in the case of the instance 8 IAMAX achieves the highest
impact reduction (86% - 88%) even though it is the largest real world data
set. However, impact reduction is minimal for the instance 1 (8% - 22%) due to
the presence of high-degree user nodes.

If an attacker has a perfect knowledge of the IAM policies, then the attacker
can cast the problem to max k-cover and solve it using a greedy approximation
algorithm. Figure 4 illustrates such a case study. IAMAX minimizes the worst
case attack impact for organizations 3, 5, and especially 8 by 41% - 89%.
However, organizations 1 and 2 remain almost unaffected due to the large
number of high-degree nodes remaining in the data set after removing the
top-30% of user nodes sorted by their degrees.

Our current simulated attacks are primarily based on the interaction patterns
between users and datastores, and the observations available to an attacker.
We note that using the rich language of constraint programming, we can also
generate automated attacks that achieve malicious goals while complying with
IAM policies. We leave this as part of the future work.

##  8 Conclusion and Future Work

Given the increasing popularity of cloud computing, associated security issues
are also increasing in severity. We developed a principled approach for the
key problem of optimizing IAM policies to reduce the attack surface of an
organization’ cloud setup (or the so-called dormant permissions that allow
users access datastores which are not needed for users’ business functions).
We presented a formulation of IAM policy optimization using constraint
programming and graph representation learning, identified key constraints
which IAM policies should satisfy, and then tested the resulting IAM policies
on 8 real world and multiple synthetic data sets. Our results show that IAMAX
is highly effective in reducing dormant permissions and generating
interpretable IAM policies. Our framework also opens the door to the
application of a host of AI methods to address additional security problems in
cloud infrastructure.

Several promising opportunities exist for enriching IAM policy generation with
more complex constraint types. For example, temporal constraints that define
the order of datastore accesses and set constraints that define what subset of
automated services can simultaneously access a given resource can
significantly limit the range of attackers’ actions after they compromise a
cloud environment. To formulate such constraints, we can use program
invariants inferred during the program analysis of automated services.
Moreover, violations of such constraints at runtime can also be used for
anomaly detection.

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Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce,
TX 75429-3011, USA Quantum Machine Learning Laboratory, Shadow Creator Inc.,
Shanghai 201208, China School of Physics and Astronomy, Shanghai Key
Laboratory for Particle Physics and Cosmology, and Key Laboratory for Particle
Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai
200240, China Department of Physics, Yuncheng University, Yuncheng 044000,
China Shanghai Advanced Research Institute, Chinese Academy of Sciences,
Shanghai 201210, and Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai 201800, China School of Physics, Southeast University,
Nanjing 211189, China

# A theoretical overview of isospin and EOS effects in heavy-ion reactions at
intermediate energies

Bao-An Liins:1 Speaker’s email address<EMAIL_ADDRESS>Bao-Jun Caiins:2
Lie-Wen Chenins:3  Wen-Jie Xieins:4  Jun Xuins:5 Nai-Bo Zhangins:6
ins:1ins:1ins:2ins:2ins:3ins:3ins:4ins:4ins:5ins:5ins:6ins:6

###### Abstract

The isospin dependence of in-medium nuclear effective interactions is a
fundamental issue in nuclear physics and has broad ramifications in
astrophysics. Its uncertainties, especially the difference of neutron-proton
interactions in the isosinglet and isotriplet channels, affect significantly
the density and momentum dependence of the isovector single-nucleon potential
and nucleon-nucleon short-range correlation in neutron-rich matter.
Consequently, the neutron-proton effective mass splitting and the density
dependence of nuclear symmetry energy are still rather uncertain. Heavy-ion
reactions especially those involving rare isotopes is a useful tool for
probing the isospin dependence of nuclear effective interactions through (1)
the neutron-skin in coordinate and proton-skin in momentum of the initial
state of colliding nuclei, (2) the density and momentum dependence of
especially the isovector nuclear mean-field as well as (3) the isospin
dependence of in-medium nucleon- nucleon cross sections. Observations of
neutron stars especially since GW1710817 have also helped us significantly in
understanding the isospin dependence of nuclear effective interactions. We
summarize here a review talk on these issues given at the 2021 International
Workshop on multi-facets of EOS and Clustering. For details we refer the
readers to the original publications and references therein.

##  1 The isospin-dependence of nuclear effective interactions and its
ramifications on the EOS of neutron-rich matter

The isospin degree of freedom and nuclear Equation of State (EOS) affect
heavy-ion reactions at low-intermediate energies through both the in-medium
nucleon-nucleon (NN) cross sections $\sigma^{*}_{NN}$ and the single-nucleon
potential $U_{\tau}(k,\rho,\delta)$. The latter for a nucleon ($\tau_{3}=\pm
1$ for neutrons/protons) of momentum $k$ in neutron-rich matter of density
$\rho$ and isospin asymmetry $\delta=(\rho_{n}-\rho_{p})/\rho$ can be written
as [1]

$U_{\tau}(k,\rho,\delta)=U_{0}(k,\rho)+\tau_{3}U_{\rm
sym,1}(k,\rho)\cdot\delta+U_{\rm sym,2}(k,\rho)\cdot\delta^{2}+\tau_{3}U_{\rm
sym,3}(k,\rho)\cdot\delta^{3}+\mathcal{O}(\delta^{4})$ (1)

in terms of the isoscalar $U_{0}(k,\rho)$ and $U_{\rm sym,2}(k,\rho)$ as well
as the isovector $U_{\rm sym,1}(k,\rho)$ and $U_{\rm sym,3}(k,\rho)$
potentials, respectively. While the energy per nucleon $E(\rho,\delta)$ in
neutron-rich matter can be approximated as a parabolic function of $\delta$ as
$E(\rho,\delta)=E_{0}(\rho)+E_{\rm{sym},2}(\rho)\delta^{2}+\mathcal{O}(\delta^{4})$
in terms of the energy per nucleon $E_{0}(\rho)\equiv E(\rho,\delta=0)$ in
symmetric nuclear matter (SNM) and the isospin-quadratic symmetry energy
$E_{\rm{sym},2}(\rho)$ (denoted often by $E_{\rm{sym}}(\rho)$ or $S$ in the
literature). The latter can be evaluated approximately from
$E_{\rm{sym}}(\rho)\equiv
E_{\rm{sym},2}(\rho)\equiv\left.\frac{1}{2}\frac{\partial^{2}E(\rho,\delta)}{\partial\delta^{2}}\right|_{\delta=0}\approx
E(\rho,1)-E(\rho,0).$ In terms of the components of the nucleon potentials,
the symmetry energy can be written as [2, 3]

$E_{\rm{sym}}(\rho)=\frac{k_{\rm{F}}^{2}}{6m^{*}_{0}(\rho,k_{\rm{F}})}+\frac{1}{2}U_{\rm{sym},1}(\rho,k_{\rm{F}}).$
(2)

where $k_{F}=(3\pi^{2}\rho/2)^{1/3}$ is the nucleon Fermi momentum and
$m^{*}_{0}/m=(1+\frac{m}{\hbar^{2}k_{\rm F}}\partial U_{0}/\partial
k)^{-1}|_{k_{F}}$ is the nucleon isoscalar effective mass of nucleons with a
free mass $m$. In terms of the $m^{*}_{0}$, according to the Hugenholtz-Van
Hove (HVH) theorem the slope $L(\rho)\equiv 3\rho\frac{dE_{\rm sym}}{d\rho}$
of the symmetry energy at an arbitrary density $\rho$ is given by [3, 4]

$L(\rho)=\frac{2}{3}\frac{\hbar^{2}k_{F}^{2}}{2m_{0}^{*}}+\frac{3}{2}U_{\rm
sym,1}(\rho,k_{F})-\frac{1}{6}\Big{(}\frac{\hbar^{2}k^{3}}{{m_{0}^{*}}^{2}}\frac{\partial
m_{0}^{*}}{\partial k}\Big{)}|_{k_{F}}+\frac{\partial U_{\rm sym,1}}{\partial
k}|_{k_{F}}k_{F}+3U_{\rm sym,2}(\rho,k_{F}).$ (3)

The single-nucleon potential of Eq. (1) is a basic input in nuclear reaction
models, e.g., various transport models for heavy-ion reactions. The above non-
relativistic decompositions of both the symmetry energy $E_{\rm{sym}}(\rho)$
and its slope $L(\rho)$ in terms of the density and momentum dependences of
the isoscalar and isovector single-nucleon potentials reveal clearly the
underlying microscopic physics. The latter can be probed by using nuclear
reactions at low-intermediate energies, especially those involving rare
isotopes at radioactive beam facilities around the world.

While the detailed density and momentum dependences of the single-nucleon
potential in neutron-rich matter can be studied consistently in microscopic
nuclear many-body theories, see, e.g., Ref. [5], to see the fundamental
physics behind the $U_{\rm{sym},1}(\rho,k)$, it is educational to simply look
at its Hartree (direct) term at $k_{F}$ in the interacting Fermi gas model
[6], namely, $U_{\rm sym,1}(k_{F},\rho)=\frac{\rho}{4}\int[V_{T1}\cdot
f^{T1}(r_{ij})-V_{T0}\cdot f^{T0}(r_{ij})]d^{3}r_{ij}$ in terms of the
isosinglet (T=0) and isotriplet (T=1) NN interactions $V_{T0}(r_{ij})$ and
$V_{T1}(r_{ij})$, as well as the corresponding NN correlation functions
$f^{T0}(r_{ij})$ and $f^{T1}(r_{ij})$, respectively. While
$V_{nn}=V_{pp}=V_{np}$ in the T=1 channel due to the charge independence of NN
interactions, the $V_{np}$ interactions and the associated NN correlations in
the T=1 and T=0 channels are not the same due to the isospin dependence of the
strong interactions. For example, it is well known that the tensor force and
the resulting short-range correlation (SRC) in the T=0 channel is much
stronger than that in the T=1 channel. However, several features of the SRC,
such as its strength, the shape of the resulting high-momentum nucleon
distributions in cold nuclear matter as well as their isospin and density
dependence, are still poorly known. Obviously, these uncertainties will affect
the density and momentum dependences of the single-nucleon potential, and thus
the corresponding density dependence of the symmetry energy and the neutron-
proton effective mass splitting. For a recent review on these issues, see,
e.g., Ref. [7].

##  2 Probing the isospin-dependence of nuclear effective interactions by
combining mutli-messengers from neutron star observations and nuclear reaction
experiments

Many studies in the literature over the last two decades have shown
consistently that some observables of neutron stars and heavy-ion reactions
bear clear signatures of the single-nucleon potential
$U_{\tau}(k,\rho,\delta)$ as well as the corresponding symmetric nuclear
matter (SNM) EOS $E_{0}(\rho)$ and the symmetry energy $E_{\rm{sym}}(\rho)$.
To see intuitively, for example, how the latter can be extracted from
combining information from both astrophysical observations and terrestrial
nuclear experiments, we note that [8] $E_{\textrm{sym}}(\rho)\approx
E_{\textrm{sym}}(\rho_{i})+\int_{\rho_{i}}^{\rho}\frac{P_{\rm{PNM}}(\rho_{v})-P_{\rm{SNM}}(\rho_{v})}{\rho_{v}^{2}}d\rho_{v}$
where $\rho_{i}$ is a reference density. The pressure $P_{\rm{SNM}}$ in SNM
over a broad density range has been extracted from terrestrial nuclear
reaction experiments [9], while recent observations of neutron stars through
multi-messengers have provided more information about the nuclear pressure in
neutron-rich matter towards pure neutron matter (PNM) [10, 11]. In the
following, we present a few examples illustrating what we have learned
recently from nuclear reactions and neutron stars about the isospin dependence
of nuclear effective interactions in neutron-rich matter.

Example1-Neutron-proton effective mass splitting in neutron-rich matter:

Figure 1: A survey of neutron-proton effective mass splitting at $\rho_{0}$
from analyzing nuclear reaction and structure experiments in comparison with
the latest chiral effective field theory prediction (its 68% confidence range
is indicated by the horizontal violet lines).

The momentum dependence of the single-nucleon potential is normally
characterized by the nucleon effective masses $M^{*}_{N}$. The study on the
effective masses of neutrons and protons in neutron-rich matter has a long
history while many interesting issues remain to be resolved. In particular,
due to our poor knowledge about the momentum dependence of isovector
interaction, the isovector nucleon effective mass normally measured using the
effective mass splitting $m^{*}_{\rm{n-p}}(\rho,\delta)\equiv(m_{\rm
n}^{*}-m_{\rm p}^{*})/m$ of neutrons and protons at the Fermi momentum $K_{F}$
has not been well constrained. The $m^{*}_{\rm{n-p}}(\rho,\delta)$ can be
written as [7] $m^{*}_{\rm{n-p}}(\rho,\delta)\approx
2\delta\frac{m}{k_{\rm{F}}}\left[-\frac{dU_{\rm{sym},1}}{dk}-\frac{k_{\rm{F}}}{3}\frac{d^{2}U_{0}}{dk^{2}}+\frac{1}{3}\frac{dU_{0}}{dk}\right]_{k_{\rm{F}}}\left(\frac{m^{*}_{0}}{m}\right)^{2}.$
At saturation density, it is related to the $E_{\rm{sym}}(\rho_{0})$ and
$L(\rho_{0})$ via [12]

$m^{*}_{\rm{n-p}}(\rho_{0},\delta)\approx\delta\cdot\frac{\displaystyle
3E_{\rm{sym}}(\rho_{0})-L(\rho_{0})-3^{-1}({m}/{m^{*}_{0}})E_{\rm{F}}(\rho_{0})}{\displaystyle
E_{\rm{F}}(\rho_{0})\left({m}/{m_{0}^{*}}\right)^{2}}.$ (4)

Therefore, the $m_{\rm n}^{*}$ is equal to, larger or smaller than the $m_{\rm
p}^{*}$ depends on the symmetry energy and its slope. For example, with the
empirical values of $E_{\rm sym}(\rho_{0})$=31 MeV, $m_{0}^{*}/m=0.7$ and
$E_{\rm{F}}(\rho_{0})=36$ MeV, a positive $m^{*}_{\rm{n-p}}(\rho_{0},\delta)$
implies that the slope L should be less than $76$ MeV. Interestingly, as we
shall show in the next example, most of the extracted values of $L(\rho_{0})$
from both terrestrial experiments and astrophysical observations satisfy this
condition.

The nuclear physics community has devoted significant efforts to constraining
the $m^{*}_{\rm{n-p}}(\rho,\delta)$ especially at the saturation density
$\rho_{0}$. Shown in Fig. 1 is a survey of $m^{*}_{\rm{n-p}}/\delta$ at
$\rho_{0}$. While there have been many, sometimes controversial, predictions
on the $m^{*}_{\rm{n-p}}(\rho,\delta)$ using various nuclear many-body
theories and interactions, only the latest prediction based on the many-body
perturbation theory using chiral effective forces ($\chi$EFT+MBPT) by
Whitehead et al [13, 14] is shown for a comparison. It is seen that most of
the results from the indicated analyses are consistent with the $\chi$EFT+MBPT
prediction within their 68% error bars. In particular, all earlier analyses of
nucleon-nucleus scattering data [15, 16, 17], the analysis based on the HVH
theorem using the 2013 systematics of nuclear symmetry energy [12] as well as
both the static and dynamical model analyses of the isoscalar and isovector
giant resonances and the electrical dipole polarizability of 208Pb [18, 19,
20, 21] indicate surely a positive $m^{*}_{\rm{n-p}}(\rho_{0},\delta)$. We
notice that the four open squares are from using the same data set but
different approaches. They give qualitatively consistent but quantitatively
appreciably different $m^{*}_{\rm{n-p}}(\rho_{0},\delta)$ values. One
interesting exception is the result of
$m^{*}_{\rm{n-p}}(\rho_{0},\delta)=(-0.05\pm 0.09)\delta$ from a Bayesian
analysis of the neutron/proton spectrum ratios in several Sn+Sn reactions at
120 MeV/nucleon at NSCL/MSU using an ImQMD transport model for nuclear
reactions with Skyrme forces [22]. A negative
$m^{*}_{\rm{n-p}}(\rho_{0},\delta)$ implies a characteristically different
momentum dependence of the nucleon isovector potential inconsistent with the
findings from optical model analyses of existing nucleon-nucleus scattering
data [7].

Figure 2: Left: The correlation between the average nucleon effective mass and
the average nucleon density (top), and the distribution of nucleon effective
masses (bottom) in the reaction of 132Sn+124 Sn at 10 fm/c with a beam energy
of 50 MeV/A and an impact parameter of 5 fm [23]. Right: The correlation
between the isospin asymmetry and density during 132Sn+124 Sn collisions at 20
fm/c with a beam energy of 400 MeV/A and an impact parameter of 1 fm and in
the core of neutron stars at $\beta$-equilibrium (inset) [25].

It is useful to note that heavy-ion reactions probe nucleon effective masses
over a wide density range while nucleon-nucleus scatterings and/or giant
resonances only probe those around $\rho_{0}$. The significant discrepancies
between the results from analyzing the heavy-ion reaction data and those from
other types of data present an interesting challenge. More studies especially
with high energy radioactive beams may help address this issue. As an example,
shown on the left in Fig. 2 are the correlation between the average nucleon
effective mass and the average nucleon density (top), and the distribution of
nucleon effective masses (bottom) at 10 fm/c in the reaction of 132Sn+124Sn at
a beam energy of 50 MeV/A and an impact parameter of 5 fm [23] within the IBUU
transport model with a modified Gogny Hartree-Fock potential [24], while on
the right are the correlations between the isospin asymmetries and densities
in central 132Sn+124Sn collision at 400 MeV/A and neutron stars at
$\beta$-equilibrium (inset) using the symmetry energy functionals shown in the
upper-right frame [25]. It is seen that during a typical heavy-ion reaction,
the nucleon effective masses and the neutron-proton effective mass splitting
depend on the density, and the matter formed can have various densities and
isospin asymmetries depending on the symmetry energy functional used.

A number of observables of heavy-ion reactions have been proposed as promising
messengers of the underlying momentum dependence of the isovector potential
and the corresponding $m^{*}_{\rm{n-p}}(\rho,\delta)$ [26]. The reason is
easily understandable. For example, the in-medium NN cross section
$\sigma^{*}_{NN}$ is proportional to the square of the reduced effective mass
of the two colliding nucleons. It will then affect the nuclear stopping power
or the nucleon mean free path (MFP)
$\lambda_{\rm{p}}^{-1}=\rho_{\rm{p}}\sigma_{\rm{pp}}^{\ast}+\rho_{\rm{n}}\sigma_{\rm{pn}}^{\ast}$
and
$\lambda_{\rm{n}}^{-1}=\rho_{\rm{n}}\sigma_{\rm{nn}}^{\ast}+\rho_{\rm{p}}\sigma_{\rm{np}}^{\ast}$.
In terms of the nucleon effective k-mass $M_{J}^{\ast,\rm{k}}$, the nucleon
MFP is given by [27]
$\lambda_{J}=k_{\rm{R}}^{J}/(2M_{J}^{\ast,\rm{k}}|W_{J}|)$ where
$k_{\rm{R}}^{J}=[2M(E-U_{J})]^{1/2}$, $U_{J}/W_{J}$ is the real/imaginary part
of the nucleon J’s optical potential. Currently, no consensus on the
$m^{*}_{\rm{n-p}}(\rho,\delta)$ has been reached from studying heavy-ion
collisions as there are still large model dependences in both the transport
models and the input single-nucleon potentials. Nevertheless, it is
encouraging to note that significant community efforts are being devoted to
improving the situation [28]. There are also strong experimental efforts to
measure nuclear stopping powers and infer the underlying in-medium NN cross
sections in neutron-rich matter, see, e.g., Ref. [29].

Example2-Progress in constraining nuclear symmetry energy around
$(1-2)\rho_{0}$:

Figure 3: The slope parameter $L$ (left) and curvature parameter
$K_{\rm{sym}}$ (right) of $E_{\rm{sym}}(\rho)$ at $\rho_{0}$ from analyses of
neutron star observables since GW170817 in comparisons with the indicated
earlier systematics. A complete list of individual analyses are given in Ref.
[11] where these plots are taken from.

Figure 4: Left: A survey of nuclear symmetry energy from heavy-ion reactions
and neutron stars, taken from Ref. [11]. Right: constraints on the high-
density symmetry energy and proton fraction using data from GW170817 and
NICER+XMM-Newton’s joint measurements of mass and radius for PSR J0740+6620,
taken from Ref. [30].

Much efforts have been made during the last two decades in constraining the
$E_{\rm{sym}}(\rho)$. The latter can be characterized by its slope parameter
$L$, curvature parameter $K_{\rm{sym}}$ and skewness parameter $J_{\rm{sym}}$
at $\rho_{0}$ according to $E_{\rm{sym}}(\rho)\approx
E_{\rm{sym}}(\rho_{0})+L(\frac{\rho-\rho_{0}}{3\rho_{0}})+\frac{K_{\rm{sym}}}{2}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{2}+\frac{J_{\rm{sym}}}{6}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{3}.$
Significant progresses have been made in constraining the $L$ and
$K_{\rm{sym}}$ parameters using data from both terrestrial experiments and
astrophysics observations, while the $J_{\rm{sym}}$ characterizing the
stiffness of $E_{\rm{sym}}(\rho)$ around $(2-3)\rho_{0}$ remains unconstrained
[11]. Shown in Fig. 3 are the slope parameter $L$ (left) and curvature
parameter $K_{\rm{sym}}$ (right) from analyses of neutron star observables
since GW170817 in comparisons with the indicated earlier systematics. A
complete list of the individual analyses are given in Ref. [11]. It is seen
that the new analyses of neutron star observables gave $L\approx 57.7\pm 19$
MeV and $K_{\rm{sym}}\approx-107\pm 88$ MeV in very good agreement with their
systematics from earlier analyses. We notice that the extracted most probable
L value from neutron star observables prefer a positive neutron-proton
effective mass splitting according to Eq. (4).

Recent neutron star observations have also led to some progresses in
constraining the symmetry energy at suprasaturation densities. Shown on the
left side of Fig. 4 is a summary of symmetry energy at $2\rho_{0}$ from eleven
analyses of heavy-ion reactions and neutron star properties in the literature.
These analyses gave a mean value of $E_{\rm{sym}}(2\rho_{0})\approx 51\pm 13$
MeV at 68% confidence level as indicated by the green line. Obviously, more
works are necessary to reduce the error bars. Most of the neutron star
constraints are extracted from the radii and tidal deformations of canonical
neutron stars. These observables are known to be sensitive mostly to pressures
around $(1-2)\rho_{0}$ in neutron stars. Their constraints on the
$E_{\rm{sym}}(\rho)$ around and above $2\rho_{0}$ are thus not so strong.
Observables from more massive neutron stars were expected to place stronger
constraints. To see how the NICER+XMM-Newton’s simultaneous measurements of
both the radius and mass of PSR J0740+6620 can help constrain the symmetry
energy at densities above $2\rho_{0}$, shown in the upper-right is the
extracted lower limits of $E_{\rm{sym}}(\rho)$ when only the mass is observed
(green line) or both the mass and radius are observed (red using the radius
reported by Riley et al. [31] and blue using the radius reported by Miller et
al [32]) from directly inverting the TOV equation within a 3-dimensional high-
density EOS parameter space [30]. While the orange line is the upper limit of
symmetry energy from analyzing the upper limit of tidal deformation of
GW170817. The lower-right window shows the corresponding proton fractions in
PSR J0740+6620. The constraining power of knowing both the mass and radius of
this currently known most massive neutron star is clearly seen by comparing
the green line with the red or blue line. While the difference between the red
and blue lines indicates the systematic error from the two independent
analyses of the same observational data. Overall, while tighter constraints on
the $E_{\rm{sym}}(\rho)$ around $(2-3)\rho_{0}$ have been obtained, there is
still a strong need for better constraining the $E_{\rm{sym}}(\rho)$ at
densities around and above $2\rho_{0}$ as indicated by the big gaps between
the inferred upper and lower limits of $E_{\rm{sym}}(\rho)$.

Example3-The incompressibility, skewness and kurtosis of SNM EOS: The
stiffness of SNM EOS can be characterized by its incompressibility $K_{0}$,
skewness $J_{0}$ and kurtosis $Z_{0}$ parameters according to
$E_{0}(\rho)\approx
E_{0}(\rho_{0})+\frac{K_{0}}{2}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{2}+\frac{J_{0}}{6}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{3}+\frac{Z_{0}}{24}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{4}$
with $E_{0}(\rho_{0})$=-15.9 MeV. The corresponding pressure can be readily
obtained from $P(\rho)=\rho^{2}\frac{dE_{0}(\rho)}{d\rho}$. While the $K_{0}$
has been constrained to the range of 220 to 260 MeV according to various
analyses of giant monopole resonances since about 1980 [33, 34], the $J_{0}$
and $Z_{0}$ characterizing the stiffness of SNM EOS at super-saturation
densities were only known roughly until very recently to be around $-200\pm
200$ and $-146\pm 1728$ MeV [35], respectively, mostly due to our poor
knowledge about the three-body force in dense matter.

Figure 5: Left: The posterior PDFs for $K_{0}$, $J_{0}$ and $Z_{0}$ and their
correlations obtained from the Bayesian analyses of the constraining bands on
the SNM pressure using Gaussian priors. Right: using the uniform prior for
$K_{0}$ in 220-260 MeV. Taken from Ref. [36].

Interestingly, some improvements were obtained very recently by a Bayesian
calibration of the SNM EOS [36] by using the constraining bands on the
pressure in cold SNM in the density range of 1.3$\rho_{0}$ to 4.5$\rho_{0}$
from transport model analyses of kaon production and nuclear collective flow
in relativistic heavy-ion collisions [9].

Figure 6: Left: $K_{0}$ and $K_{\tau}$ from a differential analysis of giant
resonance data taken at RCNP, taken from Ref. [37]. Right: the posterior PDF
of $K_{0}$ from a Bayesian analysis of giant resonance data, taken from Ref.
[38].

As shown in Fig. 5, assuming $K_{0}$, $J_{0}$ and $Z_{0}$ all have Gaussian
priors centered around $235\pm 30$, $-200\pm 200$ and $-146\pm 1728$ MeV,
respectively, the posterior most probable values of $J_{0}$ and $Z_{0}$ were
found to be $J_{0}$=-240${}^{+110}_{-120}$ MeV and
$Z_{0}$=250${}^{+300}_{-250}$ MeV, respectively, at 68% confidence level.
Their uncertainties are significantly smaller than their prior values.
Interestingly, the inferred posterior PDF (probability distribution function)
of $K_{0}$ peaks at 192 MeV, indicating that the flow data prefers a $K_{0}$
significantly smaller than its fiducial value from analyses of giant
resonances. Moreover, it was found that the posterior PDFs depend somewhat on
the prior PDF used for the $K_{0}$. For example, shown in the right window of
Fig. 5 are the posterior PDFs of $K_{0}$, $J_{0}$ and $Z_{0}$ obtained by
using a uniform prior for $K_{0}$ in 220-260 MeV while the prior PDFs for
others are kept the same. Now, the posterior PDF of $K_{0}$ peaks at the lower
boundary of its prior range, indicating again that the flow data prefers a
smaller $K_{0}$ and the prior knowledge on $K_{0}$ from giant resonances is
not sufficiently constraining. These results clearly indicate the needs for
more stringent constraints on the $K_{0}$ before tighter constraints on the
$J_{0}$ and $Z_{0}$ can be obtained.

Partially due to the reasons mentioned above, some efforts have been made
recently to infer the $K_{0}$ from the available world data of giant
resonances in new ways significantly different from the traditional “consensus
approach” (i.e., the model must first reproduce all giant resonances and also
other observables for finite nuclei before using it to predict the $K_{0}$ for
SNM). One such effort is the simultaneous inference of both the $K_{0}$ and
$K_{\tau}$ involved in the leptodermous expansion [33] of the
incompressibility of finite nuclei $K_{A}\approx
K_{0}(1+cA^{-1/3})+K_{\tau}\delta^{2}+K_{\rm{Cou}}Z^{2}A^{-4/3}$ in a
differential approach [37]. It uses the experimental $K_{A}$ values from RCNP
for any two pairs in a given chain of isotopes. Shown in the left window of
Fig. 6 are the $K_{0}$ and $K_{\tau}$ as functions of the difference
($\delta_{2}-\delta_{1}$) in isospin asymmetries of the isotope pairs used.
The approach is not good for light nuclei for which the leptodermous expansion
is less valid. While the inferred $K_{0}$ values from medium-heavy nuclei are
well within the uncertainty range of its fiducial value, the inferred
$K_{\tau}$ is much more precise than previously known [34]. It was also shown
that the nuclear pairs with the largest separations in their isospin
asymmetries give the most precise results. Another approach is the Bayesian
inference using the SHF+RPA approach [38]. Compared to the traditional
forward-modeling, this approach gives a more precisely quantified uncertainty
estimation. Among the results shown in the right window of Fig. 6, the
posterior PDFs of $K_{0}$ from using the giant resonance data from RCNP for
208Pb is consistent with that from the differential approach shown on the
left. Moreover, the slightly smaller $K_{0}$ values from analyzing the Sn data
using both approaches are also consistent. However, considering the systematic
errors from using the data from different laboratories and/or different
nuclei, the $K_{0}$ values from these two new approaches are consistent but
not more precise than its fiducial value. Thus, we have to work harder!

In summary, various isospin and EOS effects in nuclear reactions and
properties of neutron stars enabled us to explore the isospin dependence of
in-medium nuclear effective interactions. While significant progresses have
been made by the community in constraining both the SNM EOS and nuclear
symmetry energy below about $2\rho_{0}$, many interesting and challenging
questions remain to be resolved.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy, Office of
Science, under Award Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory
Institute for Physics with Exotic Nuclei) under the U.S. Department of Energy
Grant No. DE-SC0009971, the Yuncheng University Research Project under Grant
No. YQ-2017005, the National Natural Science Foundation of China under Grant
No. 11505150, No. 12005118, No. 11922514, No 11905302 and No. 11625521,
National SKA Program of China Grant No. 2020SKA0120300, the Scientific and
Technological Innovation Programs of Higher Education Institutions in Shanxi
under Grant No. 2020L0550, and the Shandong Provincial Natural Science
Foundation under Grant No. ZR2020QA085.

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