# Review of Advanced Monitoring Mechanisms in Peer-to-Peer (P2P) Botnets

1st Wong Yan Shen  National Advanced IPv6 Center
Universiti Sains Malaysia
Penang, Malaysia
<EMAIL_ADDRESS>2nd Selvakumar Manickam  National Advanced IPv6 Center
Universiti Sains Malaysia
Penang, Malaysia
<EMAIL_ADDRESS>3st Mahmood A. Al-Shareeda  National Advanced IPv6 Center
Universiti Sains Malaysia
Penang, Malaysia
<EMAIL_ADDRESS>
###### Abstract

Internet security is getting less secure because of the existing of botnet
threats. An attack plan can only be planned out to take down the botnet after
the monitoring activities to understand the behaviour of a botnet. Nowadays,
the architecture of the botnet is developed using Peer-to-Peer (P2P)
connection causing it to be harder to be monitored and track down. This paper
is mainly about existing botnet monitoring tools. The purpose of this paper is
to study the ways to monitor a botnet and how monitoring mechanism works. The
monitoring tools are categorized into active and passive mechanism. A crawler
is an active mechanism while sensor and Honeypot are the passive mechanisms.
Previous work about each mechanism is present in this paper as well.

###### Index Terms:

Peer-to-Peer (P2P), Honeypot, monitoring, architecture, Botnet.

##  I Introduction

A botnet is the network of bots. Bot is the naming for infected computers
acting as a controlled robots or machines for criminal, known as bot master to
carry out cyber-crime like infecting computers and machines with viruses and
malware, accessing a webpage at the same time causing denial of service(DOS),
important personal details thieving, spamming and the other cyber-crimes [1,
2, 3].

A user gets infected when they access to a malicious email attachment, visit a
compromised website, or unsuspectingly download the bot files onto their
computer. Once infected, the botmaster will gain access to the victim’s
computer, without acknowledgement from the victim. By accessing to a victim’s
computer will allow the botmaster to obtain two critical resources: CPU power
and IP address [4]. Combining all the CPU resources from the bots can form
powerful equipment equivalent to a modern-day supercomputer service is
provided to botmaster. The diversity of IP address will causes confusion and
increases the difficulty of a cyber-crime detection as all those IP addresses
seems to be legitimate. According to industry estimates, botnets have caused
over $110 billion in losses globally [4]. The Internet security issues caused
by botnet has even triggered the US government and American Industries
awareness and listing it as the greatest threats to cyber security [5, 6].
According to the improved speed of the Internet and the technology, the usage
of Internet of Things(IoT) gadgets has been targeted as victims to cybercrime
too [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31]. The situation went more serious when IoT
manufacturers often neglect security in the various phases of Software
development Lifecycle (SDLC) [32]. In order to take down the botnet, long-term
monitoring is required to be done before an attack strategy and method can be
determined [33]. Botnet monitoring can be done through some monitoring
mechanisms. These monitoring mechanisms are Crawler, Sensor, and Honeypot.
Each of these tools is having different functionality in botnet monitoring
activities. Since botnet is an important asset for botmaster, so, botmaster
has included some anti-monitoring mechanism in their botnet design to prevent
being tracked and removed threats from harming their botnet [34, 35, 36]. And,
these antimonitoring mechanism has increased the level of difficulty for
monitoring tools to continuously monitor on the botnet due to the existence of
those interferences and disruption [37, 38, 39].

Brief description of the botnet monitoring activities as follow. Honeypot is
normally using in monitoring activities to obtain the bootstrap nodes (first
node list) for a crawler to crawl through the botnet and the entrance point
for the sensor to be inserted into. Once all the mechanism is in place, the
monitoring activities might take up a few weeks or months. It is important to
understand how the botnet works in the design of the architecture and
mechanism. Then, an equivalent attack plan can be prepared accordingly. The
attack plan aims to bring down all the nodes in a botnet at once because even
with a few nodes leave behind might provide chances for the botnet to recover
after sometimes [40, 35].

The rest of this paper is structured as follows. Section II describes the
background of the botnet architecture and mechanism. Section III shows the
description of botnet monitoring mechanisms. Section IV discusses the
advantage and disadvantage of these mechanisms. Lastly, Section V concludes
this paper and discussion about possible future works.

##  II Background

During forming an attack plan, the network analyst needs to understand botnet
behaviour as to design a matching attack plan. Therefore, to understand botnet
behaviour, we have to know more about botnet architecture and its mechanism.
In this section, the botnet architecture and mechanisms are described. Since
P2P architecture is the main trend of botnet architecture design, so the focus
will be on P2P botnet architecture and mechanisms only.

###  II-A Botnet Architecture

There are two types of botnet architecture: centralized and decentralized or
known as Peer-to-Peer(P2P). The interaction between the bots or the connection
between the bots and the server are vary with different botnet architecture
depending on the command-and-control (C2) protocol [41]. In early botnet
design, botnet structure is designed using only centralized architecture.
However, due to centralized architecture suffers single point of failure,
Peer-to-Peer (P2P) architecture is much preferred in botnet architecture
design nowadays [42].

In P2P architecture, there are structured and unstructured categories. Unlike
centralized botnet, in P2P structured botnet, bots are able to communicate to
each other using P2P protocol for the purpose to update its neighbour peer
information. Then, the botnet connection is fixed and does not change
simplily. Gnutella and Kademlia botnet [43, 44] are examples of P2P structured
botnet. In P2P unstructured botnet, there are no fix formation for this type
of botnet. Hence, it is very hard to be monitored. Sality, ZeroAccess, Kelihos
are some examples of P2P unstructured botnet that is still active in action
till year 2013 [45].

  * •

P2P Structured Botnet: Structured P2P botnet is build up by the Distributed
Hash Table (DHT) decentralized distributed system that provide lookup service
by hash table (key, value) pairs and store to the DHT of every bots. The
botnet conenction is formed from the indexes (For example, ¡Kn,val¿). The
key(Kn) can be representing the IP address of the destination or the distance
of the node. Then, the value(val) can be the command from the botmaster. The
best example of this type of botnet is Gnutella and Kademlia botnet [43, 44].
Since the connection is based on hash table, so the architecture is structured
and connection is built up from a basis unlike unstructured P2P botnet which
does not have a fix structure.

  * •

P2P Unstructured Botnet: Unstructured botnet do not form the botnet structure
using DHT system like structured P2P botnet. The connection in the
unstructured botnet can be modified by botmaster through Membership
Maintenance (MM) mechanism to manipulate a new peer list or a Neighbour List
(NL) of the bots as to change the formation of the botnet connection [46, 36].
All bots NL is handled by Membership Maintenance (MM) mechanism which the
purpose is to control the entrance of the nodes within the bot. Thus, the
overlay connection is flexible and easily modified. When there are nodes that
are inactive, the MM mechanism will replace those inactive nodes with a more
active nodes as to maintain the botnet structure.

  * •

Challenges in P2P Botnet Monitoring: There are many challenges in P2P botnet
monitoring activities. Antimonitoring mechanism designed in the botnet by
botmaster has always causing the most troublesome experience for botnet
monitoring activities [34, 35]. Anti-monitoring mechanism purpose is to
prevent monitoring mechanism like sensor and crawler to be inserted into
botnet networks. Anti-monitoring mechanism like IP Filtering and NL
controlling are mechanisms that specifically designed to block the monitoring
mechanism from entering the botnet and causing interferences to prevent nodes
full NL information from flowing out easily. Hence, it is important to know
and upgrade the botnet monitoring mechanism to against the anti-monitoring
mechanism.

##  III Botnet Monitoring Mechanisms

There are three types of botnet monitoring tools: Honeypot, Crawler and
Sensor. Honeypot and Sensor are passive mechanism as these tools always wait
for connection from the bot. But, for active mechanism - Crawler, it will act
actively by probing the available nodes for the Neighbour List (NL) [47]. So
that, from the list, it can identify the next node to probe.

###  III-A Honeypot

Honeypot is a tool that used for trapping bots from botnet during attacks to
extract information, activities and analyses them in order to understand the
botnets [48]. It usually represent itself in the way of server or high value
asset but vulnerable in the system designed. This is to make it to become a
potential target for network attackers and then gather information for network
security researchers. From the information, researchers can understand more
about the techniques and mechanism used in the botnet but at the same time,
they need to prevent botnet from gaining access to the main systems [49].

There are 3 types of Honeypot level of interactions: low interaction honeypot,
medium-interaction honeypot and high interaction honeypot. The categories are
defined based on the services or interaction level provided by Honeypot to
potential hackers.

  * •

Low Interaction Honeypot: The risk of this Honeypot causing the whole server
to compromise to the botnet is low due to it is just a small pile from the
server and there is no operating system for the botnet to deal with [50, 51].
Some examples of the low interactive Honeypot are Honeyd, HoneyRJ [51],
BotMiner, BotGrep and BotTrack [52].

  * •

Medium Interaction Honeypot: Medium interactive Honeypot is better in terms of
the information that it can obtain from the mechanism. That is because it is
designed to have more security holes so that bots from botnet can access to
the system [51]. Hence, more information and complicated attacks from the
botnet can be gathered compare to low interactive Honeypot. Mwcollect,
Honeytrap, Nepenthes [48], HoneyBOT [51] and Kippo Honeypot Distro [53] are
some of the medium interaction honeypots that are used today.

  * •

High Interaction Honeypot: Unlike the other two, there is an operating system
in this type of Honeypot. Hence, the botnet can perform any activities in this
mechanism. Thus, more data can be gathered from the botnet activities through
high interactive Honeypot. However, it is the most risky one when it comes to
security as it provides such loosen access to the botnet and there is little
to no restrictions placed on what the bots can do once compromises the system.

The most significant weakness of this Honeypot is time consuming and difficult
to maintain [48, 53, 46, 51]. Specter and Argos are good examples of a high
interaction honeypot [53, 51]. Besides that, there are Minos, ManTrap and
Argos [48].

###  III-B Crawler

Generally, crawler is acting as an active mechanism in the botnet monitoring
tools. It starts by requesting NL from the bootstrap nodes by probing on those
nodes in active. Once the NL is received, it will continuously requesting for
more nodes from all the active nodes in the NL until all nodes are being
discovered or the action is being terminated [46]. The crawling method is
mostly developed using Breadth-First-Search (BFS) or Depth-First-Search (DFS)
method [54]. Throughout the process, it would be able to reveal the identity
of participating bots in the botnet. With all the information available, the
botnet size can now be estimated and enumerated. This is because crawler is
using graph traversal techniques to request nodes for NL connectivity. Hence,
this allows analyst to reconstruction of the connectivity graph and the
topology of the botnet [55, 56, 39]. The following subsection will be some
description on different crawlers designed by other.

  * •

P2P Graph Search Method: Graph search crawling method target is to reconstruct
the P2P botnet graph by visiting all the nodes and requesting their peer
lists. Then, further action is done through graph search to attack against the
P2P botnet [45]. It is initialized with a set of seed peers, which obtained
through reverse engineering bot samples or dynamic analysis. P2P botnet
topologies are so dynamic that it keeps on changing and affecting the output
results of the crawler. Furthermore, during crawling, it relies on the seed
peer list, some peers may not be able to be discovered via graph search method
because there might be churn [40] or peers behind Firewall and Network Address
Translation (NAT) devices.

  * •

Storm Crawler: The crawler design in [43] purpose is to understand more on the
peer’s connection in the botnet. The crawler design is based on BFS method to
locate all the involving peers. In order to avoid overlapping results from the
peer’s response, the system will send 16 route requests to every peer, and
categories them as different zones of the peer’s routing tree. For each of the
node that have response, the system will add to the peer list of its routing
tree. But the next node to crawl on will according to the sequence according
to First In First Out (FIFO) concept.

  * •

Nugache Crawler: In [57] paper, the author has designed the crawler as an
enumeration tool to estimate the botnet Nugache size while avoid getting
notice from the botnet operator. By using the DFS method, the crawler will
keep establishing with the next level nodes unless there is no more nodes in
the branch. DFS crawling method will be looking into deep down starting with
first node follow by the node in the next level, unlike BFS which go for the
next side node. DFS will always complete the routing connection of the first
node then only it will move on to the side node. The crawling algorithm is
based on Last-In-First-Out (LIFO) concept.

  * •

Less Invasive Crawling Algorithm(LICA) crawler [56]: LICA crawler design is
focusing on crawling efficiency and claim to be able to adapt to different
environment by using the parameters calculation. The crawler will crawl from
node in bootstrap list and limit the crawl number up to a parameter. The
crawling activities ended when all contactable nodes in the network have been
discovered or the limit set has reached. If the limit has not reached, LICA
crawler may repeat another iteration of crawling. However, it will use the
information from previously segregated nodes instead of the “seedpeer” list as
the crawling list. Overall, the concept of LICA crawler is to crawl over those
popular nodes and ignoring those less connected nodes as to cover the botnet
connection, as larger as possible for monitoring the botnet but not in detail.

###  III-C Sensor

Sensor node is acting passively after deployment in the botnet, waiting for
connection from other bots. If sensor node keeps presenting and responding to
other bots, it will be upgraded to a popular superpeer and will be able to mix
in to the bots prefered member list. The bots will start to share around about
this sensor node [58]. Then, it will be able to observe the messages flowing
around the botnets about the active nodes in the botnet. From these messages,
network security analyst will know which of the nodes that are involved in the
botnets [55, 46]. This is the main purpose of a sensor which is to be as
popular as possible so that to be able to increase their visibility as higher
as possible. Although sensors able to obtain information about the bots, they
cannot gather finegrained data or to obtain the interconnectivity of the bots
[55]. The following subsection will be describing some botnet monitoring
researches involving sensor node.

  * •

Monitoring Kelihos Botnet: Starting from the bootstrap nodes in Kelihos botnet
domains, they can track the growing population of hosting IPs and detect new
fast flux domains hosted by Kelihos botnet if the new domains match with the
monitored DNS authoritative traffic. This will allows the author to analyse on
various components and attributes of the infrastructure used by the Kelihos
fast flux botnet. During the monitoring, they also included filtering function
in order to avoid false positives result and confused by other non-botnet fast
flux domains [6].

  * •

Monitoring on Peer-to-Peer(P2P) Botnet: Based on Rossow et al. [45]
observation, P2P botnets peers are periodically contacted by the neighbouring
peers especially during regular peer list verification cycles. Besides that,
sensor can also be contacted by non-routable peers unlike crawler, which then
make sensor able to enumerate more nodes in a botnet. From the research
results, sensor nodes have revealed large number of bots which cannot be found
by crawlers. Another methodological difference is that, crawlers are actively
enumerating peers, while sensors are working passively that it waits to be
contacted by bots. And, sensor’s coverage relies on its popularity in the
botnet. So, the sensor node requires more time to leverage the outcome of the
monitoring result.

  * •

Passive P2P Monitoring in Storm Botnet: Passive P2P Monitoring (PPM) is a
botnet monitoring tool that designed by Kang et al. [44] that will act as peer
nodes, primarily in Storm botnet’s P2P network. The sensor node will only
listens in the Storm botnet and pretending itself as a legitimate bot and
routes messages. PPM might only have a small part of information about the
whole network, it knows only those nodes in its routing table and those nodes
that contacted it. This is because it only listens and will never contact with
other nodes by itself due to the passive nature. PPM can identify a node
location regardless it is behind a NAT or a Firewall. Then, it will further
distinguishes the node into different class. This is a function which sensor
node overtake crawler functionality [44].

##  IV Advantage and Disadvantage

Table I shows the comparison between different Honeypots according to their
advantages and disadvantages described in different researches. Nowadays
Honeypots has utilized faster networking and visualization technologies during
developing Honeypot. And, they have also deployed high interaction Honeypots
and low risk Honeynets, isolating attack traffic from connected hardware and
networks. It is good to see a lot of improvements that have been done in this
mechanism as to stop and prevent network security issues.

TABLE I: Comparison Between Different Honeypots Honeypot Types |  Advantages |  Disadvantages
---|---|---
Honeyd (Low Interactive) |  • Able to act like normal operating system. • Able to create and run with fake IP addresses simultaneously. |  • Aged and outdated NMAP fingerprint which will expose the identity of the honeypot. • The operating scripts are bound to different ports.
HoneyRJ (Low Interactive) |  • Extremely simple and easily extendable. • Multi-threaded to support multiple connections. |  • only supports string-based protocols and does not support the transmission of binary data.
Kippo (Medium Interactive) |  • Able to simulate other file systems and log all interactions with automated or non-automated attacks. • Utilized IoT device’s vulnerability as lure to hacker. |  • only provide partial implementation of services and do not allow full interaction with the system.
Nepenthes ( Medium Interactive ) |  • Having Scalability and flexibility features. • Able to create many honeypots in the system and capture data easily. |  • Many restrictions on the setting.
Honeywall ( High Interactive) |  • Hard to be hacked by hacker into Honeywall administration system. • Able to capture more useful and interesting findings. |  • Implementation will be time consuming and complicated. • Does not hide the network address.

Table II shows the comparison between different types of crawler about their
strength and weakness. Most of the crawler seems lacking of features to avoid
anti-monitoring mechanism and churn effects. But, it might be due to some of
the crawlers are designed long before the existence of those interferences.
Crawlers in future might need to include features that can tolerate with anti-
monitoring mechanism affects and have churn consideration like crawler
mentioned in [45]. These features are important for crawler to be efficient
because botnet nowadays is having those anti-monitoring mechanism and
performance of crawler without those features might get affected and the
monitoring results might be degraded as mentioned in [37, 39, 36].

TABLE II: Comparison Between Different Crawlers Crawler Types |  Advantages |  Disadvantages
---|---|---
P2P |  Covered for all P2P botnet evaluation and operated in real-time |  Not capable to handle anti-monitoring
Storm |  BFS crawler, easy to design and crawl fast |  Not capable to handle anti-monitoring mechanisms and churn effects
Nugache |  DFS crawler, easy to design |  Not capable to handle anti-monitoring mechanisms and churn effects
LICA |  Fast crawling algorithm different from BFS and DFS |  Easily missed out nodes during churn effects

Table III shows the comparison between different sensor in different botnet
monitoring. Through the experiment result in [44, 45], it shows that more than
40% of bots that contact the sensor are behind firewall or NAT devices. Since
crawler is unable to reach those nodes, it is wise to combine crawlers and
sensor nodes in a monitoring activities as to provide much more accurate
population estimation by using advantage of a sensor to complement the
weakness of crawler [45]. It will helps network analyst to obtain a better
picture of a botnet connection.

TABLE III: Comparison Between Different Sensors Sensor Types |  Advantages |  Disadvantages
---|---|---
Kelihos Botnet |  Able to monitor on fast flux spam domains and process on it. Able to eliminate the false positive result by itself. |  Not considering anti-monitoring mechanisms and churn effects during the monitoring activities
PPM in Storm Botnet |  Able to trace nodes behind NAT devices and Firewall |  Could not find nodes with short life-time
Monitoring P2P Botnet |  Able to trace nodes behind NAT devices and Firewall even in most of the P2P botnet |  Sensor’s coverage depends on its popularity

##  V Conclusion

Although it seems like the three monitoring tools are sufficient for botnet
monitoring purpose. However, there are some weakness in each of the monitoring
tools. For Honeypot, cost of maintaining a effective honeypot can be high
because the specialized skills are required to implement it. Or else, it might
causes vulnerability in the system configuration and lead to breach in the
system.

The P2P botnet crawling mechanism will only able to crawl on the nodes before
NAT devices and Firewall [40]. That is due to the mechanism of NAT devices and
Firewall will filtering out the unknown address that trying to pass through
them and block out the crawler. The disadvantage of sensor node would be its
passive behaviour. As it might take long time to have some results because it
need to become popular and always online to be able to contribute to the P2P
network [46]. Beside that, sensor is unable to determine if the node that
approach it is using spoofed or real IP address [44]. Hence, it is advisable
to combine crawler and sensor.

Furthermore, with the improvement of the botnet antimonitoring mechanism
mentioned in works done in [45, 58, 36], and it is worst when botmaster can
even backtrace the monitoring mechanisms existence in the botnet [39]. Then,
botmaster might even launch a retaliation attack on the monitoring system for
revenge [34]. There might be one day all these monitoring tools will be
blocked by the botnet. Nonetheless, there is forecast that the architecture of
advance botnet like Zeus and Sality are highly resilient to sinkholing attacks
[45]. Hence, there might be potential threats from P2P botnet where all the
existing methods might not be able to monitor or take down the botnet in
future, if all these tools and methods cannot be improved or alternate
mitigation methods cannot be discovered soon enough.

## References

  * [1] Nicolas Falliere.  Sality: Story of a peer-to-peer viral network.  Rapport technique, Symantec Corporation, 32, 2011.
  * [2] Symantec Enterprise.  Internet security threat report 2018.  Mountain View, CA, USA, 2018.
  * [3] James Wyke.  The zeroaccess botnet–mining and fraud for massive financial gain.  Sophos Technical Paper, 2, 2012.
  * [4] Nathan Goodman.  A survey of advances in botnet technologies.  arXiv preprint arXiv:1702.01132, 2017.
  * [5] Philipp M Lutscher, Nils B Weidmann, Margaret E Roberts, Mattijs Jonker, Alistair King, and Alberto Dainotti.  At home and abroad: The use of denial-of-service attacks during elections in nondemocratic regimes.  Journal of Conflict Resolution, 64(2-3):373–401, 2020.
  * [6] Renato Marinho and Raimir Holanda.  Exploring a p2p transient botnet-from discovery to enumeration.  The Journal on Cybercrime & Digital Investigations, 3(1):30–39, 2017.
  * [7] Mahmood A Al-Shareeda, Mohammed Anbar, Iznan Husainy Hasbullah, and Selvakumar Manickam.  Survey of authentication and privacy schemes in vehicular ad hoc networks.  IEEE Sensors Journal, 21(2):2422–2433, 2020.
  * [8] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, and Ali A Yassin.  Vppcs: Vanet-based privacy-preserving communication scheme.  IEEE Access, 8:150914–150928, 2020.
  * [9] Mahmood A Al-Shareeda, Mohammed Anbar, Iznan Husainy Hasbullah, Selvakumar Manickam, and Sabri M Hanshi.  Efficient conditional privacy preservation with mutual authentication in vehicular ad hoc networks.  IEEE Access, 8:144957–144968, 2020.
  * [10] Mahmoud Al Shareeda, Ayman Khalil, and Walid Fahs.  Realistic heterogeneous genetic-based rsu placement solution for v2i networks.  Int. Arab J. Inf. Technol., 16(3A):540–547, 2019.
  * [11] Mahmood A Al-shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  An efficient identity-based conditional privacy-preserving authentication scheme for secure communication in a vehicular ad hoc network.  Symmetry, 12(10):1687, 2020.
  * [12] Mahmood A Al-Shareeda, Mohammed Anbar, Murtadha A Alazzawi, Selvakumar Manickam, and Ahmed Shakir Al-Hiti.  Lswbvm: A lightweight security without using batch verification method scheme for a vehicle ad hoc network.  IEEE Access, 8:170507–170518, 2020.
  * [13] Mustafa Maad Hamdi, Lukman Audah, Sami Abduljabbar Rashid, and Mahmood Al Shareeda.  Techniques of early incident detection and traffic monitoring centre in vanets: A review.  J. Commun., 15(12):896–904, 2020.
  * [14] Mahmood A Al-shareeda, Mohammed Anbar, Iznan H Hasbullah, Selvakumar Manickam, Nibras Abdullah, and Mustafa Maad Hamdi.  Review of prevention schemes for replay attack in vehicular ad hoc networks (vanets).  In 2020 IEEE 3rd International Conference on Information Communication and Signal Processing (ICICSP), pages 394–398. IEEE, 2020.
  * [15] Mahmood A Al-shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  Review of prevention schemes for man-in-the-middle (mitm) attack in vehicular ad hoc networks.  International Journal of Engineering and Management Research, 10, 2020.
  * [16] Mahmoud Al Shareeda, Ayman Khalil, and Walid Fahs.  Towards the optimization of road side unit placement using genetic algorithm.  In 2018 International Arab Conference on Information Technology (ACIT), pages 1–5. IEEE, 2018.
  * [17] Mahmood A Al-shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  Review of prevention schemes for modification attack in vehicular ad hoc networks.  International Journal of Engineering and Management Research, 10, 2020.
  * [18] Mustafa Maad Hamdi, Ahmed Shamil Mustafa, Hussain Falih Mahd, Mohammed Salah Abood, Chanakya Kumar, and Mahmood A Al-shareeda.  Performance analysis of qos in manet based on ieee 802.11 b.  In 2020 IEEE international conference for innovation in technology (INOCON), pages 1–5. IEEE, 2020.
  * [19] Murtadha A Alazzawi, Hasanain AH Al-behadili, Mohsin N Srayyih Almalki, Aqeel Luaibi Challoob, and Mahmood A Al-shareeda.  Id-ppa: robust identity-based privacy-preserving authentication scheme for a vehicular ad-hoc network.  In International Conference on Advances in Cyber Security, pages 80–94. Springer, 2020.
  * [20] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, Ayman Khalil, and Iznan Husainy Hasbullah.  Security and privacy schemes in vehicular ad-hoc network with identity-based cryptography approach: A survey.  IEEE Access, 9:121522–121531, 2021.
  * [21] Mahmood A Al-shareeda, Mohammed Anbar, Selvakumar Manickam, Iznan H Hasbullah, Nibras Abdullah, Mustafa Maad Hamdi, and Ahmed Shakir Al-Hiti.  Ne-cppa: A new and efficient conditional privacy-preserving authentication scheme for vehicular ad hoc networks (vanets).  Appl. Math, 14(6):1–10, 2020.
  * [22] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  A secure pseudonym-based conditional privacy-preservation authentication scheme in vehicular ad hoc networks.  Sensors, 22(5):1696, 2022.
  * [23] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  Se-cppa: A secure and efficient conditional privacy-preserving authentication scheme in vehicular ad-hoc networks.  Sensors, 21(24):8206, 2021.
  * [24] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  Towards identity-based conditional privacy-preserving authentication scheme for vehicular ad hoc networks.  IEEE Access, 2021.
  * [25] MAASM Mahmood A Al-shareeda, Mohammed Anbar, Murtadha A Alazzawi, Selvakumar Manickam, and Iznan H Hasbullah.  Security schemes based conditional privacy-preserving in vehicular ad hoc networks.  Indonesian Journal of Electrical Engineering and Computer Science, 21(1), 2020.
  * [26] Mahmood A Al-Shareeda, Mohammed Anbar, Selvakumar Manickam, and Iznan H Hasbullah.  Password-guessing attack-aware authentication scheme based on chinese remainder theorem for 5g-enabled vehicular networks.  Applied Sciences, 12(3):1383, 2022.
  * [27] Mahmood A Al-shareeda, Murtadha A Alazzawi, Mohammed Anbar, Selvakumar Manickam, and Ahmed K Al-Ani.  A comprehensive survey on vehicular ad hoc networks (vanets).  In 2021 International Conference on Advanced Computer Applications (ACA), pages 156–160. IEEE, 2021.
  * [28] Mahmood A Al-shareeda, Mohammed Anbar, Selvakumar Manickam, Iznan H Hasbullah, Ayman Khalil, Murtadha A Alazzawi, and Ahmed Shakir Al-Hiti.  Proposed efficient conditional privacy-preserving authentication scheme for v2v and v2i communications based on elliptic curve cryptography in vehicular ad hoc networks.  In International Conference on Advances in Cyber Security, pages 588–603. Springer, 2020.
  * [29] Mahmood A Al-Shareeda, Selvakumar Manickam, Badiea Abdulkarem Mohammed, Zeyad Ghaleb Al-Mekhlafi, Amjad Qtaish, Abdullah J Alzahrani, Gharbi Alshammari, Amer A Sallam, and Khalil Almekhlafi.  Chebyshev polynomial-based scheme for resisting side-channel attacks in 5g-enabled vehicular networks.  Applied Sciences, 12(12):5939, 2022.
  * [30] Mahmood A Al-Shareeda, Selvakumar Manickam, Badiea Abdulkarem Mohammed, Zeyad Ghaleb Al-Mekhlafi, Amjad Qtaish, Abdullah J Alzahrani, Gharbi Alshammari, Amer A Sallam, and Khalil Almekhlafi.  Cm-cppa: Chaotic map-based conditional privacy-preserving authentication scheme in 5g-enabled vehicular networks.  Sensors, 22(13):5026, 2022.
  * [31] Mahmood A Al-Shareeda and Selvakumar Manickam.  Security methods in internet of vehicles.  arXiv preprint arXiv:2207.05269, 2022.
  * [32] Tatikayala Sai Gopal, Mallesh Meerolla, G Jyostna, P Reddy Lakshmi Eswari, and E Magesh.  Mitigating mirai malware spreading in iot environment.  In 2018 International Conference on Advances in Computing, Communications and Informatics (ICACCI), pages 2226–2230. IEEE, 2018.
  * [33] Daniel Stutzbach, Reza Rejaie, and Subhabrata Sen.  Characterizing unstructured overlay topologies in modern p2p file-sharing systems.  IEEE/ACM Transactions on Networking, 16(2):267–280, 2008.
  * [34] Dennis Andriesse, Christian Rossow, Brett Stone-Gross, Daniel Plohmann, and Herbert Bos.  Highly resilient peer-to-peer botnets are here: An analysis of gameover zeus.  In 2013 8th International Conference on Malicious and Unwanted Software:” The Americas”(MALWARE), pages 116–123. IEEE, 2013.
  * [35] Shankar Karuppayah, Stefanie Roos, Christian Rossow, Max Mühlhäuser, and Mathias Fischer.  Zeus milker: circumventing the p2p zeus neighbor list restriction mechanism.  In 2015 IEEE 35th International Conference on Distributed Computing Systems, pages 619–629. IEEE, 2015.
  * [36] Emmanouil Vasilomanolakis, Jan Helge Wolf, Leon Böck, Shankar Karuppayah, and Max Mühlhäuser.  I trust my zombies: A trust-enabled botnet.  arXiv preprint arXiv:1712.03713, 2017.
  * [37] Dennis Andriesse, Christian Rossow, and Herbert Bos.  Reliable recon in adversarial peer-to-peer botnets.  In Proceedings of the 2015 Internet Measurement Conference, pages 129–140, 2015.
  * [38] Leon Böck, Emmanouil Vasilomanolakis, Max Mühlhäuser, and Shankar Karuppayah.  Next generation p2p botnets: Monitoring under adverse conditions.  In International Symposium on Research in Attacks, Intrusions, and Defenses, pages 511–531. Springer, 2018.
  * [39] Shankar Karuppayah, Emmanouil Vasilomanolakis, Steffen Haas, Max Mühlhäuser, and Mathias Fischer.  Boobytrap: On autonomously detecting and characterizing crawlers in p2p botnets.  In 2016 IEEE International Conference on Communications (ICC), pages 1–7. IEEE, 2016.
  * [40] Daniel Plohmann, Elmar Gerhards-Padilla, and Felix Leder.  Botnets: Detection, measurement, disinfection & defence.  European Network and Information Security Agency (ENISA), 1(1):1–153, 2011.
  * [41] Ping Wang, Sherri Sparks, and Cliff C Zou.  An advanced hybrid peer-to-peer botnet.  IEEE Transactions on Dependable and Secure Computing, 7(2):113–127, 2008.
  * [42] Ramesh Singh Rawat, Emmanuel S Pilli, and Ramesh Chandra Joshi.  Survey of peer-to-peer botnets and detection frameworks.  Int. J. Netw. Secur., 20(3):547–557, 2018.
  * [43] Thorsten Holz, Moritz Steiner, Frederic Dahl, Ernst W Biersack, Felix C Freiling, et al.  Measurements and mitigation of peer-to-peer-based botnets: A case study on storm worm.  Leet, 8(1):1–9, 2008.
  * [44] Brent ByungHoon Kang, Eric Chan-Tin, Christopher P Lee, James Tyra, Hun Jeong Kang, Chris Nunnery, Zachariah Wadler, Greg Sinclair, Nicholas Hopper, David Dagon, et al.  Towards complete node enumeration in a peer-to-peer botnet.  In Proceedings of the 4th International Symposium on Information, Computer, and Communications Security, pages 23–34, 2009.
  * [45] Christian Rossow, Dennis Andriesse, Tillmann Werner, Brett Stone-Gross, Daniel Plohmann, Christian J Dietrich, and Herbert Bos.  Sok: P2pwned-modeling and evaluating the resilience of peer-to-peer botnets.  In 2013 IEEE symposium on security and privacy, pages 97–111. IEEE, 2013.
  * [46] Shankar Karuppayah.  Advanced Monitoring in P2P Botnets: A Dual Perspective.  Springer, 2018.
  * [47] Shankar Karuppayah.  Advanced monitoring in p2p botnets.  2016\.
  * [48] Deniz Akkaya and Fabien Thalgott.  Honeypots in Network Security: How to monitor and keep track of the newest cyber attacks by trapping hackers.  LAP Lambert Academic Publishing, 2012.
  * [49] Seungjin Lee, Azween Abdullah, and NZ Jhanjhi.  A review on honeypot-based botnet detection models for smart factory.  International Journal of Advanced Computer Science and Applications, 11(6), 2020.
  * [50] Seungjin Lee, Azween Abdullah, Nz Jhanjhi, and Sh Kok.  Classification of botnet attacks in iot smart factory using honeypot combined with machine learning.  PeerJ Computer Science, 7:e350, 2021.
  * [51] Eric Peter and Todd Schiller.  A practical guide to honeypots.  Washington Univerity, 2011.
  * [52] Fatih Haltaş, Erkam Uzun, Necati Şişeci, Abdulkadir Poşul, and Bakır Emre.  An automated bot detection system through honeypots for large-scale.  In 2014 6th International Conference On Cyber Conflict (CyCon 2014), pages 255–270. IEEE, 2014.
  * [53] Seamus Dowling, Michael Schukat, and Hugh Melvin.  A zigbee honeypot to assess iot cyberattack behaviour.  In 2017 28th Irish signals and systems conference (ISSC), pages 1–6. IEEE, 2017.
  * [54] Daniele G Spampinato, Upasana Sridhar, and Tze Meng Low.  Linear algebraic depth-first search.  In Proceedings of the 6th ACM SIGPLAN International Workshop on Libraries, Languages and Compilers for Array Programming, pages 93–104, 2019\.
  * [55] Steffen Haas, Shankar Karuppayah, Selvakumar Manickam, Max Mühlhäuser, and Mathias Fischer.  On the resilience of p2p-based botnet graphs.  In 2016 IEEE Conference on Communications and Network Security (CNS), pages 225–233. IEEE, 2016.
  * [56] Shankar Karuppayah, Mathias Fischer, Christian Rossow, and Max Mühlhäuser.  On advanced monitoring in resilient and unstructured p2p botnets.  In 2014 IEEE International Conference on Communications (ICC), pages 871–877. IEEE, 2014.
  * [57] David Dittrich and Sven Dietrich.  Discovery techniques for p2p botnets.  Stevens Institute of Technology CS Technical Report, 4(26):2, 2008\.
  * [58] Shankar Karuppayah, Leon Böck, Tim Grube, Selvakumar Manickam, Max Mühlhäuser, and Mathias Fischer.  Sensorbuster: On identifying sensor nodes in p2p botnets.  In Proceedings of the 12th International Conference on Availability, Reliability and Security, pages 1–6, 2017.


# Representability for small and big triangulated categories

George Ciprian Modoi  Babeş–Bolyai University, Faculty of Mathematics and
Computer Science
1, Mihail Kogălniceanu, 400084 Cluj–Napoca, Romania<EMAIL_ADDRESS>
###### Abstract.

In this note we give a variant for a Brown representability type theorem in
the spirit of Rouquier [9, Section 4.3] and Neeman [8, Section 4]. The main
advantage of our arguments are that they can be dualized.

## Introduction

Let $\mathcal{D}$ be a triangulated category. Despite the title, we are not
going to define precisely what small and big mean for such a category.
Informally the difference is that "small" categories have usually only a set
of objects, up to isomorphism (in the literature such categories are called
essentially small) whereas "big" categories have set indexed coproducts, and
consequently, they have a proper class of objects.

Brown representability theorems give conditions (necessary and sufficient) for
a functor defined on the triangulated category $\mathcal{D}$ to be
representabile. Initially, this was done by Brown itself for the homotopy
category of spectra, [2]; this involved contravariant functors which send
coproducts into products, and was generalized for other "big" triangulated
categories which can be constructed in some way from a small subcategory (see
[6, Chapter 8]). Analogous results for small categories seem to be more
intricate. To the authors best knowledge in the literature there are only
three instances [1, Theorem 1.3], [9, Theorem 4.20] and [8, Theorem 4.2] of
results of this kind. This note combines ideas form the last two, in order to
give a new proof, which is intended to be as general we can. Even if in the
formulation of the main Theorem 1.5, the noetherianness condition on the base
ring $R$ is not assumed, in the practice it seems to be quite difficult to
check the other hypotheses in the absence of it, see Corollary 1.6.
Fortunately enough the same Theorem 1.5 can be applied for big triangulated
categories which can be obtained in a finite number of steps using shifts and
triangles from the big additive closure of a set of objects, see Corollary
1.7. Moreover the arguments can be dualized without problems, obtaining a
variant of Brown representability for covariant functors.

##  1\. Results

Throughout this section $R$ is a commutative ring, $\mathcal{D}$ is an
additive $R$-linear category, and $H:\mathcal{D}\to\hbox{\rm Mod-}R$ is a
contravariant functor. Remark that the hypothesis on $\mathcal{D}$ imply that
it has finite biproducts. We call $\mathcal{D}$ karoubian, if every idempotent
endomorphism in $\mathcal{D}$ splits. In what follows, we look for some
necessary sand sufficient conditions for $H$ to be representable, that is to
be of the form $\mathcal{D}(-,X)$ for some $X\in\mathcal{D}$. The functor $H$
is called finitely generated if there is a natural epimorphism
$\mathcal{D}(-,D)\to H$, for some $D\in\mathcal{D}$. Further the functor $H$
is called finitely presented if there is an exact sequence

$\mathcal{D}(-,C)\to\mathcal{D}(-,D)\to H\to 0.$

It is routine to show that $H$ is finitely presented if and only if it is
finitely generated and for every $D\in\mathcal{D}$, the kernel of any natural
epimorphism $\mathcal{D}(-,D)\to H$ is finitely generated too.

For another contravariant functor $H^{\prime}:\mathcal{D}\to\hbox{\rm Mod-}R$,
denote by $\operatorname{Hom}_{\mathcal{D}}(H,H^{\prime})$ the class of all
natural transformations between $H$ and $H^{\prime}$. If $H$ is finitely
presented, it follows by Yoneda lemma that
$\operatorname{Hom}_{\mathcal{D}}(H,H^{\prime})$ is actually a set. Therefore
we can construct a genuine category whose objects are all finitely presented
contaravariant functors $\mathcal{D}\to\hbox{\rm Mod-}R$ and whose morphisms
are natural transformations. We denote by $\hbox{\rm mod}({\mathcal{D}})$ this
category.

Consider now a (full) subcategory $\mathcal{A}\subseteq\mathcal{D}$. We say
that $H$ is $\mathcal{A}$-finitely generated if there is $A\in\mathcal{A}$ and
a natural transformation $\alpha:\mathcal{D}(-,A)\to H$ such that

$\alpha^{X}:\mathcal{D}(X,A)\to H(X)\to 0$

is exact for all $X\in\mathcal{A}$. Remark that $H$ is finitely generated
exactly if it is $\mathcal{D}$-finitely generated and the two definitions are
consistent, in the sense that $H$ is $\mathcal{A}$-finitely generated if and
only if its restriction $H|_{\mathcal{A}}$ is finitely generated. Indeed by
Yoneda lemma both natural transformations $\mathcal{D}(-,A)\to H$,
respectively $\mathcal{A}(-,A)\to H|_{\mathcal{A}}$ are in correspondence with
an element in $H(A)$, therefore given one we can construct the other one.

In the rest of the section we suppose in addition that $\mathcal{D}$ has weak
kernels, that is given $Y\to Z$ in $\mathcal{D}$, there is $X\to Y$ such that
the sequence of functors

$\mathcal{D}(-,X)\to\mathcal{D}(-,Y)\to\mathcal{D}(-,Z)$

is exact. In this case, it is clear than that the composite morophism $X\to
Y\to Z$ vanishes, and we call it a weak kernel sequence. By [4, Theorem 1.4]
we know that the existence of weak kernels is equivalent to the fact that
$\hbox{\rm mod}({\mathcal{D}})$ is abelian (see also [3, Lemma 2.2]). Moreover
projective objects in $\hbox{\rm mod}({\mathcal{D}})$ are direct summands of
representable functors, see [3, Proposition 2.3]. If $\mathcal{A}$ is an
abelian category, and $F:\mathcal{D}\to\mathcal{A}$ is a covariant functor,
then there is a unique, up to a natural isomorphism, right exact (covariant)
functor $\widehat{F}:\hbox{\rm mod}({\mathcal{D}})\to\mathcal{A}$ such that
$\widehat{F}\mathcal{D}(-,X)=F(X)$ for all $X\in\mathcal{D}$, by [3, Universal
Property 2.1]. By a combination of [3, Lemma 2.5 and Proposition 2.3], we
infer that the functor $\widehat{F}$ is exact if and only if $F$ is weak
exact, that is it sends every weak kernel sequence $X\to Y\to Z$ to an exact
sequence $F(X)\to F(Y)\to F(Z)$ in $\mathcal{A}$. Remark that it is enough to
check the weak exactness for any particular choice of a weak kernel for $Y\to
Z$. Indeed if $X\to Y\to Z$ and $X^{\prime}\to Y\to Z$ are two weak kernel
sequences, then the maps $X\to Y$ and $X^{\prime}\to Y$ factor through each
other, therefore the images of the maps $F(X)\to F(Y)$ and $F(X^{\prime})\to
F(Y)$ coincide. Note that if $\mathcal{D}$ is triangulated, then the
completion of a map $Y\to Z$ to a triangle $X\to Y\to Z\rightsquigarrow$
provides a weak kernel, therefore $F:\mathcal{D}\to\mathcal{A}$ is weak exact
if and only if it is homological. Returning to the general case of a category
with weak kernels, if $F:\mathcal{D}\to\mathcal{A}$ is a contravariant
functor, we can apply the above statements for it seen as a covariant functor
$\mathcal{D}\to\mathcal{A}^{\mathrm{op}}$ and we get a unique contravariant
functor $\widehat{F}:\hbox{\rm mod}({\mathcal{D}})\to\mathcal{A}$ such that
$\widehat{F}\mathcal{D}(-,X)=F(X)$ for all $X\in\mathcal{D}$ and $\widehat{F}$
sends cokernels to kernels. Clearly the weak exactness for a contravariant
functor $F:\mathcal{D}\to\mathcal{A}$ means the exactness of $F(Z)\to F(Y)\to
F(X)$ for every weak kernel sequence $X\to Y\to Z$. Specializing in the
discussion above the the codomain category $\mathcal{A}=\hbox{\rm Mod-}R$, we
obtain a generalization of [5, Lemma 1]:

######  Proposition 1.1.

Let $H:\mathcal{D}\to\hbox{\rm Mod-}R$ a contravariant functor. Then there is
a natural isomorphism $\widehat{H}\cong\operatorname{Hom}_{\mathcal{D}}(-,H)$.
In particular, $H$ is finitely presented if and only if $\widehat{H}$ is
representable.

###### Proof.

The isomorphism follows by the facts that $\operatorname{Hom}_{D}(-,H)$ sends
cokernels to kernels, and
$\operatorname{Hom}_{\mathcal{D}}(\mathcal{D}(-,X),H)\cong H(X)$, according to
Yoneda lemma. The last statement is now obvious, $\widehat{H}$ being
represented by $H$; note that the notation
$\operatorname{Hom}_{\mathcal{D}}(-,H)$ make sense even for $H\notin\hbox{\rm
mod}({\mathcal{D}})$. ∎

An abelian category is said to be Frobenius, provided that projective and
injective objects coincide.

######  Corollary 1.2.

Suppose $\hbox{\rm mod}({\mathcal{D}})$ is Frobenius, and let the
contravariant functor $H:\mathcal{D}\to\hbox{\rm Mod-}R$ be weak exact. Then
$H$ is isomorphic to direct summand of a representable functor if and only if
$H$ is finitely presented. If, in addition, $\mathcal{D}$ is karoubian, then
$H$ is representable if and only if $H$ is finitely presented.

###### Proof.

A representable functor is obviously finitely presented, therefore so is any
direct summand of it. Conversely if $H$ is finitely presented, that is
$H\in\hbox{\rm mod}({\mathcal{D}})$ it follows by Proposition 1.1 that
$\widehat{H}\cong\operatorname{Hom}_{\mathcal{D}}(-,H)$ is representable. But
$H$ is weak exact, therefore $\widehat{H}$ has to be exact. This means $H$ is
an injective object in $\hbox{\rm mod}({\mathcal{D}})$. By hypothesis
injective objects in $\hbox{\rm mod}({\mathcal{D}})$ coincide with projective
objects and they are direct summand of representable functors. ∎

Let specialize us to the case of triangulated categories. If $\mathcal{D}$ is
so, we know by [6, Corollary 5.1.23], that $\hbox{\rm mod}({\mathcal{D}})$ is
Frobenius, therefore the Corollary 1.2 applies for cohomological functors
$H:\mathcal{D}\to\hbox{\rm Mod-}R$. In this way we get a new proof for [7,
Proposition 1.4]. In order to formulate it, fix a subcategory $\mathcal{S}$ of
$\mathcal{D}$ and recall that an $\mathcal{S}$-precover of $D\in\mathcal{D}$
is a morphism $S\to D$ such that the induced map
$\mathcal{D}(X,S)\to\mathcal{D}(X,D)$ is surjective for all $X\in\mathcal{S}$.
Further $\mathcal{S}$ is called precovering, provided that every
$D\in\mathcal{D}$ has an $\mathcal{S}$-precover.

######  Corollary 1.3.

Let $\mathcal{D}$ be a karoubian triangulated category and let
$\mathcal{S}\subseteq\mathcal{D}$ be a traingulated subcategory which is
closed under direct summands. If $\mathcal{S}$ is precovering, then
$\mathcal{S}$ is coreflective.

###### Proof.

Let $D\in\mathcal{D}$. Consider an $\mathcal{S}$-precover $S\to D$ and
complete it to a triangle $C\to S\to D\rightsquigarrow$. Consider further an
$\mathcal{S}$-precover $T\to C$. Together these imply that the sequence

$\mathcal{S}(X,T)\to\mathcal{S}(X,S)\to\mathcal{D}(X,D)\to 0$

is exact for all $X\in\mathcal{S}$. This means that the cohomological functor
$\mathcal{D}(-,D)|_{\mathcal{S}}:\mathcal{S}\to\hbox{\rm Mod-}R$ is finitely
presented. By hypothesis $\mathcal{S}$ is karoubian, therefore
$\mathcal{D}(-,D)|_{\mathcal{S}}$ is representable by Corollary 1.2. To find
an object $S_{D}$ representing it is equivalent to define a right adjoint
$D\mapsto S_{D}$ to the inclusion functor $\mathcal{S}\to\mathcal{D}$. ∎

From now on $\mathcal{D}$ is triangulated. For two subcategories
$\mathcal{A},\mathcal{B}\subseteq\mathcal{D}$ we define
$\mathrm{smd}\,\mathcal{A}$ to be the closure of $\mathcal{A}$ under direct
summands and

$\mathcal{A}*\mathcal{B}=\\{Y\in\mathcal{D}\mid\hbox{ there is a triangle
}X\to Y\to Z\rightsquigarrow\hbox{ with }X\in\mathcal{A},Z\in B\\}.$

For $n\geq 1$ we define inductively
$\mathcal{A}^{1}=\mathrm{smd}\,\mathcal{A}$ and
$\mathcal{A}^{*(n+1)}=\mathrm{smd}\,(\mathcal{A}^{*n}*\mathcal{A}^{1})$. Since
$0\in\mathcal{A}^{*1}$ we deduce
$\mathcal{A}^{*1}\subseteq\mathcal{A}^{*2}\subseteq\ldots$.

######  Lemma 1.4.

Let $H:\mathcal{D}\to\hbox{\rm Mod-}R$ be a cohomological functor, and let
$\mathcal{A}\subseteq\mathcal{D}$ be a (full) subcategory such that
$\mathcal{A}[1]=\mathcal{A}$. Suppose that $H$ is $\mathcal{A}$-finitely
generated and, for any $D\in\mathcal{D}$, the kernel of every natural
transformation $\mathcal{D}(-,D)\to H$ is $\mathcal{A}$-finitely generated
too. Then $H$ is $\mathcal{A}^{*n}$-finitely generated for all $n\geq 1$.

###### Proof.

We proceed by induction on $n$. For $n=1$, $H$ is $\mathcal{A}$-finitely
generated by hypothesis, so it is $\mathcal{A}^{*1}$-finitely generated too.
Suppose that $H$ is $A^{*n}$-finitely genarated for some $n\geq 1$. Then there
is $A_{n}\in\mathcal{A}^{*n}$ and $\alpha_{n}:\mathcal{D}(-,A_{n})\to H$ such
that $\alpha_{n}^{X}$ is surjective for all $X\in\mathcal{A}^{*n}$. Denote by
$K_{n}$ the kernel of $\alpha_{n}$. By hypothesis $K_{n}$ is
$\mathcal{A}$-finitely generated, so there is $B_{n}\in\mathcal{A}$ and a
natural transformation $\mathcal{D}(-,B_{n})\to K_{n}$ such that
$\mathcal{D}(X,B_{n})\to K_{n}(X)\to 0$ is exact for all
$X\in\mathcal{A}^{*1}$. By Yoneda lemma the composite $\mathcal{D}(-,B_{n})\to
K_{n}\to\mathcal{D}(-,A_{n})$ is of the form $b_{n}^{*}=\mathcal{D}(-,b_{n})$
for some (unique) $b_{n}:B_{n}\to A_{n}$ in $\mathcal{D}$. Complete it to a
triangle

$B_{n}\stackrel{{\scriptstyle
b_{n}}}{{\longrightarrow}}A_{n}\stackrel{{\scriptstyle
a_{n}}}{{\longrightarrow}}A_{n+1}\stackrel{{\scriptstyle
c_{n}}}{{\longrightarrow}}B_{n}[1].$

By construction $\alpha_{n}b_{n}^{*}=0$, which means that $H(b_{n})$ maps to
$0$ the element in $H(A_{n})$ corresponding under Yoneda isomorphism to
$\alpha_{n}$. But the sequence

$H(A_{n+1})\stackrel{{\scriptstyle
H(a_{n})}}{{\longrightarrow}}H(A_{n})\stackrel{{\scriptstyle
H(b_{n})}}{{\longrightarrow}}H(B_{n})$

is exact, since $H$ is cohomological, therefore this element in $H(A_{n})$
lies in the image of $H(a_{n})$. Applying the Yoneda lemma again, this means
that there is a natural transformation $\alpha_{n+1}:\mathcal{D}(-,A_{n+1})\to
H$ such that $\alpha_{n+1}a_{n}^{*}=\alpha_{n}$. It is clear that
$A_{n+1}\in\mathcal{A}^{*(n+1)}$ since
$B_{n}[1]\in\mathcal{A}[1]=\mathcal{A}$. In order to finish our proof we only
have to show the claim that $\alpha_{n+1}^{Y}:\mathcal{D}(Y,A_{n+1})\to H(Y)$
is surjective for all $Y\in\mathcal{A}^{*(n+1)}$. Since this property is
obviously inherited by direct summands it is enough to prove it for those
objects $Y$ for which there is a triangle

$Z[-1]\stackrel{{\scriptstyle f}}{{\longrightarrow}}X\stackrel{{\scriptstyle
g}}{{\longrightarrow}}Y\stackrel{{\scriptstyle h}}{{\longrightarrow}}Z$

with $X\in\mathcal{A}^{*n}$ and $Z\in\mathcal{A}^{*1}$.

Let $y\in H(Y)$, and put $x=H(g)(y)\in H(X)$. Since $X\in\mathcal{A}^{*n}$,
the map $\alpha_{n}^{X}$ is surjective, so there is $u:X\to A_{n}$ such that
$\alpha_{n}^{X}(u)=x$. Further $H(f)\alpha_{n}^{X}(u)=H(f)(x)=H(f)H(g)(y)=0$,
and by the naturality of $\alpha_{n}$ this is equivalent to
$\alpha_{n}^{Z[-1]}f_{A_{n}}(u)=0$, as we can see from the commutative
diagram:

$\textstyle{\mathcal{D}(Z[-1],A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n}^{Z[-1]}}$$\textstyle{\mathcal{D}(X,A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n}^{X}}$$\scriptstyle{f_{A_{n}}}$$\textstyle{H(Z[-1])}$$\textstyle{H(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H(f)}$

where $f_{*}=\mathcal{D}(f,-)$. Thus $uf=f_{A_{n}}(u)$ belongs to the kernel
$K_{n}(Z[-1])$ of $\alpha_{n}^{Z[-1]}$. But
$Z[-1]\in\mathcal{A}^{*1}[-1]=\mathcal{A}^{*1}$, therefore the sequence
$\mathcal{D}(Z[-1],B_{n})\to K_{n}(Z[-1])\to 0$ is exact, what means that
there is $v:Z[-1]\to B_{n}$ such that $uf=b^{Z[-1]}(v)=b_{n}v$. We get a
morphism $w:Y\to A_{n+1}$ completing commutative to a morphismn of triangles
the diagram:

$\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.82637pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.82639pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.53822pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 39.72232pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{v}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.16669pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.72232pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.79175pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
64.68163pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 84.68225pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.25703pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.25703pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{u}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
47.25703pt\raise-30.16669pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
84.68225pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
98.71004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
111.68472pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 134.9356pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
91.69614pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
91.69614pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{w}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
91.69614pt\raise-30.16669pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
134.9356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
141.70642pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
141.70642pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{v[1]}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 141.70642pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.72406pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.92596pt\raise-45.8611pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0pt\hbox{$\scriptstyle{b_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 38.82637pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.82637pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
55.6877pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
63.42606pt\raise-44.9375pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{a_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 79.6877pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.6877pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
103.70459pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
110.98618pt\raise-44.9375pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{c_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 127.70457pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
127.70457pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{B_{n}[1]}$}}}}}}}\ignorespaces\ignorespaces}}}}}.$

Putting together all equalities before we get:

$H(g)(y)=x=\alpha_{n}^{X}(u)=\alpha_{n+1}^{X}a_{n}^{X}(u)=\alpha_{n+1}^{X}g_{A_{n+1}}(w).$

Using the naturality of $\alpha_{n+1}$ we continue:

$H(g)(y)=H(g)\alpha_{n+1}^{Y}(w),$

therefore $H(g)(y-\alpha_{n+1}^{Y}(w))=0$. Further we know that the sequence

$H(Z)\stackrel{{\scriptstyle
H(h)}}{{\longrightarrow}}H(Y)\stackrel{{\scriptstyle
H(g)}}{{\longrightarrow}}H(X)$

is exact, so there is $z\in H(Z)$ such that $H(h)(z)=y-\alpha_{n+1}^{Y}(w)$.
But $Z\in\mathcal{A}\subseteq\mathcal{A}^{*n}$, therefore
$\alpha_{n}^{Z}:\mathcal{D}(Z,A_{n})\to H(Z)$ is surjective, therefore we get
a map $s:Z\to A_{n}$ such that $\alpha_{n}^{Z}(s)=z$. Using the naturality of
$\alpha_{n+1}$ again we get

$H(h)(z)=H(h)\alpha_{n}^{Z}(s)=H(h)\alpha_{n+1}^{Z}a_{n}^{Z}(s)=\alpha_{n+1}^{Y}h_{A_{n+1}}a_{n}^{Z}(s)=\alpha_{n+1}^{Y}(a_{n}sh).$

Finally we get

$y=\alpha_{n+1}^{Y}(a_{n}sh+w),$

proving our claim. ∎

######  Theorem 1.5.

Let $H:\mathcal{D}\to\hbox{\rm Mod-}R$ be a cohomological functor, and let
$\mathcal{A}\subseteq\mathcal{D}$ be a (full) subcategory such that
$\mathcal{A}[1]=\mathcal{A}$, $\mathcal{A}$ is precovering and
$\mathcal{A}^{*n}=\mathcal{D}$ for some $n\geq 1$. Suppose that $H$ is
$\mathcal{A}$-finitely generated and, for any $D\in\mathcal{D}$, the kernel of
every natural transformation $\mathcal{D}(-,D)\to H$ is $\mathcal{A}$-finitely
generated too. Then $H$ is isomorphic to a direct summand of a representable
functor. In particular, if $\mathcal{D}$ is karoubian, then $H$ is
representable.

###### Proof.

Let $n\geq 1$ for which $\mathcal{A}^{*n}=\mathcal{D}$. We know by Lemma 1.4
that $H$ is finitely genarated, that is there is $D\in\mathcal{D}$ and an
epimorphism $\alpha:\mathcal{D}(-,D)\to H$. The kernel $H^{\prime}$ of
$\alpha$ is cohomological and $\mathcal{A}$-finitely generated. Consider a
natural transformation $\beta:\mathcal{D}(-,C)\to H^{\prime}$ with the kernel
$K$. The composite $\mathcal{D}(-,C)\to H^{\prime}\to\mathcal{D}(-,D)$ is of
the form $f^{*}=\mathcal{D}(-,f)$, for some $f:C\to D$, by Yoneda. Complete
$f$ to a triangle $B\stackrel{{\scriptstyle
g}}{{\longrightarrow}}C\stackrel{{\scriptstyle
f}}{{\longrightarrow}}D\rightsquigarrow$. It follows that $g^{*}$ factors
trough $K$ and the sequence

$\mathcal{D}(X,B)\to\mathcal{D}(X,C)\to H^{\prime}(X)$

is exact for all $X\in\mathcal{A}$. Therefore the induced map
$\mathcal{D}(X,B)\to K(X)$ is surjective for all $X\in\mathcal{A}$. Choosing
an $\mathcal{A}$-precover, $A\to B$ we can see that $K$ is
$\mathcal{A}$-finitely generated as well. This allows us to apply Lemma again
1.4 this time for $H^{\prime}$, implying that $H^{\prime}$ is finitely
generated. Together this means $H$ is finitely presented, and the conclusion
follows by Corollary 1.2. ∎

Consider a set of objects $\mathcal{G}\in\mathcal{A}$. We denote
$\mathcal{G}[\mathbb{Z}]=\\{G[n]\mid G\in\mathcal{G},n\in\mathbb{Z}\\}$. A
case of a particular importance is when $\mathcal{G}$ is a singleton, that is
$\mathcal{G}=\\{G\\}$, with $G\in\mathcal{D}$, when we write simply
$G[\mathbb{Z}]$ instead of $\\{G\\}[\mathbb{Z}]$. The object $G$ is called
strong generator for $\mathcal{D}$ if $\mathcal{D}=G[\mathbb{Z}]^{*n}$ for
some $n\geq 1$. A functor $H\to\hbox{\rm Mod-}R$ is called $G$-finite if the
$R$-module $\bigoplus_{n\in\mathbb{Z}}H(G[n])$ is finitely generated (in
particular, this implies $H(G[n])=0$ for $|n|\gg 0$). The following gives a
new proof for [8, Theorem 4.2] or [9, Corollary 4.18]:

######  Corollary 1.6.

Assume $R$ is noetherian and let $G$ be a strong generator for $\mathcal{D}$,
such that the functor $\mathcal{D}(-,D)$ is $G$-finite, for every
$D\in\mathcal{D}$. Then a contravariant functor $H:\mathcal{D}\to\hbox{\rm
Mod-}R$ is isomorphic to a direct summand of a representable functor if and
only if it is cohomological and $G$-finite. If further $\mathcal{D}$ is
karoubian, then $H$ is representable if and only if it is cohomological and
$G$-finite.

###### Proof.

It is clear that $\mathcal{A}=\mathrm{add}G[\mathbb{Z}]$ is stable under
shifts and, by hypothesis, $\mathcal{A}^{*n}=\mathcal{D}$ for some $n\geq 1$.
The argument used in [8, Lemma 4.1] for showing that $H$ is
$\mathcal{A}$-finitely generated (the base case of the induction) goes as
follows: Let $m\geq 0$ such that $H(G[n])=0$, for $|n|>m$. Let $\\{a_{n,i}\mid
1\leq i\leq k_{n}\\}$ a (finite) set of generators for $H(G[n])$ for $|n|\leq
m$. By Yoneda we get a set of natural transformations
$\alpha_{n,i}:\mathcal{D}(-,G[n])\to H$, with $1\leq i\leq k_{n}$. Letting
$A=\bigoplus_{|n|\leq m}\bigoplus_{i=1}^{k_{n}}G[n]$, we get a natural
transformation $\alpha:\mathcal{D}(-,A)\to H$ which is an epimorphism when it
is restricted to $\mathcal{A}$. The same argument applied for the $G$-finite
cohomological functor $\mathcal{D}(-,D)$, with $D\in\mathcal{D}$ shows that
$\mathcal{A}$ is precovering. If $K$ is the kernel of a natural transformation
$\mathcal{D}(-,D)\to H$, then the same argument works again in order to show
that $K$ is $\mathcal{A}$-finitely generated. Therefore Theorem 1.5 gives the
conclusion. ∎

The following Corollary is a variant of [9, Theorem 4.20]. The main advantage
of our proof is that it can be directly dualized in order to find a criterion
for Brown representability for the dual. As in [9, Theorem 4.22], classical
Brown representability theorem follows from our result.

######  Corollary 1.7.

Assume $\mathcal{D}$ has coproducts and $H:\mathcal{D}\to\hbox{\rm Mod-}R$ is
a contravariant functor. Suppose that there is a set $\mathcal{G}$ of objects
in $\mathcal{D}$ such that $\mathcal{D}=\mathcal{A}^{*n}$, where
$\mathcal{A}=\mathrm{Add}\mathcal{G}[\mathbb{Z}]$. Then $H$ is representable
if and only if $H$ is cohomological and sends coproducts into products.

###### Proof.

Representable functors are cohomological and sends coproducts into products.
Conversely, $\mathcal{A}$ is precovering, and taking
$A=\bigoplus_{G\in\mathcal{G}[\mathbb{Z}]}\left(\bigoplus_{x\in
H(G)}G_{x}\right)$, where $G_{x}\cong G$, for all $x\in H(G)$, we get an
epimorphism $\mathcal{D}(-,A)\to H$, therefore $H$ is $\mathcal{A}$-finitely
generated. Moreover if $K$ is the kernel of a natural transformation
$\mathcal{D}(-,D)\to H$, then $K$ is cohomological and sends coproducts into
products. By the same argument $K$ is $\mathcal{A}$-finitely generated too,
hence Theorem 1.5 applies. ∎

## References

  * [1] A. Bondal, M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Math. J. 3 (2003), 1–-36.
  * [2] E. H. Brown, Cohomology theories, Annals of Math. 75 (1962), 467–484.
  * [3] H. Krause, Functors on locally finitely presented categories, Colloq. Math. 75 (1998), 105–132.
  * [4] P. Freyd, Representations in abelian categories, in: Proc. Conf. on Categorical Algebra, La Jolla, 1965, Springer, 1966, 95–120.
  * [5] G. C. Modoi, The dual of Brown representability for homotopy categories of complexes, J. Alg. 392 (2013) 115–124.
  * [6] A. Neeman, Triangulated Categories, Ann. Math. Studies 148, Princeton University Press, 2001.
  * [7] A. Neeman, Some adjoints in homotopy categories, Annals of Mathematics, 171 (2010), 2143–2155.
  * [8] A. Neeman, Triangulated categories with a single compact generator and a Brown representability theorem, preprint arXiv:1804.02240[math.CT].
  * [9] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193–256.


# Ultra-broadband quadrature squeezing with thin-film lithium niobate
nanophotonics

Pao-Kang Chen  J. C. Wyant College of Optical Sciences, University of Arizona,
1630 E. University Boulevard, Tucson, Arizona 85721, USA  Department of
Physics, University of Arizona, 1118 E. Fourth Street, Tucson, Arizona 85721,
USA  Ian Briggs  J. C. Wyant College of Optical Sciences, University of
Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA  Songyan Hou
J. C. Wyant College of Optical Sciences, University of Arizona, 1630 E.
University Boulevard, Tucson, Arizona 85721, USA  Linran Fan  J. C. Wyant
College of Optical Sciences, University of Arizona, 1630 E. University
Boulevard, Tucson, Arizona 85721, USA  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Squeezed light is a key quantum resource that enables quantum advantages for
sensing, networking, and computing applications. The scalable generation and
manipulation of squeezed light with integrated platforms are highly desired
for the development of quantum technology with continuous variables. In this
letter, we demonstrate squeezed light generation with thin-film lithium
niobate integrated photonics. Parametric down-conversion is realized with
quasi-phase matching using ferroelectric domain engineering. With sub-
wavelength mode confinement, efficient nonlinear processes can be observed
with single-pass configuration. We measure $0.56\pm 0.09$ dB quadrature
squeezing ($\sim$3 dB inferred on-chip). The single-pass configuration further
enables the generation of squeezed light with large spectral bandwidth up to 7
THz. This work represents a significant step towards the on-chip
implementation of continuous-variable quantum information processing.

††journal: optica

Squeezed light is a quintessential quantum resource with no classical
counterpart. With the recent development of quantum information, squeezed
light has become one fundamental building block for photonic quantum
technology [1, 2]. The suppression of measurement variance below shot noise
has been widely recognized and implemented for sensing applications including
gravitational wave detection [3], microscopy [4], and bio-sensing [5].
Furthermore, squeezed light also plays a critical role in quantum
communications, with demonstrations of unconditional quantum teleportation [6]
and quantum key distribution [7]. Continuous-variable quantum computing has
also been developed based on squeezed light [8, 9]. It provides an important
alternative with unique features to the discrete-variable approach using
single photons. Quantum information is encoded into the continuous quadrature
amplitudes instead of discrete degrees of freedom (such as polarization). This
renders the deterministic generation and entanglement operation of quantum
states [10, 11]. Deterministic, universal, and fault-tolerant quantum
computation can be realized [8, 9].

The further development of continuous-variable quantum technology requires a
large number of squeezed light sources and complex photonic circuits [10, 11,
12]. Integrated photonics provides the scalable fabrication of quantum
sources, with additional advantages including phase stability, near-perfect
spatial mode matching, and power efficiency [13]. Currently, squeezed light
has been generated using four-wave mixing in integrated photonic ring cavities
[14, 15, 16, 17, 18]. The small wavelength separation between pump and
squeezed light makes it challenging to remove the pump without introducing
extra loss on squeezed light. The pump efficiency is low compared with
parametric down-conversion, which is more widely used for squeezed light
generation in free-space experiments [19]. These platforms also lack electro-
optic effect. Therefore, thermo-optic or electro-optomechanical effect are
used for phase shifting, which limit the speed of photonic circuit
reconfiguration [13]. This prevents the implementation of critical functions
such as feedforward measurement, which is essential for measurement-based one-
way quantum computing [20, 21]. Squeezed light has also been generated in bulk
lithium niobate waveguide [22, 23, 24]. Parametric down-conversion and fast
reconfiguration can be achieved. However, the low index contrast and large
cross-sections prevent the scalable integration of large-scale photonic
circuits.

Figure 1: Design and fabrication of squeezed light sources on thin-film
lithium niobate. (a) Scanning electron microscope image of the fabricated
device cross-section. (b) Scanning electron microscope top-view image of the
mock-up device etched in hydrofluoric acid. (c) Simulated mode profile and
electric field of the squeezed light mode at 1550 nm. (d) Simulated mode
profile and electric field of the pump mode at 775 nm.

In this letter, we report the development of thin-film lithium niobate
integrated photonics for squeezed light generation. The strong second-order
nonlinearity and sub-wavelength confinement of optical fields facilitate
efficient parametric down-conversion processes in a single-pass configuration.
This eliminates the need of photonic cavities for pump recycling. Therefore,
squeezed light can escape the device with near-unity efficiency, in contrast
to the cavity configuration where the challenging highly over-coupled
condition is required [19]. Quasi-phase matching is realized through periodic
poling of thin-film lithium niobate. We have measured $0.56\pm 0.09$ dB
continuous-wave quadrature squeezing, with inferred $\sim$3 dB on-chip
squeezing. Our device further achieves squeezing over 7 THz bandwidth,
covering S-, C-, and L-bands for optical communications.

Figure 2: Second harmonic generation with periodically poled thin-film lithium
niobate waveguide. (a) Measured SHG efficiency versus input wavelength at room
temperature with input power 2 mW. Inset: Measured SHG efficiency versus
device temperature with fixed input wavelength 1554.2 nm. (b) Measured SHG
efficiency versus input power. Experimental data (black dot) are plotted with
a linear fit of the low power region (blue dashed line). The inset shows the
SHG signal at low input power.

The device is fabricated from a z-cut thin-film lithium niobate wafer with 600
nm device layer (Fig. 1(a)). The top width of the ridge waveguide is 1.8 µm,
defined by electron-beam lithography with hydrogen silsesquioxane resist. The
pattern is transferred to the lithium niobate device layer using argon-based
plasma processing with 400 nm etching depth. We use the fundamental
transverse-magnetic (TM00) modes for both the squeezed light around wavelength
1550 nm (Fig. 1(c)) and the pump around wavelength 775 nm (Fig. 1(d)) to
utilize the largest second-order nonlinear component $d_{33}=-25$ pm/V in
lithium niobate. In order to compensate for the wavevector mismatch between
the squeezed light $k_{s}$ and the pump $k_{p}$, an additional wavevector is
introduced by periodically inverting the crystal polarization of the thin-film
lithium niobate. The quasi-phase matching condition is fulfilled when the
domain period $\Lambda=2\pi/(2k_{s}-k_{p})$ [25]. In order to realize the
periodic domain inversion, nickel electrodes are patterned on top of the ridge
waveguide with period $\Lambda\approx 3$ µm. Multiple high-voltage pulses are
applied to nickel electrodes at an elevated temperature. Then nickel
electrodes are removed by hydrochloric acid. Figure 1(b) shows the mock-up
device after etching in hydrofluoric acid [26]. A duty cycle close to 50% can
be clearly observed. The total length of the device with domain inversion is 5
mm. Finally, the chip is cleaved to expose the waveguide facets for input and
output coupling.

Figure 3: Squeezed light measurement. (a) Experimental setup. FPC: fiber
polarization controller; PM: optical phase modulator; EDFA: erbium-doped fiber
amplifier; BS: 50/50 beamsplitter; MCC: mode cleaning cavity; PD:
photodetector; ESA: electrical spectrum analyzer; PZT: piezoelectric
transducer. (b) Normalized noise power as a function of time with phase
sweeping. Data obtained with zero span mode at frequency 22 MHz. The
resolution bandwidth is 300 KHz. The video bandwidth is 100 Hz. (c) Squeezing
and anti-squeezing levels as a function of pump power at measurement frequency
22 MHz.

We use second harmonic generation (SHG) to determine the phase-matching
condition and evaluate the efficiency of the nonlinear optical process in the
fabricated device. For SHG measurement, we use optical fields near 1550 nm as
input, and detect optical output power near 775 nm (Fig. 2(a)). At room
temperature, the device has the maximum SHG output power around 1574 nm. The
SHG efficiency 155%/W is estimated. By tuning the device temperature, the
phase-matching condition can be adjusted. At temperature 70 ${}^{\circ}C$, the
optimum wavelength is shifted to 1554.2 nm without compromising the SHG
efficiency (Fig. 2(a) inset). At low power levels, SHG power conversion
efficiency increases linearly with the input power. At high power levels, the
conversion efficiency deviates from this behavior due to pump depletion, which
is observed with our device (Fig. 2(b)). We have achieved more than 10% total
conversion efficiency, showing that efficient nonlinear processes can be
realized with single-pass configuration.

The experimental setup for squeezing measurement is shown in Fig. 3(a). A
continuous-wave (CW) laser with center wavelength 1554.2 nm is phase modulated
at 40 MHz, and amplified by an erbium-doped fiber amplifier (EDFA). The EDFA
output is collimated and divided into two beams. One beam is injected into a
commercial nonlinear crystal for SHG. The second-harmonic beam is used as the
pump for the squeezed light generation with thin-film lithium niobate
waveguides. The other beam is used as the local oscillator (LO) for homodyne
detection. The LO beam passes through a reference waveguide with identical
dimensions as the thin-film lithium niobate device for squeezed light
generation. This ensures the optimum spatial mode matching between the LO and
squeezed light. Mode cleaning cavities, which are locked to the CW laser
through Pound–Drever–Hall (PDH) technique using the 40 MHz modulation signal,
are used to remove excess noise in the pump field and LO. The homodyne signal
is sent to an electrical spectrum analyzer (ESA) for noise measurement. A
piezo-transducer is used to scan the LO phase with 1 Hz frequency. The device
temperature is tuned to maximize the nonlinear efficiency at 1554.2 nm (Fig.
2(a) inset).

Figure 3(b) shows the noise power of the squeezed light generated with the
thin-film lithium niobate waveguide. The noise power is detected by zero span
measurement at 22 MHz frequency. With 82 mW pump power, the measured squeezing
and anti-squeezing levels are $-0.56\pm 0.09$ dB and $0.83\pm 0.09$ dB,
respectively. The shot noise level is determined by blocking the output of the
thin-film lithium niobate waveguide. With 12 mW LO power, the shot noise is
9.2 dB higher than the electronic noise of the measurement setup. We further
measure the squeezing and anti-squeezing levels with respect to pump power
(Fig. 3(c)). The measured data are fitted with

$S_{\pm}=1-T+Te^{\pm 2\sqrt{\eta P}}$ (1)

where $S_{\rm\pm}$ are the anti-squeezing and squeezing levels in linear
scale; $T$ is the total detection efficiency; $\eta$ is the SHG efficiency;
$P$ is the pump power. The SHG efficiency is estimated to be 138%/W, which is
consistent with 155%/W measured in SHG experiments (Fig. 2). The total
detection efficiency of 19.4% is estimated. Therefore, an on-chip squeezing
level around 3 dB can be inferred by setting detection efficiency $T=1$. From
linear transmission measurement, we have estimated the optical losses induced
by different components. The on-chip waveguide propagation loss is 0.5 dB. The
coupling loss between the thin-film lithium niobate waveguide and output lens
is 3.5 dB. The spatial mode mismatch between LO and squeezed light introduces
0.3 dB equivalent loss. The photodetector has quantum efficiency 87%,
equivalent to 0.6 dB loss. The electric noise of the measurement setup is 9.2
dB lower than the shot noise, equivalent to 0.6 dB loss [27]. Therefore, the
total loss is 5.5 dB, corresponding to 28% detection efficiency.

Figure 4: Squeezed light bandwidth. (a) Squeezing and anti-squeezing levels as
a function of measurement frequency with homodyne detection. (b) Squeezed
light spectrum measured by optical spectrum analyzer with different pump
powers.

In order to estimate the bandwidth of squeezed light, we measure the squeezing
and anti-squeezing levels at different frequencies (Fig. 4(a)). Below 15 MHz,
noise power along both squeezing and anti-squeezing quadrature increases. This
is caused by the excess noise from amplified spontaneous emission in EDFA,
which is suppressed by the mode-cleaning cavities at high frequencies. Above
30 MHz, the measurement is limited by the finite bandwidth of the balanced
photodetector. In order to measure the full bandwidth of the squeezed light,
we directly measure the optical power generated from the parametric down-
conversion process at different wavelengths. This is achieved by replacing the
homodyne detection setup with an optical spectrum analyzer. Figure 4(b) shows
the optical spectra at different pump powers. Symmetric spectrum shape is
observed with a center wavelength 1554.2 nm, as idler and signal photons are
generated in pairs with constant total energy determined by the pump. The
half-width at half maximum is estimated to be above 7 THz (56 nm). The
generated squeezed light covers the whole C- and L-bands, and half S-band for
optical communications. Such large bandwidth will be beneficial to high-speed
quantum systems such as quantum communications with wavelength-domain
multiplexing and quantum computation with time-domain photonic cluster states
[10, 28].

Further improvement of the measured squeezing level can be readily achieved by
using photodetectors with near-unity quantum efficiency, optimizing spatial
overlap visibility, and lowering electronic noise. The squeezing level will be
ultimately limited by the coupling efficiency between integrated waveguides
and output lens/fiber. Loss below 1.7 dB has been realized between thin-film
lithium niobate waveguides and fiber [29]. This could lead to a squeezing
level above 4.8 dB, which is sufficient for two-dimensional cluster state
generation [30]. The normalized SHG efficiency of our device is $600\%$/W/cm2,
which is 10-fold higher than bulk lithium niobate [25]. Compared with
theoretical value $4100\%$/W/cm2, another 7-fold increase can be expected by
optimizing the domain inversion process and waveguide thickness uniformity.

In summary, we have demonstrated the generation of squeezed light based on
parametric down-conversion using thin-film lithium niobate waveguides. A
squeezing level of $0.56\pm 0.09$ dB has been measured with bandwidth up to 7
THz. This is enabled by the sub-wavelength mode confinement, which leads to a
10-fold increase in pump efficiency. Our sub-wavelength chip-scale platform
can enable the integration of squeezed light sources with large-scale photonic
circuits for advanced quantum functions. It also enables the realization of
second-harmonic generation and fast circuit reconfiguration on the same chip.
Therefore, this work will pave the way towards complete on-chip continuous-
variable quantum technology.

Funding U.S. Department of Energy, Office of Advanced Scientific Computing
Research, (Field Work Proposal ERKJ355); Office of Naval Research
(N00014-19-1-2190); National Science Foundation (ECCS-1842559, OIA-2040575).

Acknowledgments We thank Z. Zhang and Y. Xia for helpful discussion about
squeezing measurement.

Disclosures The authors declare no conflicts of interest.

Data availability Data may be obtained from the authors upon reasonable
request.

## References

  * [1] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Reviews of Modern Physics 84, 621 (2012).
  * [2] S. L. Braunstein and P. Van Loock, Reviews of Modern Physics 77, 513 (2005).
  * [3] The-LIGO-Scientific-Collaboration, Nature Physics 7, 962 (2011).
  * [4] T. Ono, R. Okamoto, and S. Takeuchi, Nature Communications 4, 2426 (2013).
  * [5] M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, Nature Photonics 7, 229 (2013).
  * [6] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
  * [7] L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, Nature Communications 3, 1 (2012).
  * [8] M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, Phys. Rev. A 79, 062318 (2009).
  * [9] N. C. Menicucci, Physical Review Letters 112, 120504 (2014).
  * [10] S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, Nature Photonics 7, 982 (2013).
  * [11] M. Chen, N. C. Menicucci, and O. Pfister, Physical Review Letters 112, 120505 (2014).
  * [12] X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V. Larsen, T. Gehring, M. Christandl, J. S. Neergaard-Nielsen, and U. L. Andersen, Nature Physics 16, 281 (2020).
  * [13] A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, Nature Photonics 14, 336 (2020).
  * [14] J. M. Arrazola, V. Bergholm, K. Brádler, T. R. Bromley, M. J. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. G. Helt, J. Hundal, T. Isacsson, R. B. Israel, J. Izaac, S. Jahangiri, R. Janik, N. Killoran, S. P. Kumar, J. Lavoie, A. E. Lita, D. H. Mahler, M. Menotti, B. Morrison, S. W. Nam, L. Neuhaus, H. Y. Qi, N. Quesada, A. Repingon, K. K. Sabapathy, M. Schuld, D. Su, J. Swinarton, A. Száva, K. Tan, P. Tan, V. D. Vaidya, Z. Vernon, Z. Zabaneh, and Y. Zhang, Nature 591, 54 (2021).
  * [15] V. D. Vaidya, B. Morrison, L. G. Helt, R. Shahrokshahi, D. H. Mahler, M. J. Collins, K. Tan, J. Lavoie, A. Repingon, M. Menotti, N. Quesada, R. C. Pooser, A. E. Lita, T. Gerrits, S. W. Nam, and Z. Vernon, Science Advances 6, eaba9186 (2020).
  * [16] Z. Yang, M. Jahanbozorgi, D. Jeong, S. Sun, O. Pfister, H. Lee, and X. Yi, arXiv:2103.03380 (2021).
  * [17] A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, Phys. Rev. Applied 3, 044005 (2015).
  * [18] Y. Zhao, Y. Okawachi, J. K. Jang, X. Ji, M. Lipson, and A. L. Gaeta, Phys. Rev. Lett. 124, 193601 (2020).
  * [19] U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, Physica Scripta 91, 053001 (2016).
  * [20] R. Prevedel, P. Walther, F. Tiefenbacher, P. Böhi, R. Kaltenbaek, T. Jennewein, and A. Zeilinger, Nature 445, 65 (2007).
  * [21] P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, Nature 434, 169 (2005).
  * [22] T. Kashiwazaki, N. Takanashi, T. Yamashima, T. Kazama, K. Enbutsu, R. Kasahara, T. Umeki, and A. Furusawa, APL Photonics 5, 036104 (2020).
  * [23] F. Lenzini, J. Janousek, O. Thearle, M. Villa, B. Haylock, S. Kasture, L. Cui, H.-P. Phan, D. V. Dao, H. Yonezawa, P. K. Lam, E. H. Huntington, and M. Lobino, Science Advances 4, eaat9331 (2018).
  * [24] F. Mondain, T. Lunghi, A. Zavatta, E. Gouzien, F. Doutre, M. D. Micheli, S. Tanzilli, and V. D’Auria, Photon. Res. 7, A36 (2019).
  * [25] C. Langrock, S. Kumar, J. McGeehan, A. Willner, and M. Fejer, Journal of Lightwave Technology 24, 2579 (2006).
  * [26] C. L. Sones, S. Mailis, W. S. Brocklesby, R. W. Eason, and J. R. Owen, J. Mater. Chem. 12, 295 (2002).
  * [27] J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, Phys. Rev. A 75, 035802 (2007).
  * [28] J. Roslund, R. M. De Araujo, S. Jiang, C. Fabre, and N. Treps, Nature Photonics 8, 109 (2014).
  * [29] L. He, M. Zhang, A. Shams-Ansari, R. Zhu, C. Wang, and L. Marko, Opt. Lett. 44, 2314 (2019).
  * [30] W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. N. Alexander, S. Takeda, J.-i. Yoshikawa, N. C. Menicucci, H. Yonezawa _et al._ , Science 366, 373 (2019).

bib_intro_quaninfo, bib_intro_chip, bib_LNOI_exp