Badreddine Chah

Alexandre Lombard

Anis Bkakria

Abdeljalil Abbas-Turki

Reda Yaich

H3PC: Enhanced Security and Privacy-Preserving Platoon Construction Based on Fully Homomorphic Encryption

With the increasing adoption of connected and autonomous vehicles, platooning services have emerged as a promising solution to enhance road traffic efficiency. However, the widespread deployment of platooning services raises concerns about the privacy and security of sensitive vehicle data. This paper proposes a novel privacy-preserving framework for platoon formation, named H3PC (Homomorphic Privacy-Preserving Platooning Construction), to address privacy and security challenges. The proposed approach is based on fully homomorphic encryption and order-preserving encryption, to enable secure and private operations within the platoon construction. In addition, the H3PC framework incorporates a safe vehicle control method that adheres to established norms in the literature. By striking a balance between security and computational efficiency, H3PC enables effective platooning construction. Furthermore, we present experimental results demonstrating the performance and latency of H3PC.

I. INTRODUCTION

As an important part of a smart city, Intelligent Transportation Systems (ITSs) are the key to the integration of advanced technologies and communication systems into transportation infrastructure and vehicles to improve the safety, efficiency, and sustainability of transportation networks. Indeed, ITS is fundamental to the sustainable development of urban transport, taking into account fuel consumption and traffic efficiency. ITSs are highly related to Connected and Autonomous Vehicles (CAVs) as they involve the use of various technologies such as smart sensors, cameras, artificial intelligence (AI), and connectivity (Vehicle-to-Everything (V2X) communication). In the years to come, the roads will gradually be filled with autonomous vehicles that are able to drive themselves while cooperating with each other to form sustainable transportation systems. Chah, B. et al. [START_REF] Chah | Privacy Threat Analysis for Connected and autonomous vehicles[END_REF] present and classify several innovative CAV services such as reducing road accidents, improving the quality of life, increasing the efficiency of transportation systems, and so on. In addition, ITS plays a critical role in providing the infrastructure and communication systems necessary for CAV to communicate, navigate, and assist in making informed decisions. As a representative driving pattern of CAV, the platooning use case [START_REF] Kavathekar | Vehicle platooning: A brief survey and categorization[END_REF] has great potential to improve safety, driver comfort, traffic efficiency, fuel efficiency, and emissions reduction. Imagine a scenario where a group of CAVs has an intersection in their path itinerary, as shown in Fig. 1, platooning services [START_REF] Kavathekar | Vehicle platooning: A brief survey and categorization[END_REF] allow them to travel closely together in a convoylike formation. The CAVs in a platoon must maintain a short, constant, and safe distance from each other while moving synchronously. This is possible thanks to advanced technologies such as V2X communication and control systems. In addition, platoon service is more than just convoy formation. Other platooning shapes are harnessed to improve traffic performance. For example, virtual platoons can be formed at intersections [START_REF] Abdeljalil | Autonomous Intersection Management: Optimal Trajectories and Efficient Scheduling[END_REF] and lane changes to improve traffic flow and increase the infrastructure capacity.

In order to perform this promising service, it is needed to exchange real-time sensitive information between vehicles and interfaces. In this work, we consider sensitive data to be any information that leads to information related to a physical individual. Generally in the platooning service, the sensitive manipulated data are the vehicle identification, the real-time position and speed, the itinerary, the sensors' data, and the control. The service provider may add some other sensitive data that must be protected to ensure no entity can deduce sensitive information about the drivers. A strict framework is required to prevent possible abuses. Currently, there is general data protection legislation that addresses this issue. For example, the General Data Protection Regulation (GDPR) [5] which concerns European Union (EU) countries, and The California Consumer Privacy Act (CCPA) for the United States (US). These regulations govern the use of data and protect the users' privacy and personal information, providing a single legal framework for professionals. To comply with the legal requirements of the GDPR, car manufacturers and their suppliers must ensure that their development systems protect the privacy of users. To our knowledge, until now, no study has been conducted or proposed on this topic.

The rest of the paper is organized as follows. Section II introduces our main contribution. In Section III, we provide an overview of the platooning use case and introduce the concept of fully homomorphic encryption (FHE). Section IV focuses on our proposed framework, "H3PC: Homomorphic Privacy-Preserving Platooning Construction". Section V presents the findings of our conducted experiments, including the first experiment using homomorphic max/min operations, as well as its lighter technique that leverages order-preserving encryption (OPE) methods. Finally, Section VI resumes the main findings and contributions of the study and discusses the limitations and future directions.

II. PROPOSED CONTRIBUTION

The focus of this work is on the Platooning Formation. Where platooning is initiated by selecting a vehicle leader, the group of CAVs, and computing the intersection itinerary path for all entities within the group. Our technique H3PC is employed after this initialization phase, where the vehicles come together to form the platoon 1. This formation process can be driven by either a human driver or autonomous driving capabilities. The CAVs within the group need to merge at the same x-position, which is a variable that changes based on the behavior of the CAVs. It is important to note that all CAVs within the group already know the path of the itinerary since it is their original trajectory. We represent it as a straight-ahead path in Figure 1. The CAVs only require the target information to synchronize with the other entities in the group. In a platoon, vehicles often position themselves behind the lead vehicle, maintaining a safe and constant distance between them. Vehicles establish communication links to share their position and speed and coordinate their movements. These data are sensitive information that can be linked to the drivers.

To meet the aforementioned objectives, we propose the following three main contributions:

(i) Development of a privacy-preserving protocol for Platooning Formation based on Homomorphic Encryption (HE): The study introduces a protocol that utilizes Homomorphic Encryption, a secure and widely used cryptographic technique, to perform calculations on encrypted data. This approach ensures the confidentiality of sensitive information while enabling the formation of platoons. (ii) Provision of a service without requiring access to sensitive data: The protocol offers a service that does not rely on clear access to sensitive data such as Location and Speed. This allows for the grouping of vehicles in an x-position while maintaining data privacy and security. (iii) Utilization of a Platooning Service Provider (PSP) for secure computations and platoon identification: The pro-posed protocol relies on a central server known as the Platooning Service Provider (PSP). The PSP performs heavy computations over the clients' encrypted data and delivers encrypted outputs of nearby platoons. Notably, the PSP does not have access to the client's sensitive data, ensuring the privacy of the individuals involved. (iv) Proposition of the Lightweight Linear Control Equation:

The proposed equation is a lightweight version that is based solely on addition and multiplication operations. The equation is suitable for homomorphic techniques.

III. BACKGROUND A. Platooning Use-Case Description

Let us consider a group of vehicles that drive on the same highway each day. Some drivers need to rest and others need to perform other tasks during this driving period. The platooning service [START_REF] Kavathekar | Vehicle platooning: A brief survey and categorization[END_REF] offers the possibility to automatically join and follow a nearby convoy instead of stopping for a certain time. To manage the CAVs groups, there is an entity that receives information about the group of vehicles and makes decisions, named Platooning Service Provider (PSP).

This paper focuses on the Platooning Formation (see Fig. 1). To this end, PSP uses a method that allows vehicles to merge near one point based only on their respective speeds and positions. More precisely, based on the received speeds and positions, the PSP, must compute the suitable acceleration for each CAV in the group platoon. This control method should meet the following criteria:

• Safety: Control methods need to prioritize safety, maintain a safe distance between the vehicles, and be able to react appropriately to any potential hazard or change in the environment. energy efficiency by smartly managing vehicles' accelerations. By meeting the aforementioned criteria, control methods can help establish stable behavior, ensuring safe and efficient operation while maintaining the desired formation and dynamics of the platoon. However, the platooning operation requires the exchange of specific real-time sensitive data, such as positions and speeds. The revelation of such sensitive data can therefore have severe consequences [START_REF] Chah | Privacy Threat Analysis for Connected and autonomous vehicles[END_REF]. A solution to this issue can be found by having the PSP able to compute the platooning instructions (e.g. Target Acceleration) based on encrypted data, in compliance with the GDPR principles. This can be achieved using FHE techniques, as we discuss in the following.

B. Fully Homomorphic encryption (FHE)

In 2009, Gentry [START_REF] Cheon | Introduction to homomorphic encryption and schemes[END_REF] made a discovery in the field of cryptography by defining the first fully homomorphic encryption scheme (FHE). The FHE is a powerful cryptographic primitive that allows calculations to be performed on encrypted data without having access to the secret key and without the need to decrypt. The hardness of FHE is based on a mathematical problem known as the Learning With Errors (LWE) problem. LWE is considered among the lattice-based cryptography problems, which is robust to quantum attacks. They are among the problems considered as Difficult, i.e. there is no polynomial algorithm to solve this problem.

In this work, we focus on the CKKS scheme implemented in the library OpenFHE [START_REF] Badawi | OpenFHE: Open-source fully homomorphic encryption library[END_REF]. The CKKS scheme is particularly designed for calculations on real or complex numbers, which makes it suitable for applications that involve calculations with real numbers, such as our use case (platooning services). In addition, the schema implemented in OpenFHE is the RNS variants of the CKKS scheme presented in [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF]. The RNS variant uses a different scaling factor for each level. To do that, the authors utilize the Chinese Remainder Theorem (CRT) representation to break multi-precision integers in Z q into vectors of smaller integers (q 0 , q 1 , . . . , q L ) to perform operations efficiently using native (64-bit) integer types. For L > 0. The CKKS security relies on the Ring Learning with Errors (RLWE) problem hardness.

The idea behind the construction of FHE is based on adding noise during encryption and key generation to obtain the hardness properties. In addition, the system includes multiple parameters that determine the level of security, functionality, performance, and precision supported by the scheme. These parameters are as follows: 1) Number of fractional bits f , corresponding to the accuracy of the computation. 2) Plaintext modulus p. 3) Ciphertext modulus q. 4) Ciphertext dimension n. We suppose that each plaintext value is represented as a fixed-point binary number that has f fractional bits after the radix point. Rescale operation is a tool that allows adjusting the value of f for a ciphertext, which is a distinctive feature of CKKS. It always follows a homomorphic multiplication in the OpenFHE library. To determine the computing capabilities of the system, the cipher text module q is the main functional parameter for this task. The higher the q parameter, the more operations can be performed on the encrypted data and the higher the accuracy. For a chosen value of q, the dimension of the cipher text n determines the security level of the system. A larger n represents higher security (see article for more details [START_REF]Protecting Privacy Through Homomorphic Encryption[END_REF]).

FHE allows a third party, such as a PSP, to perform calculations on encrypted data without having access to the plaintext. The result of the calculation is encrypted, and only the owner of the data can decrypt it. In [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF], the authors detail the construction of their scheme with the necessary mathematical proofs. Their algorithms for homomorphic operations satisfy the following properties:

• Homomorphic Addition -Add(c 1 , c 2 ), for given encrypts of m 1 and m 2 , output is an encryption of

m 1 + m 2 which satisfies En(m 1 + m 2 ) = En(m 1 ) + En(m 2 ). • Homomorphic Multiplication -M ult evk (c 1 , c 2 ),for
given encrypts of m 1 and m 2 , output is an encryption of m 1 * m 2 . Which satisfies En(m 1 * m 2 ) = En(m 1 ) * En(m 2 ) + e mult (mod q) for some additional error e mult ∈ R. Note that the noise grows as the number of encrypted computations increases. This is especially an issue with homomorphic multiplication. So, the number of operations needs to be limited otherwise the consequence is a false decryption. By increasing q, the modulus in the ciphertext space, there is more room for noise, which allows more operation. However, the latency of the algorithm increases exponentially with respect to the value of q. Another solution is presented in the literature, under the name of Bootstrapping Technique [START_REF] Bossuat | Efficient bootstrapping for approximate homomorphic encryption with non-sparse keys[END_REF]. This technique increases the level of a cipher text in order to perform more homomorphic operations. It is used to reset the noise and keep the value of q relatively small. As we know from the literature, FHE, even with bootstrapping technique, has a severe handicap that is confirmed in this paper: It has limited functionality, in terms of the operations to be performed. Some operations on encrypted messages such as computing the root, inverting, maximum, and minimum are possible but take a significant amount of time (see V-A).

IV. H3PC: HOMOMORPHIC PRIVACY-PRESERVING PLATOON CONSTRUCTION

Since we have discussed both the technical aspects of platooning formation and the cryptography considerations, this section provides a detailed presentation of our proposed algorithm. We begin by describing the privacy-preserving protocol, followed by an explanation of the equations used to form the platoon based on encrypted data. The system architecture studied in this work comprises two entities: the vehicles (V i ) and the PSP. The task of each vehicle is to provide the necessary data to calculate the instruction. Each vehicle is considered an honest but curious entity that provides correct data and follows the designed protocol and the received instructions. They will eventually merge into the intersection itinerary and be close enough to visually identify each other, such as the vehicle's number plate. The PSP is the entity that provides the instructions, such as the target acceleration. It is an honest but curious entity, which follows the protocol properly but is curious to deduce information about the vehicles. It provides the services and computing infrastructure that will be used to perform operations on sensitive encrypted data.

A. Threat Model and Security Requirements

Our priorities are the following:

• The PSP should not have any access to sensitive information about the vehicles (see Fig. 2).

• No entity should be able to learn the control equation used by the PSP. • Malicious entities should not be able to intercept the data flow between the vehicles and the PSP.

B. Notation

In accordance with the construction provided by the authors in [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF], the following protocol formally describes the vehicle control platoon on encrypted data. We define R q L as the residue ring of R modulo an integer q L . Its elements are polynomials of degrees less than n with coefficients in Z/q L Z. More precisely q L < n. In the CKKS construction, we focus on the following parameters: At level ℓ, the modulus is given by q ℓ = 2 p0+ℓ•p = q 0 • ∆ ℓ . In the construction presented in [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF], the authors utilize a zero-level modulus q 0 and a chain of prime modulo q 1 , q 2 , . . . , q L of the same size, satisfying q i ≡ 1 mod 2n for i = 1, . . . , L. The modulus Q ℓ is computed as the product of this modulo up to level ℓ, i.e., ℓ i=0 q i . The value of L corresponds to the highest level of the ciphertext modulus.

To simplify the notation, for given p > 0 and a modulus q 0 , we set q ℓ = q 0 • ∆ ℓ for ℓ such that 0 < ℓ ≤ L. The integer p serves as the scaling base for approximate computation. For security considerations, we select a parameter M = M (λ, q L ) representing a cyclotomic polynomial of degree n = ϕ(M ), where λ is the security parameter. These parameters play a crucial role in CKKS construction and determine the security, precision, and efficiency of the scheme. The scaling factor is denoted as ∆ = 2 p , and the zero-level modulus is defined as q 0 = 2 p . It is important to ensure that q 0 is greater than ∆ to enable correct decryption.

C. Proposed Protocol

Since we are familiar with the entities involved in the communication and the various notations used. Let's assume that each vehicle will establish individual communication channels with the PSP. So, all communications are over TLS 1.3 protocol with a unique symmetric private key for each vehicle [START_REF] Bhargavan | Verified models and reference implementations for the TLS 1.3 standard candidate[END_REF]. In the upper layer (over TLS 1.3), all vehicles need to establish a common public and private key that is exclusive to the platoon. To accomplish this, during the initial setup phase, the vehicles will engage in a collaborative communication protocol to generate a common public and private key. The protocol ensures that the keys are securely distributed among the vehicles while maintaining confidentiality. Once the common public key is derived, it will be communicated to the PSP, enabling secure interactions and information exchange between the vehicles and the PSP. As illustrated in the Fig. 3. Our construction concentrate on seven algorithms (Setup, KeyGen, Enc, EvalBootstrap, Order, SecAcc target , Dec). The protocol for generating the shared asymmetric homomorphic pair keys between vehicles is out of the scope of this paper. The details of our construction are as follows:

• Setup(1 λ , Φ) Any vehicle in the platoon must agree to respect the parameters F ixed P arm and CKKS shame chosen, so these parameters will be omitted. Let's select a FHE environment (n, p, q, σ). Set the small distributions χ key , χ err , and χ enc over R for secret, error, and encryption, respectively. The Setup algorithm returns F ixed P arm ← (n, p, q, σ, Φ). • KeyGen(1 λ ): The algorithm performs the key generation algorithm of the FHE defined [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF]. In this context, it is assumed that the vehicles collaboratively establish the common keys, while adhering to the following properties:

-Secret key (sk) generation: The secret key s choose randomly from χ key Set the secret key as sk ← (1, s). -Public key (pk) generation: The public key pk = (p 1 , p 2 ) ∈ R 2 q L is computed by p 2 ← -p 1 • sk + e (mod q L ). And by choosing p 1 ← R q L and e ← χ err . tion algorithm return ct m = v •pk +(m i +e 0 , e 1 )modq L where v is choosing randomly from χ enc and (e 0 , e 1 ) ← χ err .

• EvalBootstrap(ct i , numIterations, precision):

Let ct i = (ct i1 , ct i2 ) be a ciphertext at level ℓ = 0. The goal of the bootstrapping operation is to increase the ct i level, i.e. to increase the number of homomorphic multiplications to apply on ct i . The Bootstrapping algorithm returns the same ciphertext but with a higher level (See [START_REF] Badawi | OpenFHE: Open-source fully homomorphic encryption library[END_REF], [START_REF] Bossuat | Efficient bootstrapping for approximate homomorphic encryption with non-sparse keys[END_REF] for more details). • Order(S 1 , S 2 , ..., S m ): To order the set of speeds from minimum speed to maximum speed, we need to perform the Max or Min operation. However, it has been tested that the homomorphic Max/Min function is heavy in terms of computation time. Therefore, for each iteration t, we utilize an alternative function to secretly compute a vector of speeds in sorted order. For more information, please refer to the article [START_REF] Boldyreva | Order-preserving symmetric encryption[END_REF]. The result of the private ordering algorithm is denoted as OrderV ec Speeds ← P rivateOrder(S 1 , S 2 , ..., S m ). In addition to encrypting the speed and position homomorphically, the vehicles send the encrypted positions using Order-Preserving Encryption (OPE) to order them.

• SecAcc target (ct Si , ct Pi
′ = ct 2 + ct 1 • s mod q L , m ′ ← M P C Dec (ct, s).
The algorithm SecAcc target (ct Si , ct Pi , a, b, c, m) is based on a linear equation a linear equation, which will be presented in the following subsection.

D. H3PC Linear Control Technique

In this work, we propose a method that relies on a linear model of vehicle acceleration control to form a consistent and safe platoon. This method has the advantage to be only based on addition, multiplication, and their inverse operation to ensure that vehicles maintain a specified safety and following distance, adjust their acceleration accordingly, and react in real-time to changes in platoon configuration or external conditions. The method is inspired from [START_REF] Marden | Game-theoretic learning in distributed control[END_REF], to form a consensus between agents in a distributed system.

The equation 3 considers the position and speed of all vehicles to compute the target speed S target and the target position P target . And, the equation 5 calculates the target acceleration Acc i target for each vehicle i. It takes into account various factors, such as the difference in speed and position compared to the mean values, the deviation from the ideal platoon size, and the desired position and speed for each vehicle.

To achieve this, we first compute the average speed S Average and position P Average using follow two equations : P Average (P 0 , P 1 , ..., P m-1 ) = . Next, we determine the actual size of the ideal platoon after construction by first computing the ideal target distance h target P (see Fig. 1), using Equation 1. In this equation, p sec represents the security distance between vehicles in the ideal platoon, which can be set to 1 meter. Additionally, p conv represents the convoy position and can be 2 meters. Finally, h t denotes the driver's reflex time, which is assumed to be 2 seconds.

h target P = (p sec • S max + p sec • S Average ) (p sec + p conv ) • h t (1)
Then, the size of the ideal platoon after construction is calculated using Equation 2. After the merging of vehicles, the variable T aille P represents the desired size of the platoon (see Fig. 1).

T aille

P = h P target • m -1 (2)
Based on this information, we deduce the target speed for each vehicle, which is set to the average speed. Furthermore, To get the target position for the first vehicle P 0 target , we need to compute the equation 3.

S target = S Average P 0 target = P Average - T aille P 2 (3)
Since the PSP has knowledge of the vehicle order, we can calculate the target position of the ith-vehicle in the ideal platoon. By computing the equation 4.

F or k = m -1 to 1 :

P i target = P i+1 target + h P target (4)
And to compute the target position of the other vehicles following the 0th vehicle, we can use the following sequential calculation: starting from the last vehicle to the leader, for k = 1 to m -1, we have P i target = P i-1 target + h target P . This equation holds for the vehicles ordered as (P 0 , P 1 , ..., P m-1 ).

In the second part of the technique, we compute the target acceleration Acc i target for ith-vehicle using the results obtained so far, and the equation 5. Where α s = 2 and α p = 0, 1 are the speed and the control gains, respectively.

Acc i target = α s (S i target -S i ) + α p (P i target -P i ) (5)
Note that all operations performed thus far are computed on encrypted data, ensuring that the PSP does not gain any sensitive information about the results or even the intermediate computations. Algorithm 1 outlines our secure target acceleration algorithm, which achieves polynomial complexity O(m). Furthermore, the equations are known only by the PSP, as some companies may need to conceal their operations to maintain a competitive advantage. Regarding the constants used in the equations, they can be adjusted to achieve the desired specifications and requirements of the PSP, thereby allowing control over the final shape of the platoon.

Algorithm 1 SecAcc target algorithm

Input: (ct 0 S , ct 0 P ), (ct 1 S , ct The H3PC technique stands out from others for several reasons. Firstly, the equations used in this technique involve only multiplication and addition operations, making it highly compatible with the FHE technique. This allows for the secure computation of encrypted data without compromising the privacy of sensitive information. Secondly, the H3PC technique requires only two essential data from the vehicles: the position and the speed. By utilizing key parameters, the technique effectively captures the necessary information for controlling the platoon formation and maintaining the desired behavior. Thirdly, the H3PC technique incorporates a crucial aspect known as the maximum time to maintain stable driving behavior. This parameter is carefully set within the range of [100 ms, 600 ms], ensuring that the platoon operates smoothly and maintains safe and consistent driving patterns over time.

The technique meets most of the criteria mentioned in Section III-A (see Table I). However, it still requires some additional tests. There is only one case that did not mark: Adaptability. Our technique is currently being tested on ideal roads and platoon sizes, but it still needs to be evaluated in various traffic scenarios, and road conditions. In this section, we experimentally evaluate the performance of the H3PC construction based on two techniques. Our aim is to present and compare two experiments in terms of calculation costs, demonstrating their practicality in reallife scenarios.

In order to elect the performance and accuracy of the CKKS scheme for privacy-preserving platoon construction. We present the configuration details that were used. We fix the number of fractional bits f = 3, to represent a fixedpoint binary number. Then, We choose the ring dimension n = 1024, and plaintext modulus p = 50. n and p are fixed, then the coefficient modulus q i is chosen to be as close to 2 p as possible to minimize the error introduced by rescaling. The security parameter λ is equal to 64bits, which we build 128 -bit CKKS schemes. In other words, a malicious entity would need to perform 2 1 28 operations to exhaustively search the key space and find the correct key. We choose the lowest possible parameter because for HE, the larger the values, the longer it takes to implement the code (see [START_REF] Badawi | OpenFHE: Open-source fully homomorphic encryption library[END_REF], [START_REF] Albrecht | Homomorphic encryption standard[END_REF] for more details).

H3PC algorithm is implemented using OpenFHE cryptographic library [START_REF] Badawi | OpenFHE: Open-source fully homomorphic encryption library[END_REF] over the CKKS schema. And run on a Mac OS interface with a dual-core Intel Core i5 processor at 2.7 GHz (Turbo Boost up to 3.1 GHz), and 8 GB of integrated LPDDR3 memory at 1866 MHz. The construction is also usable on other operating systems as Linux and Windows.

For the equation 4 of the H3PC algorithm, PSP needs to determine the order of the CAVs based on their encrypted positions. The hard approach is to perform the homomorphic Max/Min operation V-A. Alternatively, a lighter approach is to utilize the OPE technique V-B.

A. Experiment 1: Homomorphic Max/Min

Max/Min is homomorphically difficult to perform and time-consuming. We have utilized mathematical skills (The Convergence) to convert it into addition and multiplication. As shown by Chon, J. et al. [START_REF] Chon | Numerical method for comparison on homomorphically encrypted numbers[END_REF], the authors employ mathematical skills to convert a complex concept into a simpler form 6. However, the conditions for a and b require them to be in the range of [0, 1), for d is the number of iterations. Knowing that positions and speeds used in this work are in the real number ensemble R.

Up until now, we have considered two new operations, namely the inverse and the square root. For the inverse operation, we relied on the algorithm presented in the article [START_REF] Goldschmidt | Applications of division by convergence[END_REF]. The idea is based on the Goldschmidt division algorithm, where the algorithm works by multiplying the divisor repeatedly by a correction factor until the quotient is obtained struction, demonstrates its effectiveness in achieving secure and privacy-preserving platooning. Several key aspects of the proposed methodology are noteworthy. H3PC enables clients participating in the platooning service to secure merge with other vehicles at intersections while maintaining safe and smooth dynamics. The control method ensures safety distances, preserves client data privacy, and upholds the integrity of operations performed by the PSP. Importantly, this service maintains confidentiality without compromising its ability to provide platoon formation.

In our future work, we aim to simulate our vehicle control technique to ensure adaptability to various traffic scenarios and road conditions. More analytical results of the platoon stability and safety will be provided to adjust the platoon according to the computation and communication performances. Additionally, on the security front, we intend to strengthen the shared common key mechanism to withstand potential attacks from malicious clients. By addressing these aspects, we can further evaluate the performance and efficacy of our techniques in real-life scenarios involving CAV. These advancements will contribute to the practical implementation and deployment of our proposed solution, making significant strides towards privacy-preserving.

Fig. 1 .

 1 Fig. 1. Platooning illustration with encryption: the PSP has no access to the speed and position of vehicles

Fig. 2 .

 2 Fig. 2. Encrypted perspective: what PSP sees

•

  Plaintext modulus (p): This parameter represents the modulus used for plaintext values. It should be a positive integer.• Ciphertext modulus (q ℓ ): The ciphertext modulus is the modulus used for encrypted values. The primes q ℓ for i = 1, ..., ℓ are chosen to be as close to 2 p as possible to minimize the error introduced rescaling (see section 2 in[START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF]). • Level ciphertext (ℓ): It refers to a vector in R k q ℓ , where k is a fixed integer.• Real value σ: It follows the χ distribution. It is discrete Gaussian with standard deviation σ if all coefficients of a ← χ are selected from discrete Gaussian distribution with standard deviation σ. • Number of fractional bits f : is the representation as a fixed-point binary number.

Fig. 3 .

 3 Fig. 3. H3PC Encrypted Communication Protocol

  , a, b, c, m): The inputs a, b, c and m are constant. Where a ∈ R qL is a ciphertext of the max car speed in real-time. b ∈ R q L is the encrypted inverse of the number of the vehicles m. c ∈ R qL is the ciphertext of the inverse of the number 3. For a given encrypted speed ct Si and position ct Si of the vehicle i. This algorithm returns the target acceleration Acc targetVi instruction to be respected, Acc targetVi ← SecAcc target (ct i S , ct i P , a, b, c, m). After several iterations, the respect of this target acceleration should lead the m vehicles to form a platoon. (see more details in the section IV-D) • Dec(m, sk): The same as the common Public key, each vehicle has the common secret key of the platoon to decrypt the result, by computing m

S

  Average (S 0 , S 1 , ..., S m-1 ) =

2 M

 2 Solution that we propose is to convert into a ′ = a/(a + b) and b ′ = b/(a + b) for (a ′ , b ′ ) ∈ [0, 1). M ax(a, b; d) = a + b 2 + sqrt((a -b) 2 ; d) in(a, b; d) = a + b 2 -sqrt((a -b) 2 ; d)2

Fig. 4 .

 4 Fig. 4. Experiment 1: Disrupted and Irregular Acceleration Behavior.

Fig. 5 .

 5 Fig. 5. Experiment 2: Encrypted Accelerations of 4 Vehicles During Their Trajectory.

Fig. 6 .

 6 Fig. 6. Experiment 2: Encrypted Speeds of 4 Vehicles During Their Trajectory.

Fig. 7 .

 7 Fig. 7. Experiment 2: Comparison of Real Positions of 4 Vehicles with their Encrypted Target Positions.

TABLE I CRITERIA

 I ASSESSMENT OF H3PC TECHNIQUE. WHERE S IS SAFETY, S&R IS STABILITY&ROBUSTNESS, SM IS SMOOTH MERGER, A IS ADAPTABILITY, SCAL IS SCALABILITY AND EE IS ENERGY EFFICIENCY V. EXPERIMENTAL RESULTS

Badreddine.chah@utbm.fr;

Alexandre.lombard@utbm.fr;

Anis.bkakria@irt-systemx.fr;

Abdeljalil.abbas-turki@utbm.fr;

Reda.yaich@irt-systemx.fr * CIAD UMR 7533, Univ. Bourgogne-Franche-Comté, UTBM, F-90010 Belfort, France.⋄ IRT SystemX, Palaiseau 91120, France.

• Enc(m i , pk): Each ith-vehicle encrypt it message m i , which is the position P i and the speed S i . The Encryp-

ACKNOWLEDGMENT

This research is funded by the German-French joint collaboration program on "Cybersecurity" funded by the MESRI (via ANR) and BMBF (via VDI/VDE IT) (ANR-20-CYAL-0008), within the frame of AUTOPSY project. In addition, we thank the anonymous reviewers for their valuable comments.

(in our simulation, d = 7 iterations for the initialized factor equal to 0,5). So we get a ′ = a * inv(a + b; d) and a ′ = a * inv(a + b; d) for (a ′ , b ′ ) ∈ [0, 1). As for the square root, we relied on the algorithm described in the article [START_REF] Chon | Numerical method for comparison on homomorphically encrypted numbers[END_REF]. The technique is based on Wilkes' method proposed in 1951 [START_REF] Wilkes | The Preparation of Programs for an Electronic Digital Computer: With special reference to the EDSAC and the Use of a Library of Subroutines[END_REF]. It allows us to converge towards the square root result after performing d iterations (in our simulation, d = 12). Final, to compare a and b we multiply the result of M ax(a ′ , b ′ ; d) with (a + b), which have the follow equation: M ax(a, b; d) = M ax(a ′ , b ′ ; d) * (a + b). These computations require at least 14 homomorphic multiplication, utilizing the Bootstrapping technique [START_REF] Kim | Approximate homomorphic encryption with reduced approximation error[END_REF] to increase the multiplication depth of the plaintext.

Unfortunately, this method suffers from significant computational overhead, leading to prolonged computation times. For the same configurations mentioned in subsection V, the computation of the homomorphic maximum between only two vehicle positions can take at least 800ms. Considering the need to compare all vehicles within the platoon group to get their order. For instance, the time taken to compute the order of 4 vehicles based on the Maximum between two values is 6 • 800ms ≈ 4800ms. In addition to this computation, the Bootstrapping, Encryption, SecAcc target , and Decryption operations take approximately 360ms, 1ms, 35ms, and 2ms, respectively II. This means the total for only one iteration of communication is ≈ 5198ms >> 600ms. Note that when the communication time (i.e., vehicle sending data and receiving target acceleration) exceeds 600ms, timely decision-making becomes crucial to maintain optimal platoon dynamics (see Figure 4). That is Why in the following section, we present a lightweight method to efficiently and quickly determine the CAVs' order.

B. Experiment 2: Order-Preserving Encryption (OPE) technique

To address the time issue of maximum calculation in our techniques, we incorporate another type of encryption called Order-Preserving Encryption (OPE) [START_REF] Boldyreva | Order-preserving symmetric encryption[END_REF]. OPE is a cryptographic technique that maintains the order of encrypted data. In other words, if two plaintext values are arranged in a particular order, their corresponding ciphertexts will also maintain the same order. Recalling our protocol 3, in addition to sending the positions encrypted homomorphically with CKKS, the vehicles included in the message the encryption of the same value P i using OPE. This allows the PSP to order the encrypted positions of the vehicles while preserving their privacy. For the configurations mentioned in subsection V, to order m have a Constant Complexity O(1).

In order to evaluate the privacy-preserving capabilities of our proposed system, we choose to conduct our simulation using homomorphic encryption and OPE encryption technique. The simulation was performed using selected example values based on our criteria. Table II presents the performance of key operations in our work. It is important to note that the time step range must be maintained at less than 600ms to have stable vehicle control. For the same parameters presented in Section V, the computation In Fig. 5, 6, and 7, we present the behaviors of four vehicles that adhere to Platooning service. The simulation is conducted with a time step of 9ms, and the initial positions and speeds of the vehicles are as follows: V ehicle 1 (S 1 , P 1 ) = (10, 120), V ehicle 2 (S 2 , P 2 ) = (12, 70), V ehicle 3 (S 3 , P 3 ) = (20, 30), and V ehicle4 (S 4 , P 4 ) = (12, 0). Fig. 5 visually depicts how the vehicles smoothly adapt their acceleration to navigate their trajectories. Each vehicle adjusts its acceleration dynamically, ensuring a seamless transition in its movement. Fig. 6 showcases the speed adaptation process, demonstrating how the vehicles synchronize their speeds to converge towards a common value. This synchronization plays a vital role in maintaining coordination among the vehicles. Lastly, in Fig. 7, we observe the vehicles gradually approaching their respective target positions while maintaining a safe distance between each other. An example of this behavior can be seen in Vehicle 2, which adapts its acceleration by decelerating when necessary. This enables it to align its speed closer to the target speed (V target ) and its position closer to the desired target position. The value of S target and P target are dynamically adjusted in each iteration to keep the smooth behavior. Overall, these figures provide insights into the effectiveness of our system in facilitating coordinated and safe vehicle movements while prioritizing privacy preservation and efficiency.

VI. CONCLUSION

In conclusion, this research presents an advanced solution to address the Platooning Formation problem by harnessing the power of homomorphic encryption. To our knowledge, this is the first approach to tackle this problem using homomorphic encryption techniques, and the proposed solution, H3PC: Homomorphic Privacy-Preserving Platooning Con-


Christophe Bonné

Anish Pahwa

Charly Picard

Marc Visseaux
email: marc.visseaux@ensc-lille.fr

Bismuth tris-silylamide: a New Non-Toxic Metal Catalyst for the Ring Opening (Co-)Polymerization of Cyclic Esters under Smooth Conditions

Keywords: Ring Opening polymerization, bismuth, lactide-lactone copolymer

Bismuth tris-silylamide: A new non-toxic metal catalyst for the ring opening (co-)polymerization of cyclic esters under smooth conditions

Introduction

Bismuth compounds have been relatively little used for catalytic purposes by comparison with other elements over the Mendeleiev classification. The bismuth element, which is present in numerous pharmacology formulations, is considered as being of very low toxicity [1]. It is a large size metal having a strong ionic character that offers a Lewis acid behaviour, therefore standing as effective candidate for the activation of a cyclic ester through Ring Opening Polymerization (ROP) process [2], to afford biodegradable and biocompatible polyesters [START_REF] Lecomte | Ring Opening Polymerization[END_REF]. This domain of the material science has seen an outstanding expansion because of environmental concerns, relating to both the life cycle of the polymer materials with their impact on ecosystems and the nature of the resources used for their preparation [4]. Polylactide (abbreviated as PLA), derived from lactide (LA), dimer of lactic acid, a renewable resource available from starch, is a major player in this area. It is nowadays industrially prepared by means of stannous octoate-mediated ROP. But tin being considered as toxic, it is of interest to propose other catalytic alternatives based on elements that would fit in a frame of biocompatibility. Bismuth derivatives have been studied quite scarcely in this context. Chart 1 shows the different bismuth compounds used for ROP that were reported in the literature: inorganic derivatives such as BiX3 (X = Cl, Br, I) Ia-c [5], the homoleptic acetate IIa [6], 2ethylhexanoate IIb [7], hexanoate IIc [8], triflate III [9,10,11] and alkoxides IVa-c [12], the heteroleptic subsalicylate V [8,13,14], (Salen)bismuth alkoxides VIa and VIb [15] and the diphenyl bismuth bromide VII [16,17]. Among them, a number were also found as efficient catalysts for the copolymerization of lactide with a lactone. Bismuth subsalicylate V, which is the main component of Pepto-Bismol®, has shown a particular interest. The recent studies cited above demonstrate its valuable performances toward the ROP of rac-lactide (LA) or caprolactone (CL), versus those of tin octoate, in terms of activity or degradation kinetics of the polymer material [18], together with a much lower toxicity [2]. But this compound has a complex polymeric structure and is insoluble in all organic solvents, which may be a limitation to its efficiency [8].

Aiming at assessing an alternate simple and soluble bismuth complex, and considering ROP catalysts based on trisamido derivatives of other metals [19], we turned our attention to Bi[N(SiMe3)2]3 (Chart 1, bottom) [20]. This complex is highly soluble in low polar solvents like toluene, which is commonly used for ROP of LA and related cyclic esters. Moreover, we were interested in the possibility to explore the reactivity of this compound in the presence of a borate co-reagent, which is known to easily generate cationic species from a metal-silylamido moiety [21]. The ROP of cyclic esters by cationic catalysts has been the recent subject of a review by Carpentier et al. in which are detailed the potential of such strategy [22].

We present in this contribution our studies devoted to the homo-polymerization of rac-lactide and -caprolactone with Bi[N(SiMe3)2]3 under various experimental conditions, and we also report some results in the statistical co-polymerization of these two monomers.

Experimental

Materials and methods

All experiments were done using Schlenk techniques or a glove box (Jacomex).

Bi[N(SiMe3)2]3 was synthesized as reported [20]. Toluene was purified through alumina column (Mbraun) and stored on molecular sieves in a glove box. E-caprolactone (CL) (Aldrich) was dried over calcium hydride, distilled twice, and stored over molecular sieves in a glove box.

Rac-lactide (LA) (Aldrich) was purified in two steps by dissolution/evaporation from dry toluene, followed by sublimation of the monomer. NMR ( 1 H, 13 C) spectra were recorded on a Bruker Avance 300 apparatus at room temperature in CDCl3. The chemical shifts were calibrated using the residual resonances of the solvent. SEC analyses were performed in THF at 40 °C with a Waters apparatus, calibrated with polystyrene standards, equipped with Waters Styragel columns HR2, HR3, HR5 and HR5E.

Polymerization procedures and characterization

In a typical polymerization (run 1 taken as an example), a flask was charged in a glove box with the Bi initiator (15 mg, 20 µmol). The solvent (toluene, 1.5 mL), the monomer (for a given monomer to initiator ratio, equal to 100 (0.288 g, 2 mmol) for run 1 and eventually benzyl alcohol (none for run 1), were added in this order. After a given time (2 h) and temperature (75 °C) under stirring, the viscous mixture was treated with a few drops of acidified methanol (methanol containing ca. 1% HCl 1M) to quench the reaction, and precipitated in a large excess (200 mL) or methanol. For experiments of LA polymerization or CL/LA co-polymerization carried out in the presence of benzyl alcohol, the reaction mixture was precipitated with a large excess (typically 200 mL) of cold (acetone / liquid nitrogen bath) diethyl ether. The resulting polymeric material was dried under vacuum at room temperature to a constant weight (0.17 g).

The yield (59 %) was determined by gravimetry or by conversion by 1 H NMR of a reaction sample withdrawn at a given time. The activity was calculated according to the following formula: [(monomer conversion)x(monomer to catalyst molar ratio)x(molar mass of monomer)]/(time of reaction in h). The copolymers composition was determined by 1

Polymerization reactions with boron compounds activation of Bi[N(SiMe3)2]3

In some experiments (runs 18-26), a boron compound: B(C6F5)3, [HNMe2Ph][B(C6F5)4] or [CPh3][(BC6F5)4] was added to Bi[N(SiMe3)2]3 (10.2, 16.2 and 18.4 mg, respectively, for 1 equiv) before addition of the toluene. Then other reagents were added as described above to proceed to polymerization reactions.

Results and discussion

Homopolymerization of cyclic esters

The bismuth compound Bi[N(SiMe3)2]3 used for this study is soluble in non-polar solvents, which fits well with our objective to carry on ROP reactions under smooth experimental conditions. It was synthesized by ionic metathesis according to equation 1 in THF followed by extraction with pentane [20]. LA polymerization experiments were conducted in toluene solution, at 75 °C, so as to achieve a compromise between LA solubility and control of the polymerization, since it was reported that undesired transesterification reactions take place when the ROP is conducted in bulk in neat monomer, which requires high temperatures [24]. These conditions were already utilized in a ROP study of Leskelä et al. with bismuth catalysts, therefore offering a possibility of comparing the results [12].

Selected results are shown in Table 1. First, it can be seen from runs 1 to 5 that Bi[N(SiMe3)2]3 displays a good efficiency toward the production of PLA, with an activity of ca. 12 kg polymer/mol(Bi)/h (run 5). This result is comparable to other reported bismuth catalysts, under similar solution conditions, and is superior to the activity of Sn(oct)3 (< 8 kg polymer/mol(Sn)/h) [12]. The experimental molecular weight is higher than the expected theoretical one (calculated for one growing chain per Bi catalyst) in most cases (runs 1-4). This can be related to a rather poor initiation efficiency -the number of polymer chains grown per catalyst -by the Bi-N moiety. With monomer to catalyst ratios higher than 100 (runs 4,5), the initiation efficiency increases up to 1.3, but along with noticeable broadening of the dispersity, in line with the likelihood of several Bi-N active bonds as already proposed in the case of homoleptic trisamido complexes of other metals [25]. For a given quantity of monomer, the dispersity is also increasing with time, which can be attributed to the likely occurrence of undesired transesterification reactions [26]. With Bi[N(SiMe3)2]3 as catalyst and in absence of alcohol (runs 1-5), the classical coordinationinsertion mechanism (Scheme 1) [29] must lead to PLA chains bearing the amide -C(O)-N(SiMe3)2. This was actually observed, with the resonance at  = 0.15 ppm in CDCl3 (0.035 ppm in C6D6) [30], which cannot be confused with the silicone grease residue ( = 0.07 ppm in CDCl3 and 0.29 ppm in C6D6) [31].

Scheme 1. Coordination-insertion mechanism affording amido-terminated PLA chains.

Reactions were in a second time conducted in the presence of an alcohol source in quantities up to a slight excess, with the aim to better control the polymerization due to anticipated faster initiation of M-O rather than M-N moiety [32] and transfer agent capability of the alcohol [33].

This approach was to our knowledge only reported once in the case of bismuth-mediated ROP [34]. To our delight, the performances were improved, with total conversion within 4 h (1-5 equiv benzyl alcohol, runs 6-8). The projected benzyl end-group of the PLA produced was evidenced by 1 H NMR ( = 7.34 ppm in CDCl3, run 8, Figure 1) [35]. However, the signal at  = 0.15 ppm was still present, as a minor contribution. From the relative integration of the latter vs that of the signal at  = 7.34 ppm, for PLA received from run 8, an estimate can be made of 17 out of 18 chains that have an ester chain end, and one chain has an amide chain end. This

shows that the benzyl alcohol has a significant effect on the initiation process, and speaks in favor of alcoholysis of a Bi-N bond, prior to the initiation of the polymerization (scheme 2), leading to a [Bi]-OCH2Ph species that is willing to initiate a PLA chain formation. Doing the same analysis for PLAs received from runs 6 and 7 confirmed the presence of both in a ratio of 5 and 13 when running the polymerization with 1 and 3 equiv BnOH, respectively. Regarding macromolecular data, a regular decrease molecular weights along with narrowing of distribution was noticed with the amount of alcohol added, good agreement with the of transfer reactions (Scheme 3). In the case of 1 equiv BnOH, the broad distribution can be attributed to several active species working at the same time, as

shown by the polymer end-group analysis above. A number of growing chains close to the one expected considering the number of alcohol equiv and one amido group is observed. Under similar conditions as for LA, a low conversion of CL was obtained (run 9, 36%).

This result is quite surprising, since CL is known to generally ring open much more easily than LA [36] even if some exceptions can be found [37]. At 25°C, the conversion was limited to 20 % within 4 h of reaction. The dispersity is very broad whatever the temperature, which is again an indication of slow initiation vs. propagation process. An end-group analysis of low Mn PCL samples performed by 1 H NMR showed the typical resonance of an amido -C(O)-NTMS2 extremity ( = 0.15 ppm in CDCl3, same as in the case of LA), thus establishing the initiation via the [Bi]-NTMS2 moiety. The polymerization was more controlled in the presence of benzyl alcohol, similarly as observed with LA, with evidence of transfer when looking at the number of polymer chains per bismuth metal (2.3, run 11). The sole end-group was this time a benzyloxy one.

LA/CL statistical copolymerization:

PLA has some limitations in terms of physico-chemical properties, and these can be upgraded by introducing a co-monomer, a well-known strategy to modulate the properties and improve the potentialities of a polymer material [38]. In general, when LA/CL copolymerizations were attempted under smooth experimental conditions (solution, low temperature) -to avoid undesired transesterification -very few CL was introduced, apart from a very little number of exceptions [23b, 39,40]. The polymerization of both monomers introduced together was thus assessed under the same experimental conditions as for homopolymerizations. Selected experiments are gathered in Table 2. With equivalent amount of monomers in the feed, some polymer was produced, but the yield was lowered when compared to the polymerization of LA alone (run 12 vs run 4). As shown by the 1 H NMR analysis [40], CL was incorporated at 6 mol% only, which highlights the classical expected preference for LA. A typical copolymer formula is shown in Scheme 4. With a 50/150 feed ratio in favour of CL, the amount of CL in the copolymer increased to 21%, but this was detrimental to the yield (run 15, 15% conversion in 16 h). The probability of CL-CL enchainment (and therefore CL-CL sequences length) was also found higher when feeding the mixture with three times more CL than LA.

Statistical LA-co-CL copolymer obtained with Bi[N(SiMe3)2]3 catalyst

In the presence of 1 equiv. benzyl alcohol, both the conversion and the CL content were improved (run 13), with twice more CL in the copolymer (13% vs. 6 %, run 13 vs. run 12, respectively). As expected also, the Mn value was significantly reduced (here divided by a factor of two). Increasing the benzyl alcohol amount afforded better conversion and an optimal of CL content of 19% (run 14, Fig. 2), which represents three times the quantity observed in absence of benzyl alcohol (run 12). This result is in line with the recent report from the group of Cui [40], who demonstrated the beneficial role of an alcohol as transfer agent in statistical LA/CL copolymerization, while a similar trend had already been notified in the case of non-polar monomers [41]. Besides, the behavior of the catalytic system resembles to that in homopolymerization: the tightening of dispersities and the gradual decrease of the molar masses decipher effective transfer of polyester alcohols formed, to the catalyst (runs 13, 14). One can note however that with a LA-CL/CL-CL ratio which remains stable at the value of 9 whatever the quantity of benzyl alcohol according to NMR, the possibility to extend the CL sequences is not attained. It seems that the presence of the alcohol will rather impact the CL/LA competition after a LA motive than a CL one. In other terms, the occurrence to insert a CL monomer after a LA is improved, but not the one to insert a second CL monomer subsequently. Running the polymerization with a CL/LA feeding ratio of three and in the presence of 3 equiv. benzyl alcohol afforded after 4 h a polymer with 23 % CL content, together with a good activity (run 16, Fig. 2). It is noteworthy that the CL motive was in this case systematically surrounded by two LA units (LA-CL/CL-CL% = 99/1), while the transesterification was limited to a few % (typical lactyl resonance at  = 171.0 ppm Figure 3 [42]). When the reaction time was prolonged (run 17, t = 22 h), a low Mn polymer material having 47 % CL incorporated with a clear increase of the number of CL-CL junctions was found. This, along with the much higher molecular weight (Mn = 47000), speaks in favor of the gradual enrichment of CL sequences in this second stage. The integration signal of the lactyl moiety significantly increased in that case.

Polymerization reactions in the presence of a boron-based activator

As demonstrated recently by the group of Diaconescu, switching between two redox forms of a metal complex may result in a change in the behavior for each monomer in statistical copolymerization reactions [43]. We thus anticipated that the competition between LA and CL would possibly be changed by running the polymerization in the presence of a catalyst having higher Lewis acid character. In this frame, the bismuth catalyst Bi[N(SiMe3)2]3 was assessed in the presence of a boron reagent, which is known to afford cationic species by the activation of a metal-amide bond [21]. The experimental conditions of polymerization were unchanged vs.

previously. Selected experiments are gathered in Table 3. e Mn(th) = yield x [monomer]/[Bi] x 144 (or 114) (calculated for one polymer chain per Bi). f Determined by 1 H NMR. g t = 16 h.

Among the three boron reagents assessed, the tri(perfluoro)phenylborane B(C6F5)3 gave the most valuable results. The catalytic system was found active, but only in the presence of at least 3 equiv. benzyl alcohol. This can be related to a poor solubility of the compound, presumably cationic, resulting from the activation with the boron derivative, or to a coordination step that wins against the ring opening of the monomer. Notably also, when benzyl alcohol was present in the mixture, the kinetic was not found as rapid as without borane (max 86% conv, run 18 vs. 98%, run 7). An interesting feature however is the narrower distribution (1.07-1.36), which can be related to a better controlled process with faster initiation vs. propagation when the bismuth compound is activated by this procedure. The Mn values are in fair adequation with high level of chain transfer in the course of the polymerization, and match quite well with a number of polymer chains close to the number of BnOH equivalents for both LA (runs 18-20) and CL (run

Chart 1 .

 1 Bismuth complexes used in Ring Opening Polymerization of cyclic polar monomers (bismuth tris(silylamido) used in this work in dot square).

  H NMR by integration of the following signals:  (ppm) = 3.30 and 4.06 (PCL), 2.64 and 4.23 (CL monomer), 5.16 (PLA), 5.04 (LA monomer), 2.39 (LA-CL enchainment), 4.13 (CL-LA enchainment) as reported in the literature [23].

  a n(Bi] = 10 mol, [M]0 = 0.67 M in toluene, 75 °C.b Determined by1 H NMR of an aliquot of the reaction mixture. c Mesured by SEC with reference to PS standards and corrected with 0.58 (LA)[27] or 0.56 (CL)[28] factor. d Mn (th) = yield x [LA]/[Bi] x 144 or yield x [CL]/[Bi] x 114 (calculated for one polymer chain per Bi). e Number of polymer chains per Bi = Mn (th)/Mn (exp). f Determined by gravimetry. g T = 25 °C.

Scheme 2 .

 2 Scheme 2. Alcoholysis by BnOH prior to initiation (for one Bi-N bond) affording benzyloxy-terminated PLA chains.

Scheme 3 .Fig. 1 .

 31 Scheme 3. Transfer reaction with benzyl alcohol affording amido-terminated PLA chains

  a n(Bi] = 10 mol, [LA]0 = [CL]0 = 0.67 M in toluene, 75 °C, t = 4h. b Total monomer conversion, determined by 1 H NMR of an aliquot of the reaction mixture. c Mesured by SEC with reference to PS standards and corrected with 0.58 factor. d Determined by 1 H NMR.

Fig. 2 .

 2 Fig. 2. 1 H NMR spectrum of the reaction mixture issued from run 14 (top, red) and run 16 (down, blue) showing the signals of the junction motives in the copolymer along with those of LA, CL, PLA, PCL (CDCl3).

Fig. 3 .

 3 Fig. 3. 13 C { 1 H] NMR region of copolymers issued from run 16 (bottom, blue) and run 17 (top, red) showing the increase of transesterification (CDCl3).

  a n(Bi] = 10 mol, [LA]0 = [CL]0 = 0.67 M in toluene, 75 °C, t = 4h. b Boron additive: B = B(C6F5)3, HNB = [HNMe2Ph][B(C6F5)4], TB = [CPh3][(BC6F5)4]. c Total monomer conversion, determined by 1 H NMR of an aliquot of the reaction mixture. d Mesured by SEC with reference to PS standards and corrected with 0.58 or 0.56 factor depending on the monomer (see above).

Table 1

 1 Solution polymerization of cyclic esters with Bi[N(SiMe3)2]3 as catalyst (runs 1-8, M = LA; runs 9-11, M = CL).

	Run a [M]/[Bi]/[BnOH] t (h)	conv. (%) b	Mn (exp) c	Mn (th) d	N e	Ð c	ester/amide
					(g/mol)	(g/mol)		chain end b
	1	100/1/0	2	59 f	17200	8500	0.5 1.58
	2	100/1/0	4	70 f	20000	10000	0.5 1.74
	3	100/1/0	8	93 f	26700	13400	0.5 1.85
	4	200/1/0	4	83 f	29100	23900	0.8 2.15
	5	400/1/0	4	85 f	36200	48800	1.3 2.26
	6	200/1/1	4	98.6	16100	28400	1.8 1.73 5
	7	200/1/3	4	98.5	6800	28400	4.2 1.54 13
	8	200/1/5	4	98.8	5600	28500	5.1 1.44 18
	9	200/1/0	4	36	16500	8200	0.5 2.32
	10 g	200/1/0	4	20	36600	4600	0.1 1.85
	11	200/1/3	4	91	9000	20800	2.3 1.28 no amide

Table 2

 2 Solution statistical co-polymerization of LA/CL with Bi[N(SiMe3)2]3 as catalyst.

	Run a	[LA]/[CL]/[Bi]/[BnOH]	t (h) conv.	Mn (exp) c	Ð c	CL % d	LA-CL/CL-CL% d
				(%) b	(g/mol)
	12	100/100/1/0	4	39	17100	1.81	6	85/15
	13	100/100/1/1	4	51	8600	1.92	13	90/10
	14	100/100/1/3	4	55	5400	1.60	19	91/9
	15	50/150/1/0	16	15	-	-	21	60/40
	16	50/150/1/3	4	57	3400	1.52	23	99/1
	17	50/150/1/3	22	65	47000	1.68	47	89/11

Table 3

 3 Solution polymerization with Bi[N(SiMe3)2]3 as catalyst activated by a boron compound.

	Run a	[LA]/[CL]/[Bi]/[BnOH]	Add b	conv.	Mn (exp) d Ð d	Mn (th) e	CL	LA-CL/CL-CL% f
				(%) c			(N)	% f
	18	200/0/1/3	B	86	7000	1.15 24800	-	-
							(3.5)
	19	200/0/1/5	B	82	4600	1.24 23600	-	-
							(5.1)
	20	200/0/1/3	2 B	16	2000	1.07 4600	-	-
							(2.3)
	21	0/200/1/3	B	62	5500	1.36 14100	-	-
							(2.6)
	22	200/0/1/3	TB	43	1800	1.13 12400	-	-
	23	200/0/1/3	HNB 21	67000	1.25 6000	-	-
	24	100/100/1/3	B	49	4200	1.16 -	5	93/7
	25 g	50/150/1/3	B	31	33700	1.24 -	21	84/16
	26 g	50/150/1/5	B	42	2000	1.24 -	11	98/2

Acknowledgement.

The authors thank the ERASMUS program (C.P. and C.B.) and the International Relations office of Université de Lille 1 (A.P.) for funding.

21) ROP experiments. It must be emphasized that CL was polymerized recently in bulk by a mixture of B(C6F5)3 with an alcohol reagent [44]. In our hands, ROP experiments conducted with Bi[N(SiMe3)2]3/B(C6F5)3 or Bi[N(SiMe3)2]3/B(C6F5)3/1 equiv BnOH failed to afford some polylactide. Regarding the LA/CL copolymerization, one can observe that in absence of boron activator the amount of incorporated CL is limited to 5 % even in the presence of benzyl alcohol (run 24), which marks a true difference with the neutral conditions (run 14, 19 %). A copolymer containing 21 % of CL is obtained by using a 50/150 LA/CL feeding ratio and 3 equiv BnOH (run 25), which appears as an optimum (run 26, 5 equiv BnOH, 11%).

Experiments aiming at a clarification of mechanistic insights with this

Bi[N(SiMe3)2]3/B(C6F5)3/BnOH catalytic system are in progress.

Conclusion

In this study, Ring Opening Polymerization of rac-lactide and -caprolactone as well as their copolymerization was achieved by means of bismuth tris-silylamide in solution and at temperatures below 75 °C. The performances of this catalyst compare well with other previously studied bismuth derivatives. When the polymerisation is carried out in presence of an alcohol as chain tranfer agent, a benefit over the control of the polymerization is noticed, with higher initiation efficiency of the Bi-O vs. the Bi-N moiety. In statistical copolymerisations of these two monomers, the rate of insertion of -caprolactone into the polylactide backbone is improved of a factor of three in that case. This bismuth catalyst can also be activated by a perfluorotriphenylborane molecule, thus affording much narrower macromolecular distributions. However, the caprolactone content remains limited to ca 20 % in the lactidecaprolactone copolymer.

----------------------------Graphical abstract.