Robert Jaromin

Barry Mullins

Jonathan Butts

Juan Lopez

Design and Implementation of Industrial Control System Emulators

Keywords: Industrial control systems, programmable logic controllers, emulation

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Understanding and mitigating attacks on industrial control systems requires researchers and operators who have extensive theoretical and practical knowledge of industrial processes and control systems. However, the cost of even a modest industrial control system can exceed 100,000, making the investment in security research and education prohibitive for most organizations. Clearly, cost-effective solutions are required to build human capital in industrial control systems security and to analyze, detect and mitigate attacks on industrial control systems that are widely used in the critical infrastructure.

Current solutions for industrial control systems research and education are inadequate in terms of cost and scalability. Full-scale, hardware-based SCADA testbeds are limited to research applications because of their capital, operational and maintenance costs, while smaller educational testbeds often lack the scalability to tackle real-world problems. Meanwhile, computer-based simulations are cost effective and useful for conducting research on certain types of large-scale, network-based systems, but they are inadequate for education and training.

This paper discusses the research and educational requirements associated with industrial control emulators. Also, it presents a novel emulation solution that is attractive in terms of its ability to accurately replicate real-world field device functionality as well as its cost and scalability.

Background

Public and private sector entities have made significant investments in experimental industrial control facilities for researching threats and vulnerabilities. Recognizing the need to develop security solutions for industrial control systems, the U.S. Department of Energy spearheaded the creation of the 114 million National SCADA Testbed at Idaho National Laboratory [START_REF] Gatens | INEEL establishes National SCADA Test Bed, Feature Story[END_REF]. As America's premier SCADA research facility, the testbed supports the identification and mitigation of vulnerabilities and the development of advanced system architectures for secure and robust SCADA systems. The testbed provides an exceptional environment for investigating real-world SCADA security threats, but similar testbeds are not feasible at academic and research institutions [START_REF]INEEL establishes National SCADA Testbed[END_REF]. Mississippi State University, for example, has implemented an educational SCADA network [START_REF] Morris | A testbed for SCADA control system cyber security research and pedagogy[END_REF], but it is limited to five control devices and two master stations. The environment is practical for educational purposes, but it does not scale up to large industrial applications.

Pure software-based simulations of SCADA networks and hybrid softwarehardware configurations have been implemented as alternatives to SCADA equipment. Examples include the Cyber Security Testbed at the University of Illinois [START_REF] Davis | SCADA cyber security testbed development[END_REF] and the SCADA Security Testbed at the Royal Melbourne Institute of Technology [START_REF] Queiroz | Building a SCADA security testbed[END_REF]. These testbeds leverage network simulation frameworks such as the Real-Time Immersive Network Simulation Environment for Network Security Exercises (RINSE) and Emulab [START_REF]Network Emulation Testbed[END_REF]. Although they are useful for network-level evaluations, the testbeds do not provide the ability to make fine adjustments to simulate SCADA device features such as firmware uploads and malware propagation Unlike most simulation efforts, which focus on network-level and/or devicelevel functionality to replicate real-world industrial control systems, this paper focuses on emulating programmable logic controllers (PLCs) that control the underlying physical processes. PLCs are specialized embedded computers with three core characteristics: (i) the ability to complete electronic circuits and measure electronic signals; (ii) the ability to be programmable to respond to inputs according to a predefined algorithm or protocol; and (iii) the ability to be networked with other devices.

Emulated industrial control systems combine the advantages of dedicated hardware and cost-effective, scalable software simulations. Emulators achieve these properties by implementing algorithms, protocols and services associated with industrial control systems using flexible programming platforms such as Linux. However, significant technical challenges are associated with emulating PLCs. A single product family, or even a single device, could involve multiple architectures and platforms. To further complicate the task, firmware can be packed or cryptographically encoded so that it can only be decoded by special hardware [START_REF] Kisserli | Self-encrypting code to protect against analysis and tampering[END_REF]. Additionally, errors or failures in the operating firmware should, by extension, be replicated in the emulator.

Industrial Control System Emulators

This section discusses the applications of industrial control system emulators in research and education, and presents the advantages of emulators over hardware testbeds, software simulations and hybrid hardware-software systems.

Research Environments

Industrial control emulators provide many of the same advantages as real hardware in a research environment. Accurate emulators not only implement the responses to user queries in the same manner as their hardware counterparts, but they also emulate the operational profile. When protocol responses are emulated accurately, the devices can be used in the same manner as real hardware for research experiments with less cost and reduced setup and maintenance times.

Accurate emulators have several advantages over real hardware devices. In addition to reduced cost, emulators can be more robust because error conditions such as buffer overflows and erroneous firmware uploads do not affect the underlying physical system. While a real PLC might be permanently damaged by such actions, it is a trivial matter to reset an emulated device and continue with the experiments.

Educational Environments

Currently, the only way to gain hands-on experience with industrial control networks is to implement an actual system, but this can be prohibitively expensive. The same is true for features that are specific to physical devices such as uploading and downloading ladder logic programs. Although simulators are available for developing and testing ladder logic programs, it is difficult, if not impossible, to adequately test network communications and functionality without using actual industrial control devices [START_REF]GuardPLC Safety Control Programming Software[END_REF]17].

In an academic setting, emulators can be used to provide a deep understanding of the tools and techniques used to conduct attacks on industrial control networks. Indeed, an emulator can be used to extend the application of tools such as Nmap [START_REF]Insecure.com[END_REF] and Metasploit [START_REF]Rapid7, Metasploit Framework[END_REF] to industrial control networks. In particular, attacks on a PLC emulator can help demonstrate traditional information technology attacks as well as their effects on an industrial control system. An emulator that responds to scans and attacks from tools such as Nmap and Metasploit in the same manner as a real PLC can provide hands-on attack experience without putting the industrial control system at risk. Also, an emulator can be remotely setup by instructors to model any number of device configurations and vulnerabilities and record the techniques used by students to exploit the device.

A PLC emulator must have high fidelity; for example, it should accurately mimic the responses of a real PLC to fingerprinting and port scanning tools. However, even a lower-accuracy emulator has value in a training environment -students could learn to identify control system devices by employing network scanning and protocol dissection techniques.

Advantages of Emulation

Industrial control devices are notoriously sensitive due their lack of input sanitization and use of non-robust communications protocols. Even seemingly benign network mapping tools can cause serious problems. A 2005 Sandia report [START_REF] Duggan | Penetration Testing of Industrial Control Systems[END_REF] describes several instances where network scans of SCADA networks disrupted industrial control systems. The reported behavior included erratic movements of a nine-foot robotic arm in an area shared with human operators, a malfunction of a microchip manufacturing process that resulted in 50,000 damage to electrical components, and a large natural gas delivery failure due to unresponsive SCADA equipment.

A significant advantage of an emulator is that it retains the robustness of the underlying system architecture. Even if the emulator locks up -i.e., it becomes "bricked" -there is no flash bootloader to accidentally overwrite or electronically-erasable programmable read-only memory (EEPROM) to accidentally erase because it is implemented purely in software. A simple reset of the software returns the emulator to its normal operating state, unlike a real PLC, which would require the use of a hardware debugging device or even the replacement of PLC components.

An industrial control emulator is cost effective compared with control system hardware, especially when a single PLC can easily cost more than 10,000. Power constraints are also a factor -incorporating large numbers of PLCs in an educational or training environment may not be possible due to power requirements.

An emulator is scalable and is easily ported to different platforms. Multiple emulators can be chained together without serious interoperability problems. Also, emulators can be configured to model different vendor platforms (e.g., Siemens, Allen-Bradley and Koyo) and myriad network parameters (e.g., MAC address, IP address, host name and ID).

Emulator Design Considerations

Despite the fact that industrial control systems are fragile, quickly antiquated from a technology standpoint and often designed without security considerations, they are expensive, complex and invariably networked with other devices. Most modern PLCs provide application-level services such as web configuration interfaces, file transfer services, remote programming services and remote monitoring services through a network interface. All these services have to be replicated in an emulator.

The accuracy of an emulator can be assessed using observable characteristics that a user would leverage to establish device authenticity. These attributes may be derived from the standard process that an attacker would use to discover, interrogate, fingerprint and exploit a networked device [START_REF] Mcclure | Hacking Exposed 7: Network Security Secrets and Solutions[END_REF]. In general, four accuracy considerations exist for an emulator:

Superficial Accuracy: This is established when a user believes that a device is authentic based on what is visually observed, such as the appearance of the web interface.

Packet-Level Accuracy: This is established when a user believes that a device is authentic based on the packets that are sent to and received from the device.

Scanning Tool Accuracy: This is established when a user believes that a device is authentic based on the results produced by an automated scanning tool.

Attack Tool Accuracy: This is established when a user believes that a device is authentic based on the success achieved by an exploitation tool.

Like a network emulation, a control system emulation must accurately reflect the operational environment. Accuracy is a measure of the realism of the emulation compared with the actual device. For example, an emulation of a web interface in a modern PLC is accurate if the webpage has the same appearance (e.g., banner, form elements and graphics) and provides the same functionality. Additionally, the packets transmitted during query and response interactions must match the packets transmitted to and received from the actual PLC.

Emulator Implementation

This section provides implementation details regarding the emulator along with an evaluation of the accuracy of the emulator.

Implementation Details

The emulator runs on a Dell Latitude D630 with 2 GB RAM, Intel Core2 Duo CPU 2.00 GHz processor running VMware with Linux Ubuntu 2.6.35. The Linux kernel was built with iptables and Netfilter NFQUEUE modules to provide the necessary Linux firewall and user-space network packet functionality.

The emulated PLC comprised the Koyo DirectLogic 405 CPU with a D4-08ND3S digital input module, D4-08TR digital relay module and H4-ECOM-100 Ethernet communications module. This PLC was selected because it demonstrated certain security vulnerabilities, including the vulnerability to an automated method that cracks device passwords [START_REF] Peck | Leveraging Ethernet card vulnerabilities in field devices[END_REF][START_REF] Wightman | Koyo/Automation direct vulnerabilities, Digital Bond[END_REF]. The emulated firmware used in our work was version H4-ECOM 4.0.1735, which contains a known vulnerability and an associated Metasploit module koyo login [START_REF] Wightman | Koyo/Automation direct vulnerabilities, Digital Bond[END_REF].

The emulator is written in C and includes several interpreted Python modules. The emulator incorporates seven simultaneously running user-space processes in addition to the iptables firewall kernel module. It offers three network services to users: (i) a web server on TCP port 80 providing a configuration interface; (ii) a Modbus TCP service on TCP port 502; and (iii) a Host Automation Products (HAP) service on UDP port 28784. The Modbus TCP service is indicative of the Koyo PLC implementation of the common industrial protocol service. The HAP service enables remote management and configuration of the Koyo PLC.

For purposes of this research, the Modbus service was replicated strictly for device fingerprinting associated with the Koyo PLC. Because the research focus was on the Koyo PLC application and not messaging protocol standards, the Modbus functionality was not evaluated. Note, however, that Modbus functionality is readily incorporated as demonstrated by Berman, et al. [START_REF] Berman | Towards a characterization of cyber attacks on industrial control systems: Emulating field devices using Gumstix technology[END_REF].

Emulator Accuracy

The emulated PLC was evaluated for consistency with the Koyo PLC in terms of superficial accuracy, packet-level accuracy, scanning tool accuracy and attack tool accuracy. Each of the two emulated services, web and HAP, had standard and non-standard types of interactions, yielding a total of four types of queries. These queries were selected for the evaluation because they are representative of the interactions associated with an attacker targeting an Internetconnected PLC.

Standard Web Queries: These queries are associated with interactions with a web server listening on TCP port 80. The queries represent normal, within-bounds requests from an Internet Explorer web browser.

Non-Standard Web Queries: These queries are associated with OS fingerprinting scans using Nmap that attempt to identify a device based on the responses to non-standard TCP, IP and ICMP packets sent to the device. By default, Nmap chooses the lowest open TCP port on a device to perform its tests. Therefore, Nmap selects TCP port 80 unless it is directed otherwise.

Standard HAP Queries: These queries are associated with interactions with the HAP industrial protocol server listening on UDP port 28784. These queries represent normal, within-bounds requests from NetEdit3 [6], a free vendor tool. Non-Standard HAP Queries: These queries are associated with attacks on HAP protocol security measures using the koyo login Metasploit module to exploit vulnerabilities.

The accuracy of the responses to the four types of queries are discussed below. The results are useful in educational environments where students learn scanning techniques and attack strategies that target industrial control systems. Additionally, the results showcase the applicability of the emulator in control system research efforts.

Standard Web Responses: To analyze the superficial accuracy, the Koyo website HTML source was used to duplicate the original content and visual representation. Although it appears to be trivial, the notion of superficial accuracy should not be underestimated. For example, in a honeypot application, even a slight visual difference can alert a potential attacker that the device is not authentic.

Web pages were transmitted byte-for-byte from the PLC to the user's browser. As such, packet-level accuracy is an important consideration in PLC emulation. A single web query to index.html of the Koyo PLC generates 2,432 bytes. Excluding the non-deterministic header fields (e.g., TCP sequence and acknowledgement numbers), there are 2,330 bytes that are expected to be identical. The emulator differs by only five bytes of the 2,330 total bytes, demonstrating a consistency of 99.8%.

Four of the five differences occur in the TCP response header as a result of the Koyo PLC TCP/IP stack implementation handling procedures for the TCP push flag and TCP congestion window. Figure 1 shows the web server responses from the emulator and the Koyo PLC for different TCP congestion window sizes. Correcting these differences requires the modification and recompilation of the Linux kernel.

The fifth difference is the result of an implementation error in the emulator that appears in the index.html web page. The emulator value is 0x3a (ASCII ":" character) while the Koyo PLC value is 0x0a (ASCII line feed character). Because this difference occurs with a whitespace character, it is likely the result of a copy-and-paste error. Whatever the cause, the difference is readily fixed by changing the index.html source code of the emulator.

Non-Standard Web Responses: An Nmap operating system scan uses TCP/IP stack fingerprinting to identify the operating system of a target device [START_REF]Nmap Network Scanning, The Official Nmap Project Guide to Network Discovery and Security Scanning, Insecure.com[END_REF]. From the point of view of scanning accuracy, the emulator must respond to an Nmap operating system scan in a manner that is consistent with the Koyo PLC. Indeed, the scan should provide no indications that the emulator is running a Linux operating system within a virtual environment.

Figure 2 shows the results obtained for non-standard web queries to the emulator and Koyo PLC. The Nmap scan results differ only in the TCP sequence number prediction fields (SP and ISR) and in the unused ICMP port unreachable field non-zero field (UN). The corresponding scanning tool accuracy is 97.25%. Note that an Nmap scan on a similar VMware configuration with no installed emulator correctly identified the Linux operating system. The results indicate that the emulator sufficiently conforms to the Koyo PLC with regard to the operating system characteristics gleaned using external communications.

Standard HAP Responses: Standard HAP requests were sent using the UDP transmission protocol. In the experiment, 305 of the 309 total bytes in the HAP response from the Koyo PLC were deterministic, and the emulator was 100% accurate for the deterministic bytes. Figure 3 shows the response bytes for HAP protocol queries to the emulator and Koyo PLC. Note that the non-deterministic fields are highlighted and include the Ethernet address, IP header checksum, IP source address and UDP header checksum.

The accurate responses produced for standard HAP requests indicate the emulator is capable of interacting with any tool in a manner that conforms with the HAP protocol standard. As such, engineering modifications and device configuration updates are appropriately handled by the emulator, consistent with the expected operations of the Koyo PLC.

Non-Standard HAP Responses: The attack tool accuracy for the Metasploit module was determined based on successful attack execution and device exploitation. Figure 4 demonstrates an execution of the koyo login Metasploit module to brute-force the emulator password. The exploit works in the same manner as the Koyo PLC and discovers the correct emulator password ("A0000322" ). Thus, the emulator can be Obviously, it is important that the emulator behave in a manner that is consistent with the Koyo PLC. Table 1 shows the emulator results for the various types of accuracy associated with the two emulated services. When emulating the standard web and standard HAP services, the emulator is accurate at the superficial and packet levels. The non-standard web service is accurate at the scanning level -when a user scans the emulator using Nmap, the scanning results confirm the authenticity of the device. Finally, the non-standard HAP service is accurate at the attack tool level because the emulator is vulnerable to a Metasploit attack consistent with the Koyo PLC.

Conclusions

The PLC emulator implemented in the VMware virtual environment provides a cost-effective and scalable emulation solution. Experimental evaluations demonstrate that the emulator accurately replicates a real PLC with respect to standard and non-standard web and HAP services.

We hope that this work stimulates renewed efforts at developing sophisticated industrial control system emulators for research and educational environments. Industrial control system emulators provide significant advantages over hardware-based testbeds and simulation environments. Their robustness, versatility and configurability are attractive for understanding, developing and analyzing defensive and offensive techniques, more so because these efforts can be conducted without concern for damaging costly equipment.

Note that the views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, U.S. Department of Defense or the U.S. Government.

Figure 1 .

 1 Figure 1. Web server responses showing differences in TCP congestion window sizes.

  Nmap OS scan result.Koyo PLC scan result.

Figure 2 .

 2 Figure 2. Nmap operating system scan results.

Figure 3 .

 3 Figure 3. Standard HAP packet responses.

Figure 4 .

 4 Figure 4. Metasploit module koyo login executing successfully on the emulator.

Table 1 .

 1 Emulator services and accuracy considerations.

	Service	Query Type	Superficial	Packet Scan	Attack
	Web	Standard (Browser)	X	X
		Non-Standard (Nmap)		X
	HAP	Standard (NetEdit3)	X	X
		Non-Standard (Metasploit)			X


Benjamin Ramsey

Barry Mullins

DEFENSIVE REKEYING STRATEGIES FOR PHYSICAL-LAYER-MONITORED LOW-RATE WIRELESS PERSONAL AREA NETWORKS

Keywords: ZigBee networks, security, RF fingerprinting, cyber-physical systems 2. Key Distribution

ZigBee networks are integrating rapidly into critical infrastructures such as the smart grid and public health centers. Numerous ZigBee-based smart meters have been installed in metropolitan areas and hospitals commonly employ ZigBee technology for patient and equipment monitoring. The associated ZigBee networks transport sensitive information and must be secured against exfiltration and denial-of-service attacks. Indeed, novel tools that exploit and disrupt ZigBee networks are already under development. Security monitors that can uniquely identify nodes by their radio frequency characteristics can be a valuable countermeasure if implemented in a practical manner. This paper investigates rekeying in response to suspected malicious devices that may be internal or external to a ZigBee network. It extends prior discussions of practical physical layer monitor implementation, and introduces a novel backward-compatible ZigBee message obfuscation technique based on preamble modifications. Experimental results demonstrate that common wireless ZigBee sniffers can be thwarted with 100% effectiveness without reducing packet reception to specific transceiver models.

Introduction

ZigBee networks provide low-rate, low-power and low-cost wireless connectivity through standards that build upon the IEEE 802.15.4 low-rate wireless personal area network (LR-WPAN) physical (PHY) and medium access control (MAC) specifications [START_REF]Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (WPANs)[END_REF]. ZigBee Smart Energy, Building Automation and Health Care standards have enabled ZigBee networks to become significant components in critical infrastructures around the world, including tens of millions of utility meters with bidirectional communications incorporated in advanced metering infrastructures [23]. Critical ZigBee networks form cyberphysical systems, where malicious activity on the networks affects the physical behavior of appliances and electrical systems. Public health networks employing ZigBee technology are also common in civilian and military hospitals. Disruptions of these networks can impact medical equipment and patient monitoring, possibly endangering lives.

The ZigBee security architecture relies on the safekeeping of symmetric keys to implement message confidentiality, message integrity, and device authentication. While the small size and low complexity of ZigBee devices make them effective to deploy in large numbers, these traits also result in tight constraints on device memory and computations. A single network key (NK) is shared by every device in a ZigBee network, although device-to-device confidentiality is also possible using link keys (LK) at the application layer. Small, inexpensive wireless sensors are unlikely to have robust defenses against theft and tampering, resulting in physical vulnerabilities to key confidentiality [START_REF] Shi | Designing secure sensor networks[END_REF]. Key extraction from first-and second-generation ZigBee chips has been shown to be relatively straightforward [START_REF] Goodspeed | Extracting keys from second generation ZigBee chips[END_REF], and inexpensive tools have been developed for locating ZigBee devices [START_REF] Kiraly | Where's the mote? Ask the MoteHunter! Proceedings of the Thirty-Seventh IEEE Conference on Local Computer Networks[END_REF][START_REF] Ramsey | Improved tools for indoor ZigBee warwalking[END_REF]. Keys may also be compromised through social engineering or keys may intercepted (if transmitted to end devices without encryption) by open source tools such as KillerBee [START_REF] Wright | KillerBee: Framework and Tools for Exploiting Zigbee and IEEE 802[END_REF] and Api-do [START_REF] Speers | Api-do: Tools for ZigBee and[END_REF].

The ZigBee specification is dependent on symmetric cryptography, which precludes key distribution without a central authority called the trust center (TC). The computational complexity of symmetric cryptography is lower than that of public key cryptography. The tradeoff, however, is a significantly more challenging key management process.

Previous research has investigated the application of public key cryptography to ZigBee devices and networks [START_REF] Blaser | Industrial-strength security for ZigBee: The case for public-key cryptography[END_REF][START_REF] Heo | Efficient and authenticated key agreement mechanism in low-rate WPAN environment[END_REF][START_REF] Huang | Fast authenticated key establishment protocols for self-organizing sensor networks[END_REF][START_REF] Huang | Energy/security scalable mobile cryptosystem[END_REF][START_REF] Nguyen | ZigBee security using identity-based cryptography[END_REF]; this research has contributed to a provision in the ZigBee Smart Energy Profile for secure LK establishment through certificates signed by a certificate authority [START_REF] Yuksel | ZigBee-2007 security essentials[END_REF]. The alternative key establishment procedure is based on a shared master key (MK) used to derive the LK. If the MK is not preloaded on the end device, the TC broadcasts it without encryption, potentially compromising the subsequent LK establishment.

Methods for detecting rogue devices in ZigBee-type networks are an active area of research, including anomalous-behavior-based fingerprinting [START_REF] Jokar | Specification-based intrusion detection for home area networks in smart grids[END_REF] and radio frequency (RF) device fingerprinting [START_REF] Danev | Transient-based identification of wireless sensor nodes[END_REF][START_REF] Dubendorfer | An RF-DNA verification process for ZigBee networks[END_REF][START_REF] Ramsey | PHY foundation for multi-factor ZigBee node authentication[END_REF]. Given the threats to ZigBee symmetric keys, an efficient and secure redistribution of keys must occur to thwart a known eavesdropper or active rogue device on the network. This paper examines rekeying strategies for a compromised ZigBee network. In particular, it investigates how PHY-based monitoring systems proposed in [START_REF] Danev | Transient-based identification of wireless sensor nodes[END_REF][START_REF] Dubendorfer | An RF-DNA verification process for ZigBee networks[END_REF][START_REF] Ramsey | PHY foundation for multi-factor ZigBee node authentication[END_REF]] can be integrated in ZigBee networks. Also, the paper describes a novel method for protecting sensitive ZigBee traffic from eavesdroppers using modified PHY preambles.

ZigBee Nodes and Topologies

The IEEE 802.15.4 standard for wireless MAC and PHY specifications defines two primary device types: (i) full function devices (FFDs); and (ii) reduced function devices (RFDs) [START_REF]Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (WPANs)[END_REF]. Mains-powered FFDs are always actively listening on the network, whereas RFDs are battery powered and primarily operate in the sleep mode, waking up only to check for pending messages or periodic updates.

ZigBee specifies three node classes within the IEEE 802.15.4 construct: (i) ZigBee coordinator (ZC); (ii) ZigBee router (ZR); and (iii) ZigBee end device (ZED). The ZCs and ZRs must be FFDs, while ZEDs can be either FFDs or RFDs. There can only be one ZC per WPAN, and it is responsible for establishing the network, allocating network layer addresses and routing traffic. The network fails without the ZC. ZRs extend the wireless range by routing messages between their child RFD ZEDs using multi-hop configurations, such as the cluster tree and mesh topologies is shown in Figure 1. Note that the star topology is shown for completeness; however, it does not support multi-hop communications. In a cluster tree topology, ZEDs have no children and can only communicate with the ZC and other ZEDs through their parent ZR. The ZigBee stack profile 0x01 limits the number of children for each ZR to N c = 20, six of which can be ZRs. The ZigBee PRO specification (stack profile 0x02) increases this limit to N c = 254 children per ZR. Mesh topologies are only allowed under ZigBee PRO, and permit FFD ZEDs to communicate directly with each another to form a self-healing network.

Stationary Networks

The term "stationary network" refers to a network in which the logical and physical topologies are both fixed. It includes the star and cluster tree topologies. ZigBee networks for building and home automation, utility meter reading, industrial control and environmental sensing typically utilize stationary networks. Unlike mobile nodes, stationary nodes can be mains powered and always active.

The ZigBee Smart Energy Profile describes standard practices suitable for securing stationary networks [START_REF]ZigBee Alliance, ZigBee Smart Energy Profile Specification, ZigBee Doc[END_REF]. In order to join a network, every new device must pass an administrator-directed commissioning process. The commissioning process is not typically automated because neighboring systems have no way to identify the devices that should be associated with one another without administrator guidance [START_REF]Daintree Networks, Understanding ZigBee Commissioning[END_REF]. At no point should any cryptographic keys be transmitted in the clear, a recommendation that is mirrored in [START_REF] Masica | Recommended Practices Guide for Securing ZigBee Wireless Networks in Process Control System Environments[END_REF]. At a high level of abstraction, a typical device commissioning process involves the following steps [START_REF]ZigBee Alliance, ZigBee Smart Energy Profile Specification, ZigBee Doc[END_REF]:

The ZC is informed via out-of-band means (e.g., remote login, handheld controller or phone call to a service center) that a new device must be added to the network.

The network enters into a permit joining ON state for a short period of time (e.g., 10 seconds).

The installer, with physical access to the new device, presses a button or navigates a menu that instructs the new device to attempt to join the network through a join procedure.

The new device attempts to join the network. After the new device is verified as being authentic by its MAC address and/or secret key, it receives new cryptographic keys and successfully joins the network.

The installer receives visual feedback from the new device that the network join was successful.

The new device may now operate in the network.

The short window of the commissioning process protects against rogue devices from joining the network. This process is also robust against tools like KillerBee's zbassocflood that transmit numerous association requests from spoofed MAC addresses to exhaust the network address pool in a denial-ofservice attack.

Smart Energy Profile standards identify three types of keys: (i) NK; (ii) application layer LKs shared by pairs of devices; and (iii) trust center link keys (TCLKs), which are LKs that are shared by a device and the TC. The NK is common to all devices in a network and protects management and control communications [START_REF]ZigBee Alliance, ZigBee Smart Energy Profile Specification, ZigBee Doc[END_REF]. LKs provide end-to-end confidentiality. The TC periodically refreshes the NK to protect the network from cryptographic attacks from outside the network (e.g., key cracking). The NK must also be refreshed in response to a suspected rogue device within the network. Rekeying is thus performed point-to-point from the TC to every trusted device on the network using TCLKs for confidentiality. The complexity of this action amounts to O(n), where n is the number of devices in the network. After every trusted device receives the new NK, the broadcasted Switch Key command instructs all devices to simultaneously switch to the new NK. The TC also revokes any LKs that were previously established to a rogue device.

Mobile Networks

The term "mobile network" refers to a network in which the logical and physical topologies change unpredictably. Examples include medical patient monitoring and inventory tracking systems. Mobile devices may drift out of communication range of other network nodes long enough to require a network rejoin, which cannot be performed manually as with stationary networks. Physical security is also significantly more difficult to maintain for mobile devices.

The ZigBee Health Care Profile [START_REF]ZigBee Alliance, ZigBee Health Care Profile Specification, ZigBee Document 075360r15[END_REF] provides key distribution recommendations for mobile networks. As with stationary networks, key delivery in the clear is prohibited. Instead, NK and LK distribution proceeds via mandatory TCLKs. Mobile nodes are also much more likely than stationary nodes to be battery powered; therefore, the nodes spend significant time in the sleep mode. If a network layer rekey occurs while a medical device is asleep, it will experience a delay in reporting its sensing data because it must first receive the new NK. To minimize such delays, devices should check periodically if they have the current NK [START_REF]ZigBee Alliance, ZigBee Health Care Profile Specification, ZigBee Document 075360r15[END_REF]. Rekeying requires all end devices to expend more of their finite energy supply than is functionally necessary, so the rate of periodic rekeying must be set based on battery longevity requirements. Nevertheless, rekeying in response to a suspected rogue device is essential.

Alternative Key Distribution Methods

Numerous key distribution mechanisms exist for ZigBee-like wireless sensor networks and ad hoc networks [START_REF] Camtepe | Key Distribution Mechanisms for Wireless Sensor Networks: A Survey[END_REF]. Each key distribution mechanism results in different computational, energy and memory burdens being placed on network nodes. The ZigBee protocol stack requires that the NK be shared by all the devices in a network. This fundamental requirement necessitates that NK rekeying be performed on every network device if the NK is suspected to be compromised.

The TC must revoke all point-to-point LKs established between a benign device and the suspected rogue device. In this case, techniques that reduce the number of required LKs also reduce the number of required LK revocations. For example, the upper limit for a ZigBee network is 2 16 ≈ 65,000 nodes, requiring n(n -1)/2 ≈ 2 billion LKs for full connectivity. In a hierarchical keying system [START_REF] Sun | Hierarchical group access control for secure multicast communications[END_REF], the number of required LKs reduces to log 2 (n) + 2 = 18 LKs. However, full connectivity is rare in practice, and LKs are frequently limited to communications between end devices and their associated data aggregation nodes. Note that spatial clustering methods, including the Hubenko Architecture [START_REF] Antosh | A comparison of keying methods in the Hubenko Architecture as applied to wireless sensor networks[END_REF], are not relevant to ZigBee LK rekeying because LKs secure unicast traffic rather than multicast traffic. Indeed, a compromised node reveals nothing about the LKs in use by any two other nodes, so TC-directed LK revocation is sufficient for threat mitigation.

ZigBee Air Monitor Integration

Recent research [START_REF] Dubendorfer | An RF-DNA verification process for ZigBee networks[END_REF][START_REF] Ramsey | PHY foundation for multi-factor ZigBee node authentication[END_REF] has proposed a ZigBee air monitor (ZAM) system to secure networks by observing wireless transmission characteristics to augment bit-layer security mechanisms. Dubendorfer, et al. [START_REF] Dubendorfer | An RF-DNA verification process for ZigBee networks[END_REF] have demonstrated the feasibility of unique device-level identification under realistic indoor office conditions. The next three sections explore how ZAMs can integrate into and defend critical ZigBee networks organized in star, cluster tree and mesh topologies.

Star Topology

The star topology is the least complex ZigBee network topology. In this topology, end devices communicate solely with the ZC. ZigBee PRO allows a ZR or ZC to have up to N c = 254 child nodes, but such a dense network can experience significant transmission congestion. A best practice is to limit the number of devices in an area to utilize spatial reuse [START_REF]Building and Operating Robust and Reliable ZigBee Networks[END_REF]. The utilization ratio U is given by:

U = 4 < Density < 16 T otal Devices (1)
where the density (i.e., number of devices per unit area where the unit used is the square of their reliable range) is between four and sixteen. A star network should, therefore, be limited to approximately fourteen end devices (15 devices -1 coordinator).

The ZAM is co-located with the ZC and observes every transmission, sending a packet reject notice to the ZC through a wired channel if the RF fingerprint of the packet does not closely match the fingerprint stored in the ZAM for the claimed origin device [START_REF] Ramsey | PHY foundation for multi-factor ZigBee node authentication[END_REF]. The ZigBee application support (APS) layer permits up to two retries for each packet. False rejections of packets from trusted devices follow a binomial distribution with the probability p of successful packet delivery given by:

p = 1 -(ρ + P ER) 1+r ( 2
)
where ρ is the probability of false rejection, P ER is the packet error rate for the network and r is the number of total retries by the APS layer. For ρ = 0.2, r = 2 and P ER = 0.1, the probability of benign packet delivery exceeds 99%. Dubendorfer, et al. [START_REF] Dubendorfer | An RF-DNA verification process for ZigBee networks[END_REF] report rejection rates of ρ ≤ 0.2 for seven like-model devices at SN R = 10 dB. For star topology networks with mains-powered devices, the energy spent on retransmission due to false packet rejections is negligible, particularly in a neighborhood advanced metering infrastructure where utility usage and pricing updates occur a few times per hour. A low utilization ratio as mentioned above mitigates network congestion introduced by frequent packet retransmissions.

The ZAM must make verification decisions within a short time, constrained by the transmission timeout settings at the APS layer. A typical value for the unicast timeout is t = 1.6 seconds per try, for a total of t total = 4.8 seconds [START_REF]Digi International, XBee/XBee-PRO ZB SMT Modules[END_REF]. Computational and memory usage requirements increase for the ZAM as the number of nodes increases, but the unicast timeout places a strict upper limit on the total packet accept/reject response time.

Cluster Tree Topology

The cluster tree topology extends the scale of a stationary ZigBee network beyond that of the star topology by leveraging spatial reuse and a data aggregation backbone of ZRs. The utilization ratio defined in Equation (1) still applies, limiting the practical number of child nodes per ZR, while the total number of nodes can increase substantially.

ZAMs co-located with every ZR provide oversight of all network traffic and provide the same per-packet rejection feedback for child nodes in their cluster as described in the star topology. The device fingerprint database in each ZAM remains at the approximate scale of the star topology scenario. No sharing of fingerprint information between ZAMs is necessary because the network nodes do not stray from their ZRs.

Mesh Topology

The mesh topology poses the greatest security challenge among the three topologies. ZAMs distributed throughout the mesh network physical topology observe all network traffic, including point-to-point traffic between ZEDs. ZAMs co-located with the core ZRs may be sufficient for full network observation, otherwise additional ZAMs must be co-located with sufficient FFD ZEDs to cover the outermost traffic. FFDs are mains powered, so power is also available for the co-located ZAMs.

FFD ZRs and ZEDs, with co-located ZAMs, receive wired per-packet feedback as in the star and cluster tree topologies. However, the logical network topology is variable and an FFD ZED unattached to a ZAM can receive a packet from a neighboring ZED. In this case, the receiving ZED requests a packet accept/reject feedback from the observing ZAM using the ZigBee network. An alternative solution is to require all packets to traverse FFDs with co-located ZAMs while still allowing the logical topology to change over time. The variable mesh topology increases the number of fingerprints that each ZAM must store to verify true packet origin, with an upper bound of n representing a fingerprint profile for every network node.

PHY-Based Sensitive Message Obfuscation

Although they are functionally consistent, transceiver implementations vary between manufacturers due to their design, components used and other unique attributes. These variations provide an opportunity to develop unique signatures for fingerprinting devices based on their implementation characteristics. This section describes a method for preventing common wireless sniffers from detecting ZigBee packets, while retaining their compatibility with available hardware. The method involves varying the length of the IEEE 802.15.4 preamble and measuring response differences that result from manufacturer implementations.

Methodology

The IEEE 802.15.4 beacon request is a standard command used by a device to locate all coordinators within transmission range. Replies by the coordinators are compulsory (i.e., unicast frames with the acknowledgment flag set garner replies from individually addressed devices).

A Tektronix TDS6124C digital storage oscilloscope was used to receive and store a single beacon request using a Ramsey LPY2 log periodic antenna with 6 dBi gain. The sampling rate R s was 1.25 GS/s and the collection length l c was 1 ms, which was long enough to capture the entire l = 512 µs beacon request. The collection yielded a vector of 1.25 million data points. All ZigBee transmissions at f c = 2.4 GHz begin with a l = 128 µs preamble of l p = 32 bits represented by eight O-QPSK symbols. As shown in Figure 2, when the first half of the preamble (four O-QPSK symbols) is absent, the entire beacon request transmission shortens by 12.5%.

A Tektronix AWG7102 arbitrary waveform generator was used to replay the original and modified beacon requests through the same Ramsey LPY2 log periodic antenna used for signal collection. The center frequency remained constant at f c = 2480 MHz to mitigate interference from neighboring IEEE 802.11 networks during the experiments. The received signal strength was approximately -79 dBm at the target devices and the response behavior remained constant through differing distances and signal strengths. All the hardware devices correctly replied to the original beacon request, but not all the devices replied to modified beacon requests. This indicates that some transceiver types cannot synchronize to the shortened preambles and are unable to interpret the message contents.

ZigBee Sniffer Hardware

Two widely available hardware platforms for wireless ZigBee sniffing are the Atmel RZUSBStick and the Microchip ZENA wireless adapter (Figure 3). Both platforms are inexpensive, contain a USB connector, include support for real-time viewing of ZigBee packets and can save the captured traffic to a local file. A limiting factor common to both devices is the lack of support for an external antenna.

The RZUSBStick includes an Atmel AT86RF230 transceiver with -101 dBm receiver sensitivity and maximum transmit power of P t = 3 dBm. Atmel Wireless Studio is the associated free application for wireless sniffing. Alternatively, open source KillerBee software and firmware fully support the RZUSBStick, enabling arbitrary ZigBee packet generation, key sniffing, denial-of-service attacks and transmitter positioning.

The ZENA wireless adapter includes a Microchip MRF24J40MA transceiver module with -94 dBm receiver sensitivity. The Microchip Wireless Development Studio, which controls wireless sniffing on the adapter, may be downloaded free of charge.

Results and Analysis

The six device types considered in this research included the two sniffers (Atmel AT86RF230, Microchip MRF24J40MA) and four other transceivers configured as coordinators (XBee XBP24CZ7PIS, Freescale MC13213, Texas Instruments (TI) CC2420 and Jennic JN5148). Beacon requests with shortened preambles emanated from the arbitrary waveform generator toward the six device types simultaneously. Beacon replies occured within milliseconds and were staggered to avoid collisions using the PHY Carrier Sensing Multiple Access Collision Avoidance Algorithm. In the case of the two sniffers, correctly interpreted packets appeared in the display windows of the sniffing control software. The four transceiver models configured as ZigBee coordinators replied to the correctly-interpreted beacon requests, and the two sniffers recorded all the replies for post-experiment analysis. Each preamble modification was transmitted a total of 100 times -five repetitions of 20 transmissions each.

The mean packet reception exceeded 98% for all device types when the beacon requests were transmitted with standard preambles. When the standard eight O-QPSK symbol preamble was shortened to five symbols, two of the six transceiver models began experiencing difficulty interpreting the packets. Figure 4 shows the mean ZigBee packet reception rates at a 95% confidence interval when only 5/8 of the preamble was present. The Atmel transceiver received a mean of 40% of the beacons, while the XBee transceiver received a mean of 87% of the beacons. No significant loss in packet reception occured for the Microchip, Freescale, TI and Jennic transceivers.

Figure 5 shows the mean ZigBee packet reception rates at a 95% confidence interval when only 4/8 of the preamble was present. In both cases, the sniffers were unable to interpret any of the packets. The XBee transceiver was thwarted, while the Freescale, TI and Jennic transceivers experienced no significant difficulty receiving packets. The Freescale MC13213 could not interpret packets with preambles shorter than four O-QPSK symbols, while the TI CC2420 and Jennic JN5148 still received all the packets. Figure 6 shows the mean ZigBee packet reception rates at a 95% confidence interval when only 2/8 of the preamble was present. Consistent with the experimental trend, only the TI and Jennic hardware correctly interpreted packets. The Jennic JN5148 continued to interpret packets without difficulty, while the TI CC2420 failed to interpret approximately half of the packets. Hundreds of O-QPSK symbol-wise modifications to the standard preamble are possible, providing a sizable search space for future research. The two modifications investigated in this paper are perforated preambles with only odd numbered symbols and only even numbered symbols present (Figure 7). Packet reception rates for the two perforated preamble modifications are shown in Figures 8 and9. When only odd-numbered symbols were present (symbols 1, 3, 5 and 7), the Atmel AT86RF230 interpreted approximately 75% of the packets, the TI CC2420 interpreted less than 10% and all the other devices were effectively thwarted. In the case of the second perforated preamble where only even-numbered symbols were present, all six device types found it difficult to interpret packets -the mean packet reception rates fell below 20% for all the transceivers. The two perforated preamble modifications were similar in composition, but they resulted in significantly different packet reception rates. The transceiver-specific responses to non-standard preambles reveal underlying differences in hardware implementations.

A promising research avenue stemming from these results is the potential for the remote identification of the make and model of ZigBee transceivers. Remote classification of transceiver models is essential for comprehensive cyber situational awareness, including identifying rogue devices and locating devices that are vulnerable to specific key extraction techniques.

Discussion

The packet reception rates suggest that interrogation packets with appropriately modified preambles can uniquely identify transceiver models based on the packet reply rates. This fingerprinting technique somewhat mirrors the network and transport layer based operating system fingerprinting techniques pioneered by Nmap. For example, based on the six device types investigated in this paper, an unknown ZigBee device that responds to packets with 4/8 preamble but not to packets with 2/8 preamble is most likely to be a Freescale MC13213 transceiver. Our future research will leverage these preliminary results to develop a remote ZigBee transceiver identification methodology to enhance cyber situational awareness.

An alternative strategy for identifying rogue devices and active eavesdroppers is to broadcast acknowledgement requests with preambles that are invisible to trusted hardware. An observed reply could only have originated from a device that is not a part of the trusted network. Such a device could then be tracked down and investigated.

Conclusions

ZigBee networks are widely used in the critical infrastructure, but an increasing number of tools are being developed to exploit and disrupt these networks. Under these circumstances, a rekeying technique that can effectively respond to suspected malicious ZigBee devices has considerable value. The rekeying technique proposed in this paper employs novel backward-compatible ZigBee message obfuscation based on preamble modifications.

Experiments involving the rekeying technique reveal that common wireless ZigBee sniffers can be thwarted with 100% effectiveness without reducing packet reception by transceiver models. In particular, specially-modified ZigBee messages sent to Freescale MC13213, TI CC2420 and Jennic JN5148 transceivers can be protected from interception by two popular sniffers and the XBee XBP24CZ7PIS, simply by shortening the IEEE 802.15.4 preamble by 50%. Packets with preambles reduced to 25% of the IEEE 802.15.4 specification completely thwart the Freescale MC13213. Alternatively, covert communications to Atmel AT86RF230 transceivers is possible using perforated preambles where only the odd-numbered O-QPSK symbols are present. When combined with effective rekeying strategies, message obfuscation can effectively complement network defense. An important advantage is that the message obfuscation is backward compatible with tens of millions of existing ZigBee devices and can be leveraged to defend key exchanges or any other sensitive message traffic in wireless networks.

Note that the views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, U.S. Department of Defense or the U.S. Government.

Figure 1 .

 1 Figure 1. ZigBee LR-WPAN topologies.

Figure 2 .

 2 Figure 2. Beacon requests with standard (top) and shortened preambles (bottom).

Figure 3 .

 3 Figure 3. Atmel RZUSBStick (left) and Microchip ZENA wireless adapter (right).

Figure 4 .

 4 Figure 4. Packet reception rate per device type (5/8 preamble).

Figure 5 .

 5 Figure 5. Packet reception rate per device type (4/8 or 12/8 preamble).

Figure 6 .

 6 Figure 6. Packet reception rate per device type (2/8 preamble).

Figure 7 .

 7 Figure 7. Beacon requests -odd (top) and even (bottom) preamble symbols.

Figure 8 .

 8 Figure 8. Packet reception rate per device type (odd preamble symbols).

Figure 9 .

 9 Figure 9. Packet reception rate per device type (even preamble symbols).

Acknowledgements

The authors wish to thank John M. Greenwell for his invaluable support with the test equipment.


Samy Skander Bahoura

HARNACK TYPE INEQUALITY FOR A NONLINEAR ELLIPTIC EQUATION

We give a sup × inf inequality for an elliptic equation.

INTRODUCTION AND MAIN RESULTS

We are on Riemannian manifold (M, g) of dimension n ≥ 3. In this paper we denote ∆ g = -∇ j (∇ j ) the Laplace-Beltrami operator and N = 2n n-2 . We consider the foolowing equation

∆ g u = V u N -1 + u α + µu, u > 0.
Where V is a function and α ∈] n n-2 , n+2 n-2 [, µ > 0.

For a, b, A > 0, α and µ > 0. We consider a sequence (u i , V i ) i of solutions of the previous equation.

Here we assume that:

0 < a ≤ V ≤ b < +∞, ||∇V || ∞ ≤ A.
Here we study some properties of this nonlinear elliptic equation. We try to find some estimates of type sup × inf. We denote by S g the scalar curvature.

There are many existence and compactness results which concern this type of equations, see for example . In particluar in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF], we can find some results about the Yamabe equation and the Prescribed scalar curvature equation. Many methods where used to solve these problems, as a variationnal approach and some other topological methods. Note that the problems come from the nonlinearity of the critical Sobolev exponent. We can find in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] some uniform estimates for various equations on the unit sphere or for the Monge-Ampere equation. Note that Tian and Siu proved uniform upper and lower bounds for the sup + inf for the Monge-Ampere equation under some condition on the Chern class, see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. In the case of the Scalar curvature equation and in dimension 2 Shafrir used the isoperimetric inequality of Alexandrov to prove an inequality of type sup + inf with only L ∞ assumption on the prescribed curvature, see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF]. The result of Shafrir is an extention of a result of Brezis and Merle, see [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] and later, Brezis-Li-Shafrir proved a sharp sup + inf inequality for the same equation with Lipschitzian assumption on the prescribed scalar curvature, see [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. Li in [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF] extend the previous last result to compact Riemannian surfaces. In the higher dimensional case, we can find in [START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF] a proof of the sup × inf inequality in the constant case for the scalar curvature equation on open set of R n . We have various estimates in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF] when we consider the nonconstant case. To prove our result, we use a blow-up analysis and the moving-plane method, based on the maximum principle and the Hopf Lemma as showed in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF][START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], and a condition on the scalar curvature is sufficient to prove the estimate.

Theorem 1.1. Assume Ricci ≡ 0 on M and µ > 0, then, for every compact K of M , there exist a positive constant c = c(α, µ, a, b, A, K, M, n, g) such that:

sup K u × inf M u ≤ c. 1
If we consider the Green function G of the Laplacian with Dirichlet condition on small balls of M , we can have a positive lower bound for G and we have the following corollary: Corollary 1.2. Assume Ricci, ≡ 0 on M , then, for every compact K of M , there exist a positive constant c ′ = c ′ (α, a, b, A, µ, K, M, n, g) such that:

K u 2n n-2 i dv g ≤ c ′ .

PROOF OF THE THEOREM.

Part I: The metric in polar coordinates.

Let (M, g) a Riemannian manifold. We note g x,ij the local expression of the metric g in the exponential map centered in x.

We are concerning by the polar coordinates expression of the metric. Using Gauss lemma, we can write:

g = ds 2 = dt 2 + g k ij (r, θ)dθ i dθ j = dt 2 + r 2 gk ij (r, θ)dθ i dθ j = g x,ij dx i dx j
, in a polar chart with origin x", ]0, ǫ 0 [×U k , with (U k , ψ) a chart of S n-1 . We can write the element volume:

dV g = r n-1 |g k |drdθ 1 . . . dθ n-1 = [det(g x,ij )]dx 1 . . . dx n , then, dV g = r n-1 [det(g x,ij )][exp x (rθ)]α k (θ)drdθ 1 . . . dθ n-1 ,
where, α k is such that, dσ Sn-1 = α k (θ)dθ 1 . . . dθ n-1 . (Riemannian volume element of the sphere in the chart (U k , ψ) ).

Then,

|g k | = α k (θ) [det(g x,ij )].
Clearly, we have the following proposition: Proposition 1: Let x 0 ∈ M , there exist ǫ 1 > 0 and if we reduce U k , we have:

|∂ r gk ij (x, r, θ)| + |∂ r ∂ θ m gk ij (x, r, θ)| ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . and, |∂ r |g k |(x, r, θ)| + ∂ r ∂ θ m |g k |(x, r, θ) ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . Remark: ∂ r [log |g k |] is a local function of θ,
and the restriction of the global function on the sphere

S n-1 , ∂ r [log det(g x,ij )]. We will note, J(x, r, θ) = det(g x,ij ).

Part II: The laplacian in polar coordinates

Let's write the laplacian in

[0, ǫ 1 ] × U k , -∆ = ∂ rr + n -1 r ∂ r + ∂ r [log |g k |]∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We have,

-∆ = ∂ rr + n -1 r ∂ r + ∂ r log J(x, r, θ)∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We write the laplacian ( radial and angular decomposition),

-∆ = ∂ rr + n -1 r ∂ r + ∂ r [log J(x, r, θ)]∂ r -∆ Sr(x) ,
where ∆ Sr (x) is the laplacian on the sphere S r (x).

We set L θ (x, r)(...) = r 2 ∆ Sr (x) (...)[exp x (rθ)], clearly, this operator is a laplacian on S n-1 for particular metric. We write,

L θ (x, r) = ∆ gx,r, S n-1 ,
and,

∆ = ∂ rr + n -1 r ∂ r + ∂ r [J(x, r, θ)]∂ r - 1 r 2 L θ (x, r). If, u is function on M , then, ū(r, θ) = u[exp x (rθ)]
is the corresponding function in polar coordinates centered in x. We have,

-∆u = ∂ rr ū + n -1 r ∂ r ū + ∂ r [J(x, r, θ)]∂ r ū - 1 r 2 L θ (x, r)ū.
Part III: "Blow-up" and "Moving-plane" methods

The "blow-up" technic Let, (u i ) i a sequence of functions on M such that,

∆u i = V i u N -1 i + u α i + µu i , u i > 0, N = 2n n -2 , (E)
We argue by contradiction and we suppose that sup × inf is not bounded. We assume that:

∀ c, R > 0 ∃ u c,R solution of (E) such that: R n-2 sup B(x0,R) u c,R × inf M u c,R ≥ c. (H) Proposition 2:
There exist a sequence of points (y i ) i , y i → x 0 and two sequences of positive real number

(l i ) i , (L i ) i , l i → 0, L i → +∞, such that if we consider v i (y) = u i [exp yi (y)] u i (y i )
, we have:

i) 0 < v i (y) ≤ β i ≤ 2 (n-2)/2 , β i → 1. ii) v i (y) → 1 1 + |y| 2 (n-2)/2
, uniformly on every compact set of R n .

iii) l

(n-2)/2 i [u i (y i )] × inf M u i → +∞ Proof:
Remark that, by using integration by part between u i and the first eigenfunction of a small ball (such that the ball is a smooth manifold with boundary), we have:

∃ c > 0, min M u i ≤ c, ∀ i.
Without loss of generality, we can assume that:

V (x 0 ) = n(n -2).
We use the hypothesis (H). We can take two sequences

R i > 0, R i → 0 and c i → +∞, such that, R i (n-2) sup B(x0,Ri) u i × inf M u i ≥ c i → +∞. Let, x i ∈ B(x 0 , R i ), such that sup B(x0,Ri) u i = u i (x i ) and s i (x) = [R i -d(x, x i )] (n-2)/2 u i (x), x ∈ B(x i , R i ). Then, x i → x 0 . We have, max B(xi,Ri) s i (x) = s i (y i ) ≥ s i (x i ) = R i (n-2)/2 u i (x i ) ≥ √ c i → +∞.
Set :

l i = R i -d(y i , x i ), ūi (y) = u i [exp yi (y)], v i (z) = u i [exp yi z/[u i (y i )] 2/(n-2) ] u i (y i ) .
Clearly, y i → x 0 . We obtain:

L i = l i (c i ) 1/2(n-2) [u i (y i )] 2/(n-2) = [s i (y i )] 2/(n-2) c 1/2(n-2) i ≥ c 1/(n-2) i c 1/2(n-2) i = c 1/2(n-2) i → +∞. If |z| ≤ L i , then y = exp yi [z/[u i (y i )] 2/(n-2) ] ∈ B(y i , δ i l i ) with δ i = 1 (c i ) 1/2(n-2) and d(y, y i ) < R i -d(y i , x i ), thus, d(y, x i ) < R i and, s i (y) ≤ s i (y i ), we can write, u i (y)[R i -d(y, y i )] (n-2)/2 ≤ u i (y i )(l i ) (n-2)/2 . But, d(y, y i ) ≤ δ i l i , R i > l i and R i -d(y, x i ) ≥ R i -d(x i , y i ) -δ i l i > l i -δ i l i = l i (1 -δ i ), hence, we obtain, 0 < v i (z) = u i (y) u i (y i ) ≤ l i l i (1 -δ i ) (n-2)/2 ≤ 2 (n-2)/2 .
We set,

β i = 1 1 -δ i (n-2)/2 , clearly β i → 1.
The function v i is solution of:

-g jk [exp yi (y)]∂ jk v i -∂ k g jk |g| [exp yi (y)]∂ j v i = 1 [u i (y i )] N -1-α v α i + µ [u i (y i )] 4/(n-2) v i +n(n-2)v i N -1 ,
By elliptic estimates and Ascoli, Ladyzenskaya theorems, (v i ) i converge uniformely on each compact to the function v solution on R n of,

∆v = n(n -2)v N -1 , v(0) = 1, 0 ≤ v ≤ 1 ≤ 2 (n-2)/2
, By using maximum principle, we have v > 0 on R n , the result of Caffarelli-Gidas-Spruck ( see [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF]) give, v(y

) = 1 1 + |y| 2 (n-2)/2
. We have the same properties for v i in the previous paper [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF].

Because R i ≥ l i we have u i (y i ) ≥ u i (x i ) using the fact that s i (y i ) ≥ s i (x i ) we obtain:

l (n-2)/2 i u i (y i ) × inf M u i → +∞.
Remark: we can replace l i by R i to have the last assertion of the proposition (our computations do not change):

R (n-2)/2 i u i (y i ) × inf M u i → +∞,

Polar coordinates and "moving-plane" method

Let, w i (t, θ) = e (n-2)/2 ūi (e t , θ) = e (n-2)t/2 u i o exp yi (e t θ), et a(y i , t, θ) = log J(y i , e t , θ).

We set δ = (n + 2) -(n -2)α 2 .

Lemma 1:

The function w i is solution of:

-∂ tt w i -∂ t a∂ t w i -L θ (y i , e t ) + cw i = V i w N -1 i + e δt w α i + µe 2t w i , with, c = c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a.

Proof:

We write:

∂ t w i = e nt/2 ∂ r ūi + n -2 2 w i , ∂ tt w i = e (n+2)t/2 ∂ rr ūi + n -1 e t ∂ r ūi + n -2 2 2 w i . ∂ t a = e t ∂ r log J(y i , e t , θ), ∂ t a∂ t w i = e (n+2)t/2 [∂ r log J∂ r ūi ] + n -2 2 ∂ t aw i .
the lemma is proved.

Now we have, ∂ t a = ∂ t b 1 b 1 , b 1 (y i , t, θ) = J(y i , e t , θ) > 0,
We can write,

- 1 √ b 1 ∂ tt ( b 1 w i ) -L θ (y i , e t )w i + [c(t) + b -1/2 1 b 2 (t, θ)]w i = n(n -2)w i N -1 , where, b 2 (t, θ) = ∂ tt ( √ b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 . Let, wi = b 1 w i ,

Lemma 2:

The function wi is solution of:

-∂ tt wi + ∆ g y i ,e t , S n-1 ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) + (c + b -1/2 1 b 2 -c 2 ) wi = = V i 1 b 1 (N -2)/2 wN-1 i + e δt 1 b 1 (α-1)/2 wα i + µe 2t wi ,
where, c 2 = [ Proof:

We have:

-∂ tt wi -b 1 ∆ g y i ,e t , S n-1 w i + (c + b 2 ) wi = V i 1 b 1 (N -2)/2 wN-1 i + +e δt 1 b 1 (α-1)/2 wα i + µe 2t wi , But, ∆ g y i ,e t , S n-1 ( b 1 w i ) = b 1 ∆ g y i ,e t , S n-1 w i -2∇ θ w i .∇ θ b 1 + w i ∆ g y i ,e t , S n-1 ( b 1 ),
and,

∇ θ ( b 1 w i ) = w i ∇ θ b 1 + b 1 ∇ θ w i , we deduce than, b 1 ∆ g y i ,e t , S n-1 w i = ∆ g y i ,e t , S n-1 ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) -c 2 wi , with c 2 = [ 1 √ b 1 ∆ g y i ,e t , S n-1 ( √ b 1 ) + |∇ θ log( √ b 1 )| 2 ].
The lemma is proved.

The "moving-plane" method:

Let ξ i a real number, and suppose ξ i ≤ t. We set t ξi = 2ξ i -t and wξi i (t, θ) = wi (t ξi , θ).

We have,

-∂ tt wξi i +∆ g y i ,e t ξ i S n-1 ( wi )+2∇ θ ( wξi i ).∇ θ log( b 1 ) wξi i +[c(t ξi )+b -1/2 1 (t ξi , .)b 2 (t ξi )-c ξi 2 ] wξi i = = V i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 + +e δt 1 b 1 (α-1)/2 wα i + µe 2t wi ,
By using the same arguments than in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have:

Proposition 3:
We have:

1) wi (λ i , θ) -wi (λ i + 4, θ) ≥ k > 0, ∀ θ ∈ S n-1 .
For all β > 0, there exist c β > 0 such that:

2) 1 c β e (n-2)t/2 ≤ wi (λ i + t, θ) ≤ c β e (n-2)t/2 , ∀ t ≤ β, ∀ θ ∈ S n-1 .
We set, Zi = -∂ tt (...) + ∆ g y i ,e t , S n-1 (...) + 2∇ θ (...).∇ θ log( b 1 ) + (c + b

-1/2 1 b 2 -c 2 )(...)
Remark: In the operator Zi , by using the proposition 3, the coeficient c + b

-1/2 1 b 2 -c 2 satisfies: c + b -1/2 1 b 2 -c 2 ≥ k ′ > 0, pour t << 0,
it is fundamental if we want to apply Hopf maximum principle.

We set δ = (n + 2) -(n -2)α 2 .

Goal:

Like in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have elliptic second order operator. Here it is Zi , the goal is to use the "movingplane" method to have a contradiction. For this, we must have:

Zi ( wξi i -wi ) ≤ 0,
-[(c + b -1/2 1 b 2 -c 2 ) ξi -(c + b -1/2 1 b 2 -c 2 )] wξi i + +V ξi i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 -V i 1 b 1 (N -2)/2 wN-1 i + +e δt ξ i b ξi 1 (1-α)/2 ( wξi i ) α -e δt b
(1-α)/2 1

( wi ) α + µe 2t ξ i wξi i -e 2t wi ( * * * 1)

Clearly, we have:

Lemma 3 : b 1 (y i , t, θ) = 1 - 1 3
Ricci yi (θ, θ)e 2t + . . . , R g (e t θ) = R g (y i )+ < ∇R g (y i )|θ > e t + . . . . According to proposition 1 and lemma 3, Propostion 4 :

Zi ( wξi i -wi ) ≤ Ã(e t -e t ξ )( wξi i ) N -1 ) + (1/2)(e δt ξ i -e δt )(w ξi i ) α + +C|e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| + o(1)[ wξi i + ( wξi i ) N -1 ] + µ wξi i + C ′ |e 3t ξ i -e 3t |.
Proof:

We use proposition 1, we have:

a(y i , t, θ) = log J(y i , e t , θ) = log b 1 , |∂ t b 1 (t)| + |∂ tt b 1 (t)| + |∂ tt a(t)| ≤ Ce 2t ,
and,

|∂ θj b 1 | + |∂ θj,θ k b 1 | + ∂ t,θj b 1 | + |∂ t,θj,θ k b 1 | ≤ Ce 2t , then, |∂ t b 1 (t ξi ) -∂ t b 1 (t)| ≤ C ′ |e 2t -e 2t ξ i |, on ] -∞, log ǫ 1 ] × S n-1 , ∀ x ∈ B(x 0 , ǫ 1 )
Locally,

∆ g y i ,e t , S n-1 = L θ (y i , e t ) = - 1 |g k (e t , θ)| ∂ θ l [g θ l θ j (e t , θ) |g k (e t , θ)|∂ θ j ].
Thus, in [0, ǫ 1 ] × U k , we have,

A i =   1 |g k | ∂ θ l (g θ l θ j |g k |∂ θ j ) ξi - 1 |g k | ∂ θ l (g θ l θ j |g k |∂ θ j )   ( wξi i ) then, A i = B i + D i with,
B i = gθ l θ j (e t ξ i , θ) -gθ l θ j (e t , θ) ∂ θ l θ j wξi i (t, θ), and,

D i = 1 |g k |(e t ξ i , θ) ∂ θ l [g θ l θ j (e t ξ i , θ) |g k |(e t ξ i , θ)] - 1
|g k |(e t , θ) ∂ θ l [g θ l θ j (e t , θ) |g k |(e t , θ)] ∂ θ j wξi i (t, θ),
we deduce,

A i ≤ C k |e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| , We take C = max{C i , 1 ≤ i ≤ q}
and if we use ( * * * 1), we obtain proposition 4.

We have:

∂ θj w λ i (t, θ) w λ i = e (n-2)[(λ-λi)+(λ-t)]/2 e [(λ-λi)+(λ-t)] (∂ θj v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci ,
and,

∂ θj ,θ l w λ i (t, θ) w λ i = e (n-2)[(λ-λi)+(λ-t)]/2 e 2[(λ-λi)+(λ-t)] (∂ θj ,θ l v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci ,
with Ci tending to 0 and does not depend on λ ≤ λ i + 2.

We have,

c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a + R g e 2t , ( α
|∂ t c 2 | + |∂ t b 1 | + |∂ t b 2 | + |∂ t c| ≤ K 1 e 2t .
We have:

w i (2ξ i -t, θ) = w i [(ξ i -t + ξ i -λ i -2) + (λ i + 2)],
Thus, w i (2ξ i -t, θ) = e [(n-2)(ξi-t+ξi-λi-2)]/2 e n-2 v i [θe 2 e (ξi-t)+(ξi-λi-2) ] ≤ 2 (n-2)/2 e n-2 = c.

We set δ = (n + 2) -(n -2)α 2 . The left right side are denoted Z 1 et Z 2 , we can write:

Z 1 = ( V ξi i -Vi )(w ξi i ) N -1 + Vi [(w ξi i ) N -1 -w i N -1 ],
and, Z 2 = e δt [(w ξi i ) α -w α i ] + (w ξi i ) α (e δt ξ i -e δt ). We can write the part with nonlinear terms as: (w ξi i ) α [(A w ξi i N -1-α + B) (e t -e t ξ i ) + c (e δt ξ i -e δt )].

Because w ξi i ≤ c, we have:

-Zi (w ξi i -w i ) ≤ (w ξi i ) α [(Ac N -1-α + B) (e t -e t ξ i ) + c (e δt ξ i -e δt )] + w ξi i (µ/2)(e 2t ξ i -e 2t )]

But α ∈] n n -2 , n + 2 n -2 [, δ = n + 2 -(n -2)α 2 ∈]0, 1[.
We obtain for t ≤ t 0 < 0: e t ≤ e (1-δ)t0 e δt , pour tout t ≤ t 0 . and, t ξi ≤ t (ξ i ≤ t), we integrate:

(e δt ξ i -e δt ) ≤ δ e (1-δ)t0 (e t ξ i -e t ). Finaly:

1 ) b 2 √ b 1 ∂ 2 ) 1 ∆

 12121 (t, θ) = ∂ tt ( b 1 ) = 1 2 tt b 1 -1 4(b 1 ) 3/2 (∂ t b 1 ) 2 , (α g y i ,e t , S n-1 ( b 1 ) + |∇ θ log( b 1 )| 2 ], (α 3 )Then,∂ t c(y i , t, θ) = (n -2) 2 ∂ tt a,by proposition 1,

  θ,e t ξ i -∇ θ,e t )(w ξi i ).∇ θ,e t ξ i log( b ξi 1 ) + 2∇ θ,e t ( wξi i ).∇ θ,e t ξ i [log( b ξi 1 ) -log b 1 ]+ +2∇ θ,e t w ξi i .(∇ θ,e t ξ i -∇ θ,e t ) log b 1

	We write:	if wξi i -wi ≤ 0.
		Zi ( wξi i -wi ) = (∆ g y i ,e t ξ i , S n-1	-∆ g y i ,e t , S n-1	)( wξi i )+
	+2(∇

We apply proposition 3. We take t i = log √ l i with l i like in proposition 2. The fact

We define ξ i by:

If we use proposition 4 and the similar method that in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF] we can deduce by Hopf maximum principle,

It is in contradiction with proposition 2.

Then we have,