Nicolai Kuntze

Carsten Rudolph

On the secure distribution of vendor-specific keys in deployment scenarios

Product counterfeit is a tremendous challenge for vendors in many areas. Particularly important is a prevention of product counterfeit where products like telecommunication devices interact with other systems and thus a malfunctioning of a single device can jeopardize the complete system. This can also deteriorate the reputation of the vendor. Furthermore, violation of intellectual properties can cause financial losses. Detection of product counterfeit can be based on tracking back each device to the production process of the vendor to ensure the product origin. Devices without a verified source can then be considered counterfeit with a high potential to be malicious or of low quality. Vendors already apply vendor-specific security technologies protecting the distribution. These often employ special hardware-based security mechanisms specifically designed for a particular range of products. This publication shows the usage of the already available Trusted Platform Module to allow for distribution channel protection and to leverage overall security by allowing the secure identification of a specific device. It also explains a few additional Trusted Platform Module functionalities that can be used.

Introduction

Today's international collaboration and outsourcing of production processes results in a separation of device design and device production. One example is the production of network equipment such as routers or switches. The vendor controls the design but other companies (the manufacturers) are subcontracted to actually produce and assemble the devices. Thus, device blueprints and production is not under the direct control of the vendor, but controlled by the manufacturer. One core concern of the vendor is to be able to control that no additional devices are produced and sold bypassing the vendor. The manufacturer could in principle produce more devices and directly bring them to the market. Thus, the manufacturer would diminish the revenue of the vendor. Even worse, clients might believe they buy original products and request support and maintenance from the vendor. This creates additional costs.

Furthermore, outsourcing the production and other steps in the actual creation of the final product results in perils to the vendor's business and to the end users [START_REF] Stradley | The electronic part supply chain and risks of counterfeit parts in defense applications[END_REF]. The manufacturerers have the knowledge and technology to also build counterfeit products that cannot be easily distinguished from the original devices, but may have less quality, do not comply to the specification, or have back-doors and other malicious software installed. Counterfeit devices with low quality or malicious behaviour also cause a loss of reputation and endanger the brand experience if they cannot be reliably recognized.

The vendor is in a paradoxical situation. All information on how to build the device needs to be given to the manufacturer, while at the same time the vendor wants to keep control on the number and properties of devices being built and sold under the vendor's name. One possible approach to this catch-22 situation is to create individual registrations of each single device and track devices from manufacturer to the customer. The identification of each single device can be based on individual vendor-specific keys to be inserted into device or on a subsequent registration process using a vendor key not known to the manufacturer. Such a key needs to be protected against duplication. One aspect hereby is that a vendor specific key is only issued to devices after production and final tests to restrict keys to devices being sold. However this would require each device to be inspected by the vendor in a controlled environment, which adds a lot of additional costs.

The state of the art is based on vendor-specific or commercial off-the-shelf hardware tokens providing random-number generation and asymmetric cryptography. These tokens are developed by security hardware vendors and included by the contracted manufacturer into the devices. Tokens can be pre-equipped with unique keys, keys can be generated within the token or the vendor can directly inject a device identity into this hardware token. Through this process each device can be uniquely identified and the vendor can track these devices without requiring direct access to the complete final device in a controlled environment.

By controlling the hardware tokens, the vendor also keeps control over the number and identity of devices being shipped or getting into the market. This approach can provide some protection against counterfeit products, but it also has some disadvantages. First, an additional piece of hardware needs to be included that has no other benefit in the product. These hardware tokens are in many cases single purpose chips and in comparison to a multi-purpose commercial off the shelf chip relatively expensive due to smaller production numbers and additional overhead for logistics. Further, the process for establishing keys creates additional overhead, because the used security hardware does not support protocols suitable for a zero-touch registration and initialization. Another option is to use radio frequency identification (RFID) tags to create unique device identities (see [START_REF] Devadas | Design and implementation of puf-based "unclonable" rfid ics for anti-counterfeiting and security applications[END_REF]). The drawback with a RFID based solution is that any kind of (remote) enforcement is not applicable and that it requires RFID readers at various places in the value creation chain. It might be more efficient to use general purpose security hardware in the devices that also provides more generic security functionality useful for all other parts of the device's life-cycle.

Security industry has developed various approaches for hardware security anchors. Examples include TrustZone, SmartCards or the Trusted Platform Module (TPM) as specified by the TCG. The value of TrustZone strongly depends on additional security functions provided by the platform, while SmartCards are usually not integrated into a platform but considered to be removable devices suitable e.g. for user authentication or digital signatures. In contrast to this, the Trusted Platform Module is a self-contained chip integrated into the device which offers an interesting basis to implement protection schemes suitable for industry. It provides a hardware-protected identity, various security functionalities and the possibility to establish device-specific keys that cannot be extracted or migrated to other devices. The goal of this paper is to present an approach to use one type of such generic commercial of the shelf security hardware, namely the TPM, to satisfy the industry demand for IP protection and control on logistic processes. Further, this publication will show additional benefits when using the TPM for authentication-only approaches.

This paper presents a concept for the establishment of vendor specific cryptographic keys using the Trusted Platform Module. In the remainder of the paper a more detailed scenario and first requirements towards the distribution process are defined in Section 2. Section 3 presents necessary basic concepts of the TPM for the context of this use case. Based on the requirements and TPM concepts a solution architecture is detailed in Section 4, followed by an analysis in Section 5. The publication concludes in Section 6 showing advanced uses for strengthening the customer's experience as an additional use of the security functionality.

Scenario

Given the setting of a vendor (i.e. the equipment manufacturer) outsourcing production to the actual manufacturer (i.e. the assembler) as described above, this section provides a structured and more detailed analysis of the addressed scenario. To counter product counterfeit an understanding of the involved stake-holders and their interactions in the course of the production of equipment is required.

Stake-holders

The value creation chain of a product involves mainly Original Equipment Manufacturer (OEM ), Device Assembler, Distribution Partner and End User. The role of the OEM differs in various business cases which impacts the counterfeit techniques suitable. In more detail, the workflow of producing a device can involve the following stake-holders:

OEM represents the publicly know vendor of the device. The OEM often creates the design, owns the intellectual property, and provides product liability and device support to the End User. Actual production of the equipment is very often outsourced to contractors to allow the OEM to concentrate on development and marketing and to reduce production costs. Device Assembler represents the contractor of the OEM that produces and/or assembles the hardware device. This may actually be several companies not even in the same country (and not under the same jurisdiction) as the OEM. Distribution Partner receives the devices directly from the Device Assembler in the name of the OEM and sells the device to the End User. Devices in stock are not handled by the OEM but by subcontractors.

End User buys and uses the device which is perceived as a product of OEM.

In addition to these stake-holders, the secure distribution of devices introduces additional security-critical stake-holders to the described scenario. For the proposed solution, the TPM Supplier is a vendor that produces and sells TPM hardware modules. The TPM can come with a security certification and pre-established key so-called Endorsement Key that also should be certified by the TPM Supplier.

Given the security critical focus of these hardware chips, these vendors need to be trustworthy to the OEM and usually already have implemented secure production arrangements.

Functional Production Workflow Ignoring the steps for anti-counterfeit solutions, the following workflow describes the production of a device from a functional perspective. The OEM designs a new device using its resources and intellectual property. Then the OEM contracts a Device Assembler to actually build a larger number of devices. This includes committing the blueprints of the new design and giving all information necessary to build the device to the Device Assembler. The Device Assembler produces devices according to the blueprints of the OEM 's design and might provide evidence, test results, production logs, etc. to the OEM. The devices are delivered from the Device Assembler 's site directly to the Distribution Partner. The Distribution Partner sell and ship the final device to the End User who then can register the device with the OEM.

Security Analysis Given the above work flow, obviously, by contracting of the Device Assembler, the OEM loses control of the production process. In this step, the OEM provides the Device Assembler with all blueprints necessary to produce the device. After this step, the OEM has no direct control over the amount of devices produced or sold. It completely relies upon the Device Assembler 's and/or Distribution Partner ' accounting of devices shipped. The Device Assembler possibly colluding with one of the Distribution Partner has various options to cheat in this process.

Threats The Device Assembler possesses the required blueprints and capabilities to manufacture devices. It might sell these to counterfeit producers or itself produce and sell more devices than it accounts for with the OEM. An involvement of Distribution Partner and the requirement for call for bids further increase potential for counterfeits purchases by End User. Neither End User nor OEM would be able to reliably recognize counterfeits (beyond potential alterations in production blueprints). Thus, several types of malicious actions by the Device Assembler can be distinguished. (i) Device Assembler builds a higher number of devices in accordance with the blueprint of the OEM. These devices cannot be distinguished from original devices. The only difference is, that the OEM does not get any revenue from these devices and the Device Assembler gets a much bigger share. Clearly, the OEM loses revenue and at the same time is responsible for maintenance, warranty, etc. (ii) Device Assembler builds a higher number of the same devices, but sells them under a different brand name, stealing the OEM 's IPR. (iii) Device Assembler builds devices that are of lower quality, use cheaper hardware elements or have other (possibly intentionally malicious) changes to increase its revenue. This decreases the OEM 's revenue and also creates additional risks for the End User using the devices.

Security Requirements The threats described above yield the following security requirements. (i) The OEM needs to be able to identify non-counterfeit products and to track all devices sold with the OEM 's brand. (ii) Counterfeit devices must be distinguishable to the OEM and (depending on the type of device), remote detection of deployed counterfeit products can be necessary. (iii) End User need to be able to distinguish counterfeits from original products. This might require the help of the OEM. (iv) The End User should to be able to recognize counterfeits using an offline scheme without active work of the OEM.

Trusted Computing Basics

According to the mission statement of the Trusted Computing Group (TCG), Trusted Computing based on hardware roots of trust has been developed by industry to protect computing infrastructure and end points. The TPM provides the core security functions and serves as a root of trust for each individual device.

TCG defines the currently used and market available TPM in version 1.2 but also provides a specification for the next generation of TPMs in form of the version 2.0. Both versions of the TPM design share underlying principles regarding the functionality and functions provided. The TPM is regarded as a trust anchor bound to an individual system. Trust is defined within the TCG to convey an expectation of behaviour. Hereby it needs to be emphasized that a predictable behaviour does not constitute behaviour that is secure or worth to be trusted. However, various security properties for platforms can be satisfied and controlled based on the TPM. For example, to determine the trust in a certain platform, it is required to identify the identity of the platform. The TPM provides a unique identity for a platform which can either used to directly identify a specific platform or provide for pseudonymous identification. The next sections give a more detailed presentation of how a TPM is used to implement its dedicated role as a root of trust. Specifically, it is shown how different roots of trust in a system design complement each other to build platforms with particular security properties. A more detailed description and also explanation of the TPM commands used can be found in the book by Chris Mitchell [START_REF] Mitchell | Trusted computing[END_REF].

Roots of Trust

The high level concept of Trusted Computing as defined by the TCG introduces different roots of trust in the system design providing complementary security functions (see also [START_REF] Parno | Bootstrapping trust in commodity computers[END_REF]). To attest on the health of a system each software component needs to be measured beginning from the initialization of the device. In the reference design according to TCG, the initial start of a system begins with the Root of Trust for Measurement (RTM). A RTM measures itself and is implemented using platform features to ensure the tamper resistance of the RTM. The RTM measures the next stage in the boot process and transfers control to it. By this, the RTM already behaves according to the measure before execute approach underpinning the overall measurement of the boot process and of software subsequently started on the system. The second component is the Root of Trust for Storage (RTS), which is usually implemented by protected non-volatile storage within the TPM hardware. The RTS mainly has three roles. First it needs to provide secure storage for the cryptographic identity of the TPM. Second, it also provides secure storage for the keys that are used to encrypt data to be securely stored on (insecure) storage media on the platform or outside the platform (e.g. symmetric keys for bulk encryption). Finally, the third role includes particular protected registers used to store the measurement information as hash-values (i.e. chains of hash values represented by the final value in the chain.). These registers are called Platform Configuration Registers (PCR). The final root of trust is a securely stored cryptographic key (Endorsement Key EK), the Root of Trust for Reporting (RTR). This key is used to create so called Attestation Identity Keys (AIKs) that are then used to digitally sign PCR content in a TPM Quote and to certify non-migrateable keys generated by the TPM. AIKs can be used to identify a platform with different levels of pseudonymity [START_REF] Brickell | Direct anonymous attestation[END_REF].

Basic Trusted Platform Features

The TPM provides a large set of security functionalities. The secure distribution of vendor-specific keys in deployment scenarios builds on and uses authentication, attestation and protected location functionalities. Authentication can be easily based on the Endorsement Key (EK) that is either established in the TPM by the producer of the TPM or implanted to the TPM in a later stage of the production process. Together with a certificate for the EK, this key is used to build subsequent authentication processes. Attestation in this context describes the process of reliably reporting the platform status. The TPM provides the functionality for secure remote attestation. Thus, remote entities can get digitally signed information on the current content of PCRs. Protocols for remote attestation are defined in the TCG standards. Protected Location for keys and other data transferred to the platform is provided by the TPM. In the TPM context, the process of encrypting data with a key protected by the TPM and binding this encrypted data to a particular state of the platform (i.e. particular PCR values) is called Sealing. Thus, sealed data can only be decrypted when two conditions are satisfied. First, the same TPM needs to be used with the correct key loaded to the TPM (and optional the correct authentication value/password for the key is given) and second, the platform is in the correct state that the PCR values match.

Vendor specific key establishment

As described in the previous section, TPMs provide a different kind of unique identities. These identities allow to establish secure channels to the TPMs or respectively a given device. These unique identities are implemented as cryptographic keys that are protected by hardware inside the TPMs. This ensures a higher protection level compared to solely software based solutions.

Based on the concepts of the TPM, there are several ways to establish vendor and end user specific identification keys. These keys may be generated externally and injected into the TPM, where they are stored securely, or the TPM's internal Random Number Generator and Key Generator can be used to create keys and their binding to a certain TPM can be validated. Both of these schemes are presented in this section using different characteristics. The logistics processes involved with each of these approaches are also outlined.

TPM Generated versus Vendor Injected Keys

The usage of the Endorsement Key is restricted to only very few operations. In particular, it cannot be used for digital signatures at all and for decryption only during a very constrained set of operations. These restrictions are in place, in order to use the TPM in scenarios (such as Consumer Electronics) where a unique identification of devices is not desired. However, TPMs provide additional means to establish context related identities that can be used for the given scenario of counterfeit protection. Generally, vendor established keys can be based on either of two approaches -Generated externally and injected into a device / TPM or generated by the TPM and recorded / certified in a secure way.

Key Injection

In the case of key injection, the OEM or End User create keys to be used as (context related) identification externally and inject them into the TPM. After the key injection, the keys are protected by the TPM and can be utilized as a primary identity credential for a given device. Based on these keys, the device can proof its origin and identity. This can also be utilized in other technical as well as organizational processes. To ease the process, the OEM or End User may issue certificates for these injected keys.

TPM created Keys

The TPM has the ability to create keys on its own, using a strong random number generator. These keys are then protected by the TPM chip, such that the private portion is never known or usable outside of the specific TPM that generated it. In order to achieve an authentication against the OEM or End User, the public portions of these keys may be read out in a protected environment and stored in a database or the OEM or End User may issue corresponding certificates. Furthermore, it is possible to certify the origin of these keys using the TPM's functionalities themselves based upon previously established identity keys. An establishment of TPM generated keys is even possible remotely without a protected environment or channel, given that an identity relation has been established before using another key.

Local Establishment of a TPM 1.2

The workflow for the local establishment of vendor specific key material with a certain TPM works similarly to the deployment of custom security tokens. The OEM or TPM Supplier generates key material and injects it into the security chip (in this case the TPM) in a protected environment. The public portion is stored for identification and the private portion is usually deleted, such that it only exists inside the TPM.

Logistics The logistics process of physical objects (focused on the TPM chip) in this approach acts are as follows. (i) The TPM Supplier produces a TPM (optionally without an Endorsement Key (EK) and EK certificate). (ii) The TPM chip is delivered to a protected environment at the TPM Supplier or the OEM facilities. (iii) Local, in the protected environment, keys are created by or injected into the TPM and their public portions recorded in the OEM 's database. (iv) The TPM chip is delivered to the Device Assembler. (v) The TPM chip is assembled into the target device. (vi) The target device is shipped and distributed. (vii) Remotely, at any time after assembly (from testing at the Device Assembler to the End User ), additional keys can be created by the TPM or injected into the TPM using the previously established keys.

Identity Establishment

In this usage scenario an establishment of the TPM provided identity means is not necessary, since it is used in a protected environment. The logistics process and trustworthiness between the TPM Supplier and the OEM need to ensure that a specification conform TPM is used. The TPM needs to be owned (using TPM TakeOwnership) in order to enable its Root of Trust for Storage, which also activates the ownership of the EK for identity reporting. The End User may change the credentials for identity usage after deployment without invalidating the keys. The only constraint is that the ownership may not be cleared, since this would invalidate the vendor supplied keys.

Local Injection of Key Whilst the TPM is in the protected environment at the TPM Supplier or OEM facilities, it is possible to inject vendor specific keys (for storage, signing and binding; not for identity) into the TPM key hierarchy. To do so, the TPM Supplier or OEM generate a new key-pair and construct a MigrationBlob that is targeted at the TPM's Storage Root Key (SRK) using TPM ConvertMigrationBlob for importing. In order to do so, the TPM must have been taken into ownership before (the SRK's authorization value can be set to the "well-known secret" as defined by the TCG). This process can be repeated several times. Also, if one of the newly created keys is a storage key, this may be used for further key injections as parent. The public portions of these keys (or their fingerprint) are securely saved in the OEM 's database for established keys, such that they can be validated in the future. Alternatively, the TPM Supplier or OEM may issue according certificates to be shipped with the TPM and later stored on the assembled device. The private portions can be deleted, as they are wrapped in TPM bound key blobs. These wrapped key blobs must be shipped with the TPM to the Device Assembler to be accessible inside the assembled device (the (limited) TPM non-volatile storage area can be used for this).

Local TPM-based Key Creation Whilst the TPM is in the protected environment, it is possible to let the TPM generate new keys using its internal Random Number Generator (using TPM CreateWrapKey or TPM MakeIdentity) for use as vendor specific keys (for storage, signing, binding as well as identity and sealing). To do so, the TPM must have been taken into ownership before. The SRK is one of these TPM generated keys and can be used as vendor specific storage key itself (without having to be stored in an additional wrap key blob). The public portions of these TPM generated keys can be read (from the public portion of the KeyStorageBlobs) and must be stored in the OEM 's database for established keys or an according certificate must be shipped. The private portions are only known to this TPM and are wrapped in TPM bound key blobs. These wrapped key blobs must be shipped with the TPM to the Device Assembler (except for the SRK; the TPM's NV storage area can be used for the other keys).

Remote TPM-based Key Generation or Injection after Assembly If one of the keys that were created in the protected environment is a signing key, it is possible to remotely create new keys and use these vendor specific signing keys to certify (using TPM CertifyKey) that the newly created keys were generated inside the same TPM and the signing key. This is similar to the Remote Establishment of a TPM 1.2. If one of the keys that were created in the protected environment is a storage key, it is possible to remotely inject new keys and use the vendor specific storage key as target for a MigrationBlob containing this newly created key via the TPM ConvertMigrationBlob. (This is similar to the Remote Establishment of a TPM 1.2) Both of these processes can happen while the device is at the Device Assembler, Distribution Partner or even End User.

Remote Establishment of a TPM 1.2

The workflow for the Remote Establishment of vendor specific key material in a TPM 1.2 does not require the existence of a protected environment under the control of the OEM. Instead, TPM chips are shipped directly to the Device Assembler and the key material is established after device assembly (potentially during a testing and registration phase). It does however require and active network session from the newly assembled device to the OEM.

Logistics The logistics process of physical objects (focused on the TPM chip) in this approach acts are as follows. (i) The TPM Supplier produces a TPM (typically shipped with an Endorsement Key (EK) and EK certificate). (ii) The TPM chip is directly delivered to the Device Assembler. (iii) The TPM chip is assembled into the target device. (iv) Remotely, keys are created by or injected into the TPM by the OEM and their public portions recorded in the OEM 's database. (v) The target device is shipped and distributed. (vi) Remotely, at any time after assembly (from testing at the Device Assembler to the End User ), additional keys can be created by the TPM or injected into the TPM using the previously established keys.

Identity Establishment

The approach requires an establishment of a (pseudonym) identity inside the TPM. This can be done with standard TPMs that provide an EK certificate via several approaches (such as [START_REF] Kuntze | On the automatic establishment of security relations for devices[END_REF]) or the OEM may have requested certain custom identity establishment from the TPM Supplier. This identity only needs to account that it actually belongs to a TPM and is free of collisions. The counting of TPM identities is used to cross check the Device Assembler 's production accountings. As a result of this step, there exists an Attestation Identity Key with the TPM that can be used for further operations.

TPM-based Key Creation

It is now possible to remotely use the TPM to generate key pairs (via TPM CreateWrapKey and TPM MakeIdentity) using the TPM's internal Random Number Generator (for storage, signing, binding, identity and sealing). Then the Identity Key can be used to certify (via TPM CertifyKey) that the newly generated key was actually generated by the same TPM that the Identity Key is bound to. From the certificate, the OEM may store the public portion (or the fingerprint) of the certified key in the database for established vendor keys or issue a certificate to be shipped with the device. This process may also be performed again after sales of the device at the End User, even with newly created identity keys instead of the previously established Identity Key. The original Identity Key can even be deleted.

Injection of Key

It is also possible to remotely inject keys generated by the OEM (for storage, signing and binding; not for identity). To do so, the OEM generates a key pair and creates a MigrationBlob using one of the certified TPM residing key. This may also include using the Storage Root Key directly. These MigrationBlobs can be imported into the TPM key hierarchy using TPM ConvertMigrationsBlob and serve as additional vendor specific keys. The OEM would save the public portions of these key pairs in its database and delete the private portions, that would be saved in wrapped key blobs on the device. This process may also be performed again after sales of the device at the End User, even with newly created identity keys (must be TPM-based Created) instead of the previously established Identity Key. The original Identity Key can even be deleted.

Local Establishment of a TPM 2.0

With the specification of TPM 2.0 an additional key hierarchy has been introduced that can be used by OEM s without interfering with End User ' use of the TPM. This key hierarchy is called platform hierarchy. With the OEM being the designer and administrator of the platform, this key hierarchy is exclusively used by him. The presented approach again requires a protected environment for secure establishment of some keys. Additional keys may be introduced in the future. We do not provide a remote establishment here as the necessary identity establishment is out of scope for this contribution.

Logistics The logistics process of physical objects (focused on the TPM chip) in this approach acts are as follows. (i) The TPM Supplier produces a TPM 2.0 (optionally without an Endorsement Key (EK) and EK certificate). (ii) The TPM chip is delivered to a protected environment at the TPM Supplier or the OEM facilities. (iii) Local, in the protected environment, a Platform Hierarchy primary key is generated in the TPM the public portion recorded in the OEM 's database. (iv) Local, in the protected environment, any number of additional keys in the Platform Hierarchy can be created and their public portions recorded in the OEM 's database. (v) The TPM chip is delivered to the Device Assembler. (vi) The TPM chip is assembled into the target device. (vii) The target device is shipped and distributed. (viii) Remotely, at any time after assembly (from testing at the Device Assembler to the End User ), additional keys can be created by the TPM or injected into the TPM using the previously established keys.

Identity Establishment In this scenario, an establishment of TPM identity is not necessary, since it is operated in a protected environment. In the protected environment at the site of the TPM Supplier or OEM the primary key for the Platform Hierarchy is created using TPM2 CreatePrimary. This key serves as a root key for this hierarchy and protects all subsequently created OEM keys. The OEM reads and stored the public portion of this key in its database via TPM2 ReadPublic. This can be used for the remote key creation in the future.

Injection of Key Under the Platform Hierarchy the OEM can import any number of keys for different purposes using TPM2 LoadExternal and TPM2 Import. In contrast to TPM 1.2 it is possible to also assign advanced usage policies to these keys. It is now possible to import symmetric keys into the TPM.

TPM-based Key Creation Under the Platform Hierarchy, the OEM can create any number of keys for different purposes using TPM2 Create. It can then read out the public keys inside the TPM and store it in the database for vendor keys.

Injection of Keys remotely Using TPM2 LoadExternal and TPM2 Import of the TPM 2.0, it is possible to encrypt a new key with an established key. This way, it is possible to ensure that only the TPM that was targeted can decrypt and use a newly generated key. This allows the OEM to inject additional keys into the TPM even after assembly, when the TPM is not in the protected environment.

TPM-based Key Creation remotely

The TPMs Key Generation capabilities can be used remotelyusing TPM2 Create. The TPM is instructed to generate a new key under the Platform Hierarchy. Then an established storage key is used with the concept of TPM2 ActivateCredential to attest that the newly created key belongs to the same TPM as the previously established key.

Analysis

The OEM now has the ability to establish vendor specific keys with its devices that can be used for many applications. This section revisits the requirements and analyses their fulfilment. Further, it discusses additional functionalities of TPMs that can be utilized and analyses potential residual threats in this approach.

Requirement Fulfillment

Given the requirements from Section 2 we come to the following analysis of requirement fulfilment. First, the OEM needs to be able to identify non-counterfeit products and can track all devices sold with the OEM 's brand name. The OEM is able to establish an identity key with the TPMs that allows it to identify an original device and track them. Second, counterfeit devices must be distinguishable to the OEM and (depending on the type of device) remote detection of deployed counterfeit products can be necessary. The OEM is able to distinguish counterfeit devices from original ones even remotely. Note the potential for Relay Attacks on this. Third, End User need to be able to distinguish counterfeits from original products. This might require the help of the OEM. Similar to the above requirements, the End User can request a device validation by the OEM, e.g. via a device management platform. End User help via e.g. a management platform helps in the detection of Relay Attacks. Fourth, the End User should to be able to recognize counterfeits using an offline scheme without active contribution of the OEM. If the OEM provides certificates for the public portions of the TPM established identity keys, then the End User can validate the originality of a given device without active contribution of the OEM.

Additional TPM functionalities

Given the installation of a TPM there are additional functionalities that can be leveraged by the OEM as well as End User. As described in Section 3, one of the most valuable features of the TPM are the RTM and the RTR. These features allow the OEM and the End User to assert that a given version of the firmware or configuration is booted on a given device. These may be utilized for reporting but also in order to protect credentials that the device possesses against disclosure or misuse. Note that these depend on the integrity of the RTM as discussed in the section on Residual Attacks below. Furthermore, the TPM offers a certain amount of secure storage space (Non-Volatile memory). This memory can be used for arbitrary data and protected via several means, including passwords, software configuration and advanced policies in case of the TPM 2.0. Finally, there exist slight differences between the possibilities of TPM ownership between TPM 1.2 and TPM 2.0. TPM 1.2 Ownership In the case of a TPM 1.2 deployment, the OEM is required to take ownership of the TPM in order to deploy its keys. Though this still allows End User to change the authentication values and use the TPM for their own purposes additionally, it means that a clearing of the ownership is not possible. This may pose a problem for resale of used devices, since a clearing of ownership is the only way to protect all TPM-bound resources from reuse by the future new device owner. However, the scheme for Remote Establishment of TPM1.2 as described above could be reenacted for a resale scenario. The OEM would need to employ an according functionality in its management software. TPM 2.0 Ownership With the introduction of the Platform Hierarchy in TPM 2.0, it is now possible to completely separate the ownerships between OEM and End User. Accordingly, the End User may clear their ownership for resale and a new device owner make claim ownership of the TPM again without gaining access to the previous owner's secret material. Given the standard scheme for TPMs in consumer electronics, the End User and OEM interactions with the different platforms are distributed between OS and firmware. For devices as discussed in this scenario however these distributions may be realized differently, since there is no actual general purpose OS installation planned.

Residual Threats

Given the presented solutions, the following residual attacks still pose a (minor) threat. These threats are now discussed and their practical relevance evaluated.

Root of Trust for Measurement Manipulation

The integrity of the RTM is under the direct responsibility of the Device Assembler. Since the Device Assembler acts as one of the potential attackers in this scenario, it seems contradictory to entrust it with this integrity requirement. However, the given scenario in which the Device Assembler is a potential attacker was solved without the use of the RTM. The RTM is intended for leveraging the additional TPM functionalities as described above. For these scenarios, a potential mistrust in the Device Assembler cannot be technically solved anyways, since the Device Assembler may always add additional components into a given device that provide so-called "backdoors". This can create a majour risk for devices in critical infrastructures and cannot be prevented by a TPM. However, the identification of the device based on the TPM will not be affected and, accordingly, the risk potential for successful attacks on the mechanism introduced in teh paper is not as high, but should of course be assessed.

Relay Attacks A relay attack could be performed from counterfeit products in such a way that devices are assembled without a TPM and any TPM-related request to this device if forwarded to an original device. In this case, however the counterfeiter would require one original device per counterfeited device, which would increase the attack costs beyond the counterfeiter's gain, and could be detected by the End User by the additional communication channel from the counterfeited device to the original device. If the counterfeiter would attempt to relay several counterfeited devices to the same original device (which would be profitable), then the End User could still detect the additional communication and the OEM would recognize the n-to-1 mapping between devices and TPMs, since e.g. a firmware update would be applied n times for only one device.

Conclusion and Outlook

TPM chips in the current version 1.2 as well as in the upcoming generation 2.0 provides a large set of hardware-based security functionality. It is a generic massmarket product and billions of devices are estimated to be currently equipped with a TPM. Many of these TPMs are installed in PCs, where they are mostly not used. The counterfeit scenario concentrates on other types of devices, such as routers, embedded systems and other special purpose devices, where hardwarebased security functionalities are are often not readily available. The TPM-based approach to counterfeit protection describes a new use-case for this security chip. The process shows that the security functionality provided by the TPM is suitable to establish counterfeit prevention and detection for outsourcing scenarios.

It should be noted that the integration of a TPM as well as the implementation of the initial key enrollment process as part of the anti-counterfeit scheme can be the basis for the efficient and cost-effective realisation of various additional security processes. One example is the remote attestation of software actually running on the system by feeding measurement information into the TPM through a Integrity Measurement Architecture (IMA) [START_REF] Sailer | Design and implementation of a tcg-based integrity measurement architecture[END_REF] and using TPM Quote to generate securely report these measurements. This information can then be validated by comparing it to the expected status available through current remote software management systems that only know about software installed through the regular process. Changes due to attacks to the device will be recognized by changed hash values for the software that is loaded. into memory and reported by the TPM. Protocols like Trusted Network Connect or trusted mobile ad-hoc networks [START_REF] Oberle | Integrating Trust Establishment into Routing Protocols of Today's MANETs[END_REF] use this feature to continuously or regularly determine the health by checking the software running with reference values.

Another example of an efficient management process based on the TPM results from the ability to automatically distribute initial configurations to the device without the need to touch it. This zero touch configuration protocol [START_REF] Kuntze | On the automatic establishment of security relations for devices[END_REF] only requires network access and can be implemented without any pre-configuration for particular clients. Also credentials for user authentication can be protected using the TPM. In the context of future production environments and crossorganization processes, a multi purpose trust anchor like the TPM can also be used to implement strong concepts for intellectual property protection.


Alex Amenta
email: amenta@fastmail.fm

Jonas Teuwen
email: jonasteuwen@gmail.com

L p -L q off-diagonal estimates for the Ornstein-Uhlenbeck semigroup: some positive and negative results

We investigate L p (γ)-L q (γ) off-diagonal estimates for the Ornstein-Uhlenbeck semigroup (e tL )t>0. For sufficiently large t (quantified in terms of p and q) these estimates hold in an unrestricted sense, while for sufficiently small t they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

Introduction

Consider the Gaussian measure dγ(x) := π -n/2 e -|x| 2 dx

(1)

on the Euclidean space R n , where n ≥ 1. Naturally associated with this measure space is the Ornstein-Uhlenbeck operator

L := 1 2 ∆ -x, ∇ = - 1 2 ∇ * ∇,
where ∇ * is the adjoint of the gradient operator ∇ with respect to the Gaussian measure. This operator generates a heat semigroup (e tL ) t>0 on L 2 (γ), called the Ornstein-Uhlenbeck semigroup, with an explicit kernel: for all u ∈ L 2 (R n , γ) and all x ∈ R n we have

e tL u(x) = R n M t (x, y)u(y) dγ(y),
where

M t (x, y) = 1 (1 -e -2t
) n/2 exp -e -t |x -y| 2 1 -e -2t exp 2e -t x, y

1 + e -t (2)
is the Mehler kernel. If we equip R n with the Euclidean distance and the Gaussian measure, and if we consider operators associated with the Ornstein-Uhlenbeck operator, we find ourselves within the realm of Gaussian harmonic analysis: here, the Ornstein-Uhlenbeck operator takes the place of the Laplace operator ∆. 1 For a deeper introduction to Gaussian harmonic analysis see the review of Sjögren [START_REF] Sjögren | Operators associated with the Hermite semigroup-a survey[END_REF] and the introduction of [START_REF] Teuwen | A note on Gaussian maximal functions[END_REF].

In this article we investigate whether the Ornstein-Uhlenbeck semigroup satisfies L p (γ)-L q (γ) off-diagonal estimates: that is, estimates of (or similar to) the form

F |e tL 1 E f | q dγ 1/q t -θ exp -c dist(E, F ) 2 t E |f | p dγ 1/p , (3)
for some parameters c > 0 and θ ≥ 0, where 1 ≤ p < q ≤ ∞, f ∈ L p (γ), and for some class of testing sets E, F ⊂ X. Often such estimates hold whenever E and F are Borel, but in applications we generally only need E to be a ball and F to be an annulus associated with E. Such estimates serve as a replacement for pointwise kernel estimates in the harmonic analysis of operators whose heat semigroups have rough kernels, or no kernels at all, most notably in the solution to the Kato square root problem [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF] (see also [START_REF] Axelsson | Quadratic estimates and functional calculi of perturbed Dirac operators[END_REF]). Even though the Ornstein-Uhlenbeck semigroup has a smooth kernel, it would be useful to show that it satisfies some form of off-diagonal estimates, as this would suggest potential generalisation to perturbations of the Ornstein-Uhlenbeck operator, whose heat semigroups need not have nice kernels. Various notions of off-diagonal estimates, including (3), have been considered by Auscher and Martell [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators part ii: Off-diagonal estimates on spaces of homogeneous type[END_REF]. However, they only consider doubling metric measure spaces, ruling out the non-doubling Gaussian measure. Mauceri and Meda [START_REF] Mauceri | BMO and H 1 for the Ornstein-Uhlenbeck operator[END_REF] observed that γ is doubling when restricted to admissible balls, in the sense that γ(B(x, 2r)) γ(B(x, r)) when r ≤ min(1, |x| -1 ). Therefore it is reasonable to expect that the Ornstein-Uhlenbeck semigroup may satisfy some form of L p (γ)-L q (γ) off-diagonal estimates if we restrict the testing sets E, F to admissible balls and sufficiently small annuli.

Here we demonstrate both the success and failure of off-diagonal estimates of the form (3), as a first step in the search for the 'right' off-diagonal estimates.

First we give a simple positive result (Theorem 2.3): for p ∈ (1, 2), and for t sufficiently large (depending on p), (3) is satisfied for all Borel E, F ⊂ R n . This is proven by interpolating between L2 -L 2 Davies-Gaffney-type estimates and Nelson's L p -L 2 hypercontractivity. We follow with a negative result (Theorem 3.1): for 1 ≤ p < q < ∞ and for t sufficiently small (again depending on p and q), (3) fails when E is a 'maximal' admissible ball B(c B , |c B | -1 ) and when F is a sufficiently small annulus C k (B), in the sense that the implicit constant in (3) must blow up exponentially in |c B |. This is shown by direct estimates of the Mehler kernel.

Notation

Throughout the article we will work in finite dimension n ≥ 2. 2 We will write

L p (γ) = L p (R n , γ). Every ball B ⊂ R n is of the form B = B(c B , r B ) = {x ∈ R n : |x -c B | < r B }
for some unique centre c B ∈ R n and radius r B > 0. For each ball B and each scalar λ > 0 we define the expansion λB = λB(c B , r B ) := B(c B , λr B ), and we define annuli

(C k (B)) k∈N by C k (B) := 2B k = 0, 2 k+1 B \ 2 k B k ≥ 1. For two sets E, F ⊂ R n we write dist(E, F ) := sup{|x -y| : x ∈ E, y ∈ F }.
For two non-negative numbers A and B, we write A a1,a2,... B to mean that A ≤ CB, where C is a positive constant depending on the quantities a 1 , a 2 , . . .. This constant will generally change from line to line.

A positive result

The Ornstein-Uhlenbeck semigroup satisfies the following 'Davies-Gaffney-type' L 2 (γ)-L 2 (γ) off-diagonal estimates. These appear in [START_REF] Van Neerven | Finite speed of propagation and off-diagonal bounds for Ornstein-Uhlenbeck operators in infinite dimensions[END_REF]Example 6.1], where they are attributed to Alan McIntosh.

Theorem 2.1 (McIntosh)

. There exists a constant C > 0 such that for all Borel subsets E, F of R n and all u ∈ L 2 (γ),

1 F e tL (1 E u) L 2 (γ) ≤ C t dist(E, F ) exp - dist(E, F ) 2 2t 1 E u L 2 (γ) .
Furthermore, Nelson [START_REF] Nelson | A quartic interaction in two dimensions[END_REF] established the following hypercontractive behaviour of the semigroup. 3Theorem 2.2 (Nelson). Let t > 0 and p ∈ (1 + e -2t , 2]. Then e tL is a contraction from L p (γ) to L 2 (γ).

Note that p > 1+e -2t if and only if t > 1 2 log 1 p-1 . Thus the hypercontractive behaviour of the Ornstein-Uhlenbeck semigroup is much more delicate than that of the usual heat semigroup e t∆ on R n , which is a contraction from L p (R n ) into L q (R n ) for all 1 ≤ p ≤ q ≤ ∞ and all t > 0.

As indicated in the proof of [1, Proposition 3.2], one can interpolate between Theorems 2.1 and 2.2 to deduce certain L p (γ)-L 2 (γ) off-diagonal estimates for the Ornstein-Uhlenbeck semigroup.

Theorem 2.3. Suppose that E, F are Borel subsets of R n . Let t > 0 and p ∈ (1 + e -2t , 2). Then for all u ∈ L p (γ),

1 F e tL (1 E u) L 2 (γ) ≤ Ct dist(E, F ) exp - dist(E, F ) 2 2t
1-δ(p,t)

1 E u L p (γ) ,
where C is the constant from Theorem 2.1 and where

δ(p, t) := 1 2 -1 p 1 2 -1 1+e -2t ∈ (0, 1). Proof. Write C M := Ct dist(E, F ) exp dist(E, F ) 2 2t . Theorem 2.1 says that e tL L 2 (γ,E)→L 2 (γ,F ) ≤ C M .
For all p 0 ∈ (1 + e -2t , p) we have

e tL L p 0 (γ,E)→L 2 (γ,F ) ≤ e tL L p 0 (γ)→L 2 (γ) ≤ 1
by Theorem 2.2. Therefore by the Riesz-Thorin theorem we get

e tL L p (γ,E)→L p (γ,F ) ≤ C θ(p0) M
, where p -1 = (1 -θ(p 0 ))/p 0 + θ(p 0 )/2, or equivalently

θ(p 0 ) = 1 p -1 p0 1 2 -1 p0 = 1 - 1 2 -1 p 1 2 -1 p0 .
Taking the limit as p 0 → 1 + e -2t gives

e tL L p (γ,E)→L p (γ,F ) ≤ C 1-δ(p,t) M
and completes the proof.

Remark 2.4. For 1 < p < q < ∞, a L p (γ)-L q (γ) version of Theorem 2.3 could be proven by first establishing L q (γ)-L q (γ) off-diagonal estimates (which may be obtained by interpolating between boundedness on L q (γ) and the Davies-Gaffney type estimates) and then arguing by the L p (γ)-L q (γ) version of Nelson's theorem.

This positive result does not rule out the possibility of some restricted L p (γ)-L 2 (γ) off-diagonal estimates for p ≤ 1 + e -2t . In the next section we show one way in which these can fail.

Lower bounds and negative results

In this section we show that the L p (γ)-L q (γ) off-diagonal estimates of (3) are not satisfied for admissible balls and small annuli when t is sufficiently small (depending on p and q). More precisely, we show that (3) fails when E is a maximal admissible ball B (i.e. a ball for which r B = min(1, |c B | -1 )), and F is an annulus C k (B) with k small. These sets typically appear in applications of off-diagonal estimates. Theorem 3.1. Suppose that 1 ≤ p < q < ∞, and that

2 e t + 1 > 1 - 1 p - 1 q , (4)
or equivalently that

t < log 1 + ( 1 p -1 q ) 1 -( 1 p -1 q )
. Then the estimate (3) does not hold for the class of testing sets

{(E, F ) : E = B(c B , |c B | -1 ), F = C k (B), 1 ≤ k < ⌊log 2 |c B |⌋ -2}.
Note that 1 p -1 q ∈ (0, 1), so we always obtain some range of t for which the off-diagonal estimates (3) fail.

Let us compare Theorems 3.1 and 2.3. Having fixed p ∈ (1, 2), we get failure of L p (γ)-L 2 (γ) off-diagonal estimates for maximal admissible balls and small annuli for e tL when t < log

1+( 1 p -1 2 ) 1-( 1 p - 1
2 ) , and when t > 1 2 log 1 p-1 the offdiagonal estimates hold for all Borel sets. We do not know what happens for the remaining values of t. Some notation will make our lives easier. Write P (X) a1,a2,... to denote an arbitrary polynomial in the variable X with coefficients that depend on the quantities a 1 , a 2 , . . ., and which may change from line to line. We will need to deal with various expressions of the form

b 2 |c B | 2 + b 0 + b -1 |c B | -1 + • • • + b -m |c B | -m ,
but the only information that we will actually need is the coefficient b 2 of |c B | 2 . Thus our notation allows us to write

b 2 |c B | 2 + b 0 + b -1 |c B | -1 + • • • + b -m |c B | -m = b 2 |c B | 2 + P (|c B | -1 ),
keeping track only of the important coefficient.

To prove Theorem 3.1 we rely on the following lower bound.

Lemma 3.2. Suppose k ≥ 1 is a natural number, 1 < q < ∞, and let B be a maximal admissible ball with |c B | > 2 k+2 . Then

C k (B)
|(e tL 1 B )(y)| q dγ(y)

1/q k,n,t |c B | -n(1+ 1 q ) exp |c B | 2 2 e t + 1 -1 - 1 q + P (|c B | -1 ) k,q,t .
Proof of Lemma 3.2. Suppose x ∈ B and y ∈ C j (B). We argue by computing a lower bound for the Mehler kernel M t (x, y) as given in (2). First we focus on the factor which involves

x, y = |x||y| cos θ x,y , where θ x,y is the angle between x and y. For 0 ≤ θ x,y < π 2 , cos θ x,y is positive and order-reversing in θ x,y . For x ∈ B and y ∈ C k (B) we have θ x,y ≤ θ x0,y0 < π 2 , where x 0 and y 0 are depicted in Figure 1; the second inequality follows from |c B | > 2 k+2 . 4 Thus we have min

x∈B y∈C k (B) cos θ x,y ≥ cos θ x0,y0 .
Using the cosine rule and the definitions of C j (B) and (x 0 , y 0 ), we can compute

c B B θ x0,y0 x 0 C k (B) y 0 Figure 1: The angle θ x0,y0 . Not to scale. cos θ x0,y0 = 2(|c B | 2 -2 k+2 |c B |r B + 2 • 4 k+1 r 2 B ) -4 k+2 r 2 B 2(|c B | 2 -2 k+2 |c B |r B + 2 • 4 k+1 r 2 B ) = |c B | 2 -2 k+2 |c B | 2 -2 k+2 + 2 • 4 k+1 |c B | -2 , using r B = |c B | -1 . Since |x| ≥ |c B | -r -1 B and |y| ≥ |c B | -2 k+1 r -1 B ,
we get

x, y = |x||y| cos θ x,y

≥ (|c B | -r B )(|c B | -2 k+1 r B ) |c B | 2 -2 k+2 |c B | 2 -2 k+2 + 2 • 4 k+1 |c B | -2 = (|c B | 2 -2 k+1 -1 + 2 k |c B | -2 ) |c B | 2 -2 k+2 |c B | 2 -2 k+2 + 2 • 4 k+1 |c B | -2 = |c B | 2 + P (|c B | -1 ) k , again using r B = |c B | -1 ,
n |c B | -n exp 2|c B | 2 e t + 1 + P (|c B | -1 ) k,t -(|c B | + |c B | -1 ) 2 = |c B | -n exp |c B | 2 2 e t + 1 -1 + P (|c B | -1 ) k,t (5)
using a straightforward estimate on γ(B). Next, we estimate

γ(C k (B)) n |C k (B)|e -(|cB |+2 k+1 rB) 2 ≃ n 2 kn r n B exp -(|c B | 2 + 2 k+2 |c B |r B + 2 k+2 r 2 B ) ≃ k,n |c B | -n exp -|c B | 2 + P (|c B | -1 ) k .
Combining this with [START_REF] Bakry | Dimension dependent hypercontractivity for Gaussian kernels[END_REF] gives

C k (B)
|(e tL 1 B (y))| q dγ(y)

1/q n,t |c B | -n exp |c B | 2 2 e t + 1 -1 + P (|c B | -1 ) k,t γ(C k (B)) 1/q k,n |c B | -n(1+ 1 q ) exp |c B | 2 2 e t + 1 -1 - 1 q + P (|c B | -1 ) k,q,t ,
as claimed.

Remark 3.3. When n = 1 the proof is simpler: we automatically have cos θ x,y = 1, and the necessary restriction on

|c B | is |c B | > 2 (k+1)/2 (or equivalently, 0 / ∈ 2 k+1 B).
Proof of Theorem 3.1. We argue by contradiction. Suppose that e tL satifies the L p (γ)-L q (γ) off-diagonal estimates (3) for some θ ≥ 0, and for (E, F ) as stated. Fix a natural number k ≥ 1 and let B be a maximal admissible ball with B | > 2 k . Lemma 3.2 and the off-diagonal estimates for E = B, F = C k (B), and f = 1 B then imply

|c B | -n(1+ 1 q ) exp |c B | 2 2 e t + 1 -1 - 1 q + P (|c B | -1 ) k,q,t k,n,t,θ exp -c (2 k+1 -1) 2 r 2 B t γ(B) 1/p = exp P (|c B | -1 ) k,t γ(B) 1/p
for some c > 0. Since

γ(B) 1/p n |B| 1/p e -1 p (|cB |-rB) 2 ≃ n |c B | -n/p exp - |c B | 2 p + P (|c B | -1 ) p , this implies exp |c B | 2 2 e t + 1 -1 + 1 p - 1 q + P (|c B | -1 ) k,p,q,t k,n,t,θ |c B | n(1-( 1 p -1 q )) .
The left hand side grows exponentially in |c B | when (4) is satisfied. However, the right hand side only grows polynomially in |c B |. Thus we have a contradiction.

Remark 3.4. By the same argument we can prove failure of L p (γ)-L q (γ) offdiagonal estimates for the derivatives (L m e tL ) m∈N of the Ornstein-Uhlenbeck semigroup, with the same conditions on (p, q, t) and the same class of testing sets (E, F ). This relies on an identification of the kernel of L m e tL , which has been done by the second author [START_REF]On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroup[END_REF].

Remark 3.5. In this article we only considered off-diagonal estimates with respect to the Gaussian measure γ. In future work it would be very interesting to consider appropriate weighted measures, following in particular [START_REF] Bakry | Dimension dependent hypercontractivity for Gaussian kernels[END_REF] and [START_REF] Bakry | Weighted Nash inequalities[END_REF], in which (among many other things) it is shown that estimates of the form e tL f L 2 (γ) f V t L 1 (γ) hold, where V t is a certain weight depending on t. Thus the Ornstein-Uhlenbeck semigroup does satisfy a form of 'ultracontractivity', but with the caveat that one must keep track of t-dependent weights. It seems that this has not yet been explored in the context of Gaussian harmonic analysis.

  and using the polynomial notation introduced earlier. Therefore, using the Mehler kernel representation of e tL , for all y ∈ C k (B) we have Since |x -y| < 2 k+1 r B < 2 (using r B = |c B | -1 < 2 -k ), this gives

	e tL 1 B (y) n,t exp	2|c B | 2 e t + 1	+ P (|c B | -1 ) k,t γ(B)
	e tL 1 B (y) n,t	B	exp -e -t |x -y| 2 1 -e -2t exp	2|c B | 2 e t + 1	+ P (|c B | -1 ) k,t dγ(x).

The multiplicative factor 1/2, which is not present in the usual definition of the Laplacian, arises naturally from the probabilistic interpretation of the Ornstein-Uhlenbeck operator.

The same results hold when n = 1, and in fact the proofs are slightly simpler. We fix n ≥ 2 because the geometry is more apparent in this case.

This is done only for n = 1 in this reference, and a full proof for general n is given in Nelson's seminal 1973 paper[START_REF]Construction of quantum fields from Markoff fields[END_REF]. These papers won him the 1995 Steele prize.

We could compute a sharper upper bound, but the estimate here is sufficient for our applications.

Acknowledgements

The first author acknowledges financial support from the Australian Research Council Discovery Grant DP120103692 and the ANR project "Harmonic analysis at its boundaries" ANR-12-BS01-0013. Part of this work was done while the first author was visiting the second author at the Delft University of Technology, and he thanks them for their support. The authors thank Mikko Kemppainen, Jan van Neerven, and Pierre Portal for valuable discussions and encouragement on this topic.


W Oukil

A Kessi

Ph Thieullen

Synchronization hypothesis in the Winfree model

Keywords: Coupled oscillators, Winfree Model, synchronization, desynchronization, periodic orbit

We consider N oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength κ and the spectrum width γ of the frequencies of each oscillator centered at 1. In the uncoupled regime, κ = 0, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. In the zero-width regime for the spectrum, the oscillators are simultaneously in the death state if and only if κ is above some positive value κ * . We say that N oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in (γ, κ) of synchronization contains {0} × [0, κ * ] in its closure. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We show numerically that the above hypothesis is necessary for the existence of synchronization.

Introduction

In 1967 Winfree [START_REF] Winfree | Biological rhythms and the behavior of populations of coupled oscillators[END_REF] proposed a model describing the synchronization of a population of organisms or oscillators that interact simultaneously. We assume that the state of each oscillator is described by a real number, the 1 phase, that corresponds to a point on the circle. We call natural frequency, the frequency of each oscillator, as if it were isolated from the others. The natural frequencies are localized inside an interval [1 -γ, 1 + γ] for some constant γ that we call spectrum width. The interaction of the surrounding oscillators on a particular oscillator has a shape independent of this oscillator. The term mean field is used for this kind of interaction or coupling. We denote by N the number of oscillators and by κ a parameter called coupling strength that measures the coupling between these oscillators.

A numerical investigation and a mathematical analysis of the transitions between the phases show a diagram of 4 major states (see Ariaratnam and Strogatz [START_REF] Ariaratnam | Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators[END_REF]). A phase called death state where each oscillator is at rest, a phase called locking state where all the oscillators possess the same nonzero frequency, a phase called incoherence state where any two oscillators have different frequencies, and a phase called partial death state which is a mixed state where a part of the oscillators are in death state and a part is in incoherence state. For small values of κ, the Winfree model is reduced to a different model, the Kuramoto model that will not be studied in this article. For large values of κ, the oscillators are in the death state. We denote by κ * the bifurcation parameter for the death state in the case γ = 0.

We say that N oscillators are synchronized if max 1≤i,j≤N |x i (t) -x j (t)| is bounded from above uniformly in time. Notice that we do not consider the phase of the oscillators but rather their lifts. For example, in the case γ = 0, our definition implies that N oscillators are always synchronized, independently of their initial conditions, thanks to the uniqueness property of periodic solutions of an ODE. The existence of synchronization has been studied in [START_REF] Kuramoto | Coexistence of coherence and incoherence in nonlocally coupled phase oscillators[END_REF][START_REF] Omelchenko | Coherence-incoherence patterns in a ring of nonlocally coupled phase oscillators[END_REF][START_REF] Panaggio | Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators[END_REF] for κ close to 0. We identify a new hypothesis of synchronization that produces a domain of parameters (γ, κ) containing {0}×[0, κ * ] in its closure. We show in addition in a numerical example that the existence of synchronization disappears precisely when the hypothesis is not anymore realized.

The Winfree model is given by the following differential equation

ẋi = ω i -κ 1 N N j=1 P (x j )R(x i ) (1)
where P and R are C 2 2π-periodic functions, x i (t) is the phase of the i-th oscillator, and X(t) = (x 1 (t), . . . , x N (t)) is the global state of the system. Although x i (t) represents a scalar in the torus R/2πZ, we actually consider its unique continuous lift in R, that we continue to call it x i (t). When N = 1

and ω 1 = 1, the Winfree model reduces to the equation

ẋ = 1 -κP (x)R(x). (2)
We call locking bifurcation critical parameter, the coupling critical parameter κ * which separates the death state and the locking state in the reduced Winfree model:

κ * := max{κ > 0 : 1 -κP (x)R(x) > 0, ∀x ∈ R}. (3)
(Notice that κ * = +∞ if max x P (x)R(x) ≤ 0.) We assume through out this work the following hypotheses:

H1 the coupling strength κ is taken in the interval (0, κ * ), H2 the natural frequencies ω i are chosen in (1 -γ, 1 + γ) with γ ∈ (0, 1), H3 synchronization hypothesis:

2π 0 P (s)R (s) 1 -κP (s)R(s) ds > 0, ∀κ ∈ (0, κ * ).
For numerical results we use a simplified version of the Winfree model

ẋi = ω i - κ N N j=1 P β (x j ) sin(x i ), where P β (x) = 1 + cos(x + β), (4)
and β ∈ [0, π]. It is important to notice that we do not make any assumptions on the number N of oscillators; in particular we do not assume that N → +∞. We do not assume either that the natural frequencies are distributed according to a particular law as it is done in [START_REF] Ariaratnam | Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators[END_REF]. Our main objective is to exhibit a domain in the parameter space (γ, κ) independent of N and any choice of the natural frequencies satisfying H2 such that the oscillators stay synchronized for all time and for any initial conditions sufficiently packed.

Synchronization and locking may have several meanings or definitions depending on the authors. We choose the following definitions.

Definition 1 (Synchronization). We say that the oscillators x i (t) are synchronized if sup 1≤i,j≤N |x i (t) -x j (t)| is bounded from above uniformly in time t ≥ 0. Definition 2 (Periodic locking). We say that the oscillators x i (t) are periodically locked to the frequency Ω if they are synchronized and if there exist 2π/Ω-periodic functions Ψ i (t) such that

x i (t) = Ωt + Ψ i (t), ∀i = 1 . . . N, ∀t ≥ 0.
We notice that in the periodic locking case the average frequency x i (t) t admits the limit Ω as t → +∞ and that this limit is independent of the ith oscillator. The fact that the limit does exist has never been addressed mathematically (except of course in the death state). Our main result is a partial result in that direction in the locking case when κ is any parameter in (0, κ * ) and γ ∈ (0, Γ(κ)) for some Γ : [0, κ * ] → R + . Theorem 3. We consider the Winfree model given by (1) and satisfying the hypotheses H1-H3. Then, there exists an open set U in the space of parameters (γ, κ) ∈ (0, 1) × (0, κ * ), independent of N , containing in its closure {0} × [0, κ * ], such that for every parameter (γ, κ) ∈ U , for every N ≥ 1 and every choice of natural frequencies

(ω i ) N i=1 , 1.
There exists an open set C N γ,κ invariant by the flow, and of the form,

C N γ,κ := X = (x i ) N i=1 ∈ R N : max i,j |x j -x i | < ∆ γ,κ 1 N N i=1 x i
where ∆ γ,κ : R → (0, 1) is a 2π-periodic C 2 function independent of N . In other words, for every parameters (γ, κ) ∈ U , the oscillators are synchronized for any initial conditions x i (0) with sufficiently small dispersion.

2. There exists a particular initial condition (x * i (0)) N i=1 ∈ C N γ,κ , and a common frequency Ω γ,κ > 0 such that

x * i (t) = Ω γ,κ t + Ψ N i,γ,κ (t), ∀i = 1, . . . , N, ∀t ≥ 0,
where Ψ N i,γ,κ : R + → R are C 2 , 2π/Ω γ,κ -periodic functions, uniformly bounded with respect to N . In other words, the oscillators are periodically locked for a particular initial condition.

Notice that the simplified Winfree model (4) satisfies the hypothesis H3 when β = 0 as it is proved in section 3.

For small values of κ << 1, Quinn and Strogatz [START_REF] Quinn | Singular unlocking transition in the Winfree model of coupled oscillators[END_REF] use the Lindstedt's Method and find an approximation value of the tangent bifurcation curve of the synchronization sate. We mention relative results about the Winfree model in [START_REF] Popovych | Phase chaos in coupled oscillators[END_REF][START_REF] Giannuzzi | Phase diagram of a generalized Winfree model[END_REF][START_REF] Basnarkov | Critical exponents of the transition from incoherence to partial oscillation death in the Winfree model[END_REF][START_REF] Atay | Spatially structured networks of pulsecoupled phase oscillators on metric spaces[END_REF][START_REF] Pazó | Low-dimensional dynamics of populations of pulse-coupled oscillators[END_REF]. Recently Ha, Park and Ryoo studied in [START_REF] Ha | Emergence of phase-locked states for the Winfree model in a large coupling regime[END_REF] the stability of the stationary state for large coupling.

Proof of theorem 3

Let X = (x 1 , . . . , x N ) be a solution of [START_REF] Ariaratnam | Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators[END_REF]. We call mean µ(t) and dispersion δ(t) of X the quantities

µ(t) := 1 N N j=1 x j (t), δ(t) := max i,j |δ i,j (t)|, δ i,j (t) := x i (t) -x j (t). (5)
We want to build a space of parameters U of the form

U = {(γ, κ) ∈ (0, 1) × (0, κ * ) : 0 < γ < Γ(κ)},
and a 2π-periodic function ∆ γ,κ , called dispersion curve such that, if X(t) is a solution of the Winfree model ( 1) satisfying H1 -H3,

δ(0) < ∆ γ,κ (µ(0)) =⇒ δ(t) < ∆ γ,κ (µ(t)), ∀t ≥ 0.
The dispersion curve is obtained by solving a non-autonomous affine differential equation with periodic coefficients. The following lemma is standard.

Lemma 4. We consider the affine differential equation

d ds ∆(s) = α -β(s)∆(s), (6)
where α > 0 and β is a C 1 , 2π-periodic function satisfying

2π 0 β(s) ds > 0.
Then there exists a unique C 2 , positive and 2π-periodic function solution of (6) given by

∆(s) := α s+2π s exp t s β(u) du dt exp 2π 0 β(u) du -1 . Moreover max s∈R ∆(s) ≤ α2π exp 2π 0 β -(u) du 1 -exp - 2π 0 β(u) du
where β -:= max(0, -β), and β + := max(0, β).

Proof. The differential equation ( 6) is a first-order non-homogeneous linear differential equation. The general form of solution z(t) of ( 6) with initial condition z(s 0 ) ∈ R at time s 0 ∈ R is given by

z(s) = exp s s 0 -β(u) du z(s 0 ) + α s s 0 exp s t -β(u) du dt.
If z(s) is 2π-periodic, using the 2π-periodicity of β, one obtains a unique z(s 0 ) given by

z(s 0 ) = α s 0 +2π s 0 exp s 0 +2π t -β(u) du dt 1 -exp s 0 +2π s 0 -β(u) du
and a unique 2π-periodic solution, positive thanks to α > 0, given by

z(s) = α s+2π s exp s+2π t -β(u) du dt 1 -exp 2π 0 -β(u) du .
The maximum of ∆ is obtained using the following estimates

t s β(s) ds = t s (β + -β -)(s) ds ≤ t s β + (s) ds ≤ 2π 0 β + (s) ds, ≤ 2π 0 (β + -β -)(s) ds + 2π 0 β -(s) ds, ≤ 2π 0 β(s) ds + 2π 0 β -(s) ds.
The following lemma shows that, if the dispersion of a solution X(t) of ( 1) is a priori uniformly bounded, δ(t) < D, ∀t ∈ [0, t * ], then each δ i,j (t) is a sub-solution strict of an affine differential equation as in lemma 4.

Lemma 5. Let us assume that for some (γ, κ) ∈ (0, 1) × (0, κ * ), there exist D > 0 and t * > 0 such that for every t ∈ [0, t * ], the solution X(t) of (1) satisfies δ(t) < D. Then for every 1 ≤ i, j ≤ N , and t ∈ [0, t * ]

d dt δ i,j < (2γ + CκD 2 ) -κP (µ)R (µ)δ i,j , d dt δ i,j ≤ 2γ + CκD 2 + CκD, where δ i,j (t) := x i (t) -x j (t), µ(t) = 1 N N k=1 x k (t), C := P ∞ R ∞ + P ∞ R ∞ , and C := P ∞ R ∞ + P ∞ R ∞ .
Proof. By substracting two equations ( 1) we obtain

dδ i,j dt = ω i -ω j - κ N N k=1 P (x k ) R(x i ) -R(x j ) = ω i -ω j -κP (µ)R (µ)δ i,j + κ N N k=1 E i,j,k
where

E i,j,k = P (µ)R (µ)(x i -x j ) -P (x k )[R(x i ) -R(x j )
] that we bound from above using the following estimates

P (x k )[R(x i ) -R(x j )] = P (µ)[R(x i ) -R(x j )] + Ẽi,j,k , | Ẽi,j,k | ≤ P ∞ R ∞ |x k -µ||x i -x j | < P ∞ R ∞ D 2 , P (µ)[R(x i ) -R(x j )] = P (µ)R (µ)(x i -x j ) + Êi,j,k , | Êi,j,k | < P ∞ R ∞ D 2 ,
and

|ω i -ω j | ≤ 2γ.
We estimate in the following lemma the velocity of µ. We will later find conditions on (γ, κ) so that dµ dt > 0. The constant C is defined in lemma 5. Lemma 6. Assuming the same hypotheses as in lemma 5, we have

1 -κP (µ)R(µ) - d dt µ ≤ γ + CκD, ∀t ∈ [0, t * ].
Proof. We use indeed the estimates

|x i -µ| ≤ D, |1 -1 N N k=1 ω k | ≤ γ, dµ dt = 1 -κP (µ)R(µ) -1 - 1 N N k=1 ω k + κ N N k=1 E k , where E k = 1 N N i=1 P (µ) R(µ) -R(x i ) -P (x k ) -P (µ) R(x i ) .
We now assume that (γ, κ) have been chosen so that

1 -γ -CκD - κ κ * > 0. ( 7
)
Notice that the equation dx i dt = 1 -κP (x i )R(x i ) corresponds to γ = 0 and identical initial conditions in equation [START_REF] Ariaratnam | Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators[END_REF]. By a small perturbation of γ = 0, and a small perturbation of the initial condition

x 1 (0) = • • • = x N (0), one obtain dµ dt = 1 ± -κP (µ)R(µ) and d dt (x i -x j ) = ± i,j -κP (µ)R (µ)(x i - x j )
. The synchronization hypothesis H3 may be interpreted as a stability hypothesis after changing the time variable t to s = µ(t).

We now proceed formally. Thanks to the definition of κ * in (3), condition (7) implies dµ dt > 0 on [0, t * ]. We then consider the change of variable s = µ(t) from [0, t * ] to [s 0 , s * ] whre s 0 = µ(0) and s * = µ(t * ). Let τ be the inverse function

τ : [s 0 , s * ] → [0, t * ] s → τ (s) . (8)
Define

x * i (s) = x i • τ (s), X * (s) = (x * 1 (s), . . . , x * N (s)), δ * i,j (s) = x * i (s) -x * j (s)
. With respect to the new variable s, lemma 5 admits the following equivalent form.

Lemma 7. We assume the same hypotheses as in lemma 5 and the new condition [START_REF] Atay | Spatially structured networks of pulsecoupled phase oscillators on metric spaces[END_REF]. Then

d ds δ * i,j (s) < α(γ, κ, D) -β κ (s)δ * i,j (s), ∀s ∈ [s 0 , s * ],
where

α(γ, κ, D) := 2γ + CκD 2 1 -κ/κ * + (2γ + CκD 2 + CκD)(γ + CκD) (1 -γ -CκD -κ/κ * )(1 -κ/κ * ) , β κ (s) := κP (s)R (s) 1 -κP (s)R(s)
.

Proof. The chain rule gives

d dt δ i,j = d ds δ * i,j d dt µ, or d ds δ * i,j 1 -κP (µ)R(µ) = d dt δ i,j + d ds δ * i,j 1 -κP (s)R(s) - d dt µ .
The definition of κ * implies 1 -κP (µ)R(µ) ≥ 1 -κ/κ * > 0. From lemma 5, one obtains

d ds δ * i,j ≤ (2γ + CκD 2 ) + d ds δ * i,j 1 -κP (s)R(s) -d dt µ 1 -κP (µ)R(µ) -β κ (s)δ i,j .
From lemma 6, one obtains

d dt µ ≥ 1 -κP (µ)R(µ) -γ -CκD ≥ 1 -γ -CκD -κ/κ * , | d ds δ * i,j | ≤ 2γ + CκD 2 + CκD 1 -γ -CκD -κ/κ * .
Let ∆ γ,κ,D (s) be the unique C 2 2π-periodic solution of the equation

d∆ ds = α(γ, κ, D) -β κ (s)∆
given by lemma 4. The following lemma gives sufficient conditions on (γ, κ, D) so that max s ∆ γ,κ,D (s) < D.

Lemma 8.

There exists an open set U of parameters (γ, κ) whose closure contains {0} × [0, κ * ], defined in the following way

U := {(γ, κ) ∈ (0, 1) × (0, κ * ) : 0 < γ < κD 2 (κ)}, where
D(κ) := min 1, L(κ) 2(2 + C)/(1 -κ/κ * ) + 2 C(1 + C)κ/(1 -κ/κ * ) 2 , L(κ) := 1 -exp(- 2π 0 β κ (s) ds) 2πκ exp( 2π 0 β - κ (s) ds) , such that, for every (γ, κ) ∈ U , 1 -γ -CκD(κ) - κ κ * > 0, and max s∈[0,2π] ∆ γ,κ,D(κ) (s) < D(κ).
Proof. Thanks to lemma 4, it is enough to check that α(γ, κ, D(κ)) κL(κ) < D(κ).

We have

D(κ) < 1 -κ/κ * 2κ(1 + C) , γ + CκD(κ) < κD(κ)(1 + C) < 1 2 1 - κ κ * , 1 -γ -CκD(κ) -κ/κ * > 1 2 1 - κ κ * , α(γ, κ, D(κ)) < 2(2γ + CκD(κ) 2 ) 1 -κ/κ * + CκD(κ)(γ + CκD(κ)) 1 2 (1 -κ/κ * ) 2 , < κD(κ) 2 2(2 + C) 1 -κ/κ * + 2 C(1 + C)κ (1 -κ/κ * ) 2 < κD(κ)L(κ).
Definition 9. We call dispersion curve the periodic function ∆ γ,κ (s) := ∆ γ,κ,D(κ) (s), ∀s ∈ R.

We prove the first part of Theorem 3. The set C N γ,κ defined in Theorem 3 is obviously open and has the shape of a tubular neighborhood about the line (1, 1, . . . , 1)R with bounded convex transverse section µ(X) = µ 0 . We want to prove that C N γ,κ is invariant by the flow of (1). We recall that U denotes the set of parameters (γ, κ) defined in lemma 8.

Proof of Theorem

3 -Item 1. Let be X(0) ∈ C N γ,κ and t * := sup{t ≥ 0 : ∀ 0 < t < t, δ(t ) < ∆ γ,κ (µ(t ))}.
Assume by contradiction that t * < +∞. We use the change of variable s = µ(t) for every s ∈ [s 0 , s * ], s 0 = µ(0) and s * = µ(t * ). Then there exist 1 ≤ i 0 , j 0 ≤ N such that δ * i 0 ,j 0 (s * ) = ∆ γ,κ (s * ). Since max ∆ γ,κ < D(κ) and 1 -γ -CκD(κ) -κ/κ * > 0, lemma 7 implies

d ds δ * i 0 ,j 0 (s * ) < α(γ, κ, D(κ)) -β κ (s)δ * i 0 ,j 0 (s * ), = α(γ, κ, D(κ)) -β κ (s)∆ γ,k (s * ) = d ds ∆ γ,κ (s * ).
There exists s < s * close enough to s * such that δ * i 0 ,j 0 (s) > ∆ γ,κ (s) or in other words there exists t < t * close enough to t * such that δ i 0 ,j 0 (t) > ∆ γ,κ (µ(t)). We have obtained a contradiction.

We now prove the second part of Theorem 3. We show there exists a periodically locked solution X(t) = (x 1 (t), . . . , x N (t)) of the system (1) with some initial condition in C N γ,κ , that is, there exists a common rotation number Ω γ,κ > 0 such that x i (t) = Ω γ,κ t + Ψ i (t), ∀i = 1, • • • , N where Ψ i (t) are periodic functions of period 2π/Ω γ,κ . Our strategy consists in constructing, by fixing the mean of X(0), a compact and convex transverse section Σ γ,κ to the closure CN γ,κ , and a continuous Poincaré map P γ,κ : Σ γ,κ → Σ γ,κ by waiting the first time the mean of X(t) return to 0. We then use Brouwer fixed point theorem to prove the existence of a fixed point of P γ,κ . We denote by Φ t (X) the flow of the equation ( 1).

Lemma 10. Let be (γ, κ) ∈ U where U is defined in lemma 8. Define

Σ γ,κ = {X ∈ C N γ,κ : µ(X) = 0} and 1 = (1, • • • , 1)
. Then there exist a C 2 map (the Poincaré map) P γ,κ : Σ γ,κ → Σ γ,κ and a C 2 function (the return time map) θ γ,κ : Σ γ,κ → R + such that,

Φ θ(X) (X) = P γ,κ (X) + 2π1, ∀X ∈ Σ γ,κ , 2π 1 + γ + κ P ∞ R ∞ < θ(X) < 2π 1 -γ -CκD(κ) -κ κ * . Proof. Let be X ∈ C N
γ,κ such that µ(X) = 0. Let be µ(t) := µ(Φ t (X)) and τ (s) be the inverse function of µ(t) as it has been defined in ( 5) and [START_REF] Omelchenko | Coherence-incoherence patterns in a ring of nonlocally coupled phase oscillators[END_REF]. We prefer to write the explicit dependence on X: µ X (t) = µ(t) and τ X (s) = τ (s). Thanks to lemma 6, we obtain

1 -γ -CκD(κ) - κ κ * < μX (t) < 1 + γ + κ P ∞ R ∞ .
Define θ(X) := τ X (2π). Then τ X (2π) 0 μX (t)dt = 2π implies the second estimate of the lemma. Let be P γ,κ (X) := Φ θ(X) (X) -2π1. Then

µ(P γ,κ (X)) = µ X (θ(X)) -2π = µ X • τ X (2π)) -2π = 0, δ(P γ,κ (X)) = δ(Φ θ(X) (X)) < ∆ γ,κ (µ X (θ(X)) = ∆ γ,κ (2π) = ∆ γ,κ (0).
We have shown that P γ,κ is a map from Σ γ,κ into itself.

Corollary 11. The Poincaré map P γ,κ defined in lemma 10 admits a fixed point X * ∈ Σ γ,κ .

Proof. Σγ,κ is compact and convex; P γ,κ : Σγ,κ → Σγ,κ is continuous. By Brouwer fixed point theorem, P γ,κ admits a fixed point in X * ∈ Σγ,κ . We claim that X * ∈ ∂ Σγ,κ . Suppose by contradiction X * ∈ ∂ Σγ,κ . Let be

δ * i,j (t) := x * i (t) -x * j (t), δ * (t) := max i,j δ * i,j (t) and µ * (t) := 1 N N i=1
x * i (t).

Then there exist 1 ≤ i 0 , j 0 ≤ N , such that δ * i 0 ,j 0 (0) = ∆ γ,κ (0). As in the proof of item 1 of Theorem 3, there exists t 0 > 0 small enough such that, for every 0 < t < t 0 , δ * i 0 ,j 0 (t) < ∆ γ,κ (µ * (t)). By repeating this argument for every 1 ≤ i 1 , j 1 ≤ N satisfying the equality δ * i 1 ,j 1 (t) = ∆ γ,κ (µ * (t)), we obtain for some t > 0, δ * (t ) < ∆ γ,κ (µ * (t )). The invariance of C N γ,κ implies δ * (t) < ∆ γ,κ (µ * (t)) for every t > t . We have obtained a contradiction with the fact that

δ * (θ(X * )) = δ * (0) = ∆ γ,κ (0) = ∆ γ,κ (2π) = ∆ γ,κ (µ * (θ(X * )).
Proof of Theorem 3 -Item 2. Corollary 11 implies the existence of a point X * ∈ C N γ,κ and a time θ * > 0 such that Φ θ * (X * ) = X * + 2π1. By the uniqueness property of the solutions of an ordinary differential equation, we have Φ θ * +t (X * ) = Φ t (X * ) + 2π1, ∀t ≥ 0.

Let be Ψ(s

) := Φ s (X * ) -2πs θ * 1 = (Ψ 1 (s), • • • , Ψ N (s)), ∀s ≥ 0. Then Ψ is periodic of period θ * . We have indeed Ψ(s + θ * ) = Φ s+θ * (X * ) -2π s + θ * θ * 1 = Φ s (X * ) + 2π1 -2π s + θ * θ * 1 = Ψ(s).
Moreover the return time θ * is uniformly bounded from above with respect to N as in lemma 10.

Numerical results

The study of organized populations has been modeled by Winfree in [START_REF] Winfree | Biological rhythms and the behavior of populations of coupled oscillators[END_REF] in 1967. He observed that a week coupling between independent individuals tends to synchronize them. An example of biological synchronization is given by flashes of tropical fireflies. The synchronization appears when the individuals emit their flashes simultaneously with the same frequency. In this section we analyze a simplified Winfree model given by equation ( 4) with P β (x) = 1 + cos(x + β) and R(x) = sin(x). When β = 0, the synchronization hypothesis H3 is satisfied. Indeed

d ds ln(1 -κP 0 R) = -κ(P 0 R + P 0 R ) 1 -κP 0 R , and
2π 0 P 0 R 1 -κP 0 R ds = - 2π 0 P 0 R 1 -κP 0 R ds = 2π 0 sin 2 (s) 1 -κ(1 + cos(s)) sin(s) ds > π 3 .
The numerical value of κ * is 4 3 √ 3 ∼ 0.769 and the two constants C and C of lemmas 5 and 6 are bounded from above by 3. The theoretical domain of parameters U for which the Winfree model is synchronized is built in lemma 8 using D(κ) and L(κ). Its size is very small, 10 -3 smaller than the size of the numerical domain one can expect as in figure 1(a).

In order to analyze numerically the hypothesis H3, we discuss the model We show in figure 2(a) the variation of the critical parameter κ * (β) for β ∈ [0, π]; we notice that the minimum is obtained at κ * (0) ≈ 0.769. Let be

H κ (β) := 2π 0 P β (s)R (s) 1 -κP β (s)R(s)
ds.

We also show numerically in 2(a), that the gray region corresponds to the domain of (β, κ) satisfying β ∈ [0, π], κ ∈ [0, κ * (β)], and H κ (β) > 0, that is to the domain β < π 2 . As noticed by the referee, the symmetry

H κ (β) = -H κ (π -β) implies H κ ( π 2 ) = 0 for every κ ∈ [0, κ * ( π 2 )
]. We choose κ = 0.6 in figure 2(b) and compute the largest γ satisfying |d X (T )| < 3π, that is the boundary of U β at κ = 0.6. We notice that the numerical domain of synchronization is negligible if and only if β ∈ [ π 2 , π]. We study in figure 3 the variation of the order parameter r X (β) as β increases in [0, π] at a given time T and coupling strength κ. A numerical value r X (β) ≈ 1 suggests a very tight cluster of almost all oscillators on the circle, a value r X (β) ≈ 0 suggests on the contrary symmetrically distributed oscillators. We observe a sharp decrease of r X (β) at β = π 2 that is when H κ (β) becomes negative. The other parameters are chosen as in figure 1.

Figure 4 shows the graph of d X (t) as a function of t ∈ [0, 3 × 10 4 ] for different values of γ and β at a given κ = 0.6. It also shows the location of the oscillators on the circle at the final time T = 3 × 10 4 . Synchronization has been defined by the condition sup t>0 d X (t) < +∞. For β > π 2 , except at γ ≈ 0, t → d X (t) does not seem to be bounded: the oscillators are desynchronized. This observation corroborates the disappearance of the synchronization domain in figure 2 when β > π 2 . For γ = 0, the oscillators have identical natural frequencies; the comparison principle of ODE's in the periodic case implies, if max 1≤i,j≤N |x i (0) -x j (0)| ≤ 2π, that d X (t) ≤ 2π for every t > 0: the oscillators are always synchronized. For β < π 2 , we choose two values of γ which are close, but above, the desynchronization curve: γ 1 > γ( π 2 -0.25) and γ 2 > γ( π 2 -0.5). The plots suggest a non decreasing function t → d X (t) going to infinity with flat parts.

Conclusion

We defined synchronization as a state where the dispersion of the oscillators is uniformly bounded in time. For every given coupling strength κ, corresponding to the non death state, we showed analytically the existence of synchronized oscillators when the hypothesis H3 is satisfied and when the spectrum width γ is sufficiently smaller than some value depending on κ. In addition, under the same hypothesis, we showed the existence of N periodic synchronized oscillators with the same strong rotation number. We also showed numerically, for a parametrized simplified Winfree model at a given coupling strength, that the hypothesis H3 seems to be necessary for the existence of synchronization.

( 4 )

 4 for different values of β ∈ [0, π]. We use two different order parameters to measure the synchronization r X (β) := 1 N N j=1 e ix j (T ) , d X (T ) := max i |x i (T ) -µ(T )|.

Figure 1 Figure 1 :

 11 Figure 1 shows the numerical domains U β of synchronization for three different values of β. The domain decreases along the γ-direction as β increases to π 2 as we shall see. The critical parameter giving the transition to the death state is defined as in equation (3) by κ * (β) := max{P β (x)R(x) : x ∈ R} -1 .

Figure 2 :Figure 3 :

 23 Figure 2: We plot in 2(a) the critical parameter of the death state transition κ * (β). We choose in 2(b) κ = 0.6, T = 3 × 10 4 and draw the desynchronization curve, the largest γ for which d X (T ) < 3π. We use the same experiment parameters as in figure 1.

(a) γ 2 Figure 4 :

 24 Figure 4: We choose κ = 0.6, random initial conditions in [-π, π], and the other parameters as in figure 1. We choose vertically from botton to top, γ 0 = 0, γ 1 = 0.011 et γ 2 = 0.0412, horizontally from left to right, β = 0, β = π 2 -0.5, β = π 2 -0.25, β = π 2 + 0.25 and β = π 2 + 0.5. We plot the order parameter d X (t) for t ∈ [0, 3 × 10 4 ], the oscillators on the circle (bold dots) and the order parameter r X (β) (circle dot) at time T = 3 × 10 4 .

Acknowledgements

The first author would like to thank Professor Strogatz for his support to answering many questions by mail. We also would like to thank both referees for their precise remarks that helped us to improve the text.