Sami Debbichi
email: sami.debichi@esct.rnu.tn

Walid Hichri
email: hichri@gate.cnrs.fr

Market Power and Collusion on Interconnection Phone Market in Tunisia: What Lessons from International Experiences

Keywords: Termination rate, Market power, Competition, Mobile phone Market. JEL Classification: D41, L96, L71

 to compare the critical threshold of collusion. We show that the "conduct parameter" measuring the intensity of competition is not null during the period 1993-2011. Results show also that collusion is easier on the Tunisian market that on the Algerian, Jordanian, or Moroccan one.

Introduction

Liberalization of the telecommunications sector consists of opening all its segments to competition and privatization ( [START_REF] Wallsten | An Econometric Analysis of Telecom Competition, Privatization, and Regulation in Africa and Latin America[END_REF]). Economic issues in the telecommunications sector and regulation are very important [START_REF] Flacher | Réguler le secteur des Télécommunications ? Enjeux et perspectives[END_REF]). In fact, several economic issues are resolved, as the barriers to entry [START_REF] Baranes | Interconnexion de réseaux et qualité de linfrastructure comme barrière à l'entrée : Quels Instruments de Régulation[END_REF]), interconnection networks [START_REF] Bulatovic | Les Enjeux Economiques de l'Interconnexion des Réseaux de Télécommunications[END_REF], [START_REF] Colombier | Une analyse empirique des stratégies d'interconnexion des opérateurs internet[END_REF], [START_REF] Schiff | Three Essays in Network Economics: Two-Way Interconnection, Two-Sided Networks, and Reputation Systems[END_REF]), the level of pricing [START_REF] Berger | Bill-and-keep vs. cost-based access pricing revisited[END_REF], [START_REF] Dessein | Network competition in nonlinear pricing[END_REF]), privatization [START_REF] Wallsten | Does Sequencing Matter? Regulation Privatization in Telecommunications Reforms[END_REF]), market structure [START_REF] Donder | L'entreprise publique en concurrence : les oligopoles mixtes[END_REF]) and strategic behavior competitors such as competition [START_REF] Laffont | Competition in Telecommunications[END_REF]), collusion [START_REF] Debbichi | Collusion et structure des coûts dans un marché de duopole mixte vs privé de téléphonie mobile[END_REF], [START_REF] Parker | Collusive conduct in duopolies: multimarket contact and cross-ownership in the mobile telephone industry[END_REF], [START_REF] Pénard | Structures du marché et pratiques facilitant la collusion : une approche par la théorie des jeux répétés[END_REF], [START_REF] Souam | Collusion et politique de la concurrence en information asymétrique[END_REF], [START_REF] Berger | Bill-and-keep vs. cost-based access pricing revisited[END_REF]) , agreement merging [START_REF] Artz | The merger paradox in a mixed oligopoly[END_REF]), entry on the market [START_REF] Bourreau | La boucle locale radio comme vecteur d'entrée dans les télécommunications[END_REF]) or deviation.

The mobile phone market had known recently a dynamic and changing structure in most of the countries who have undertaken to reform their telecommunications sectors. Depending on the characteristics of the local market (private or public sector) resulting from the restructuration and the implementation of progressive market liberalization policies, the number of actors (duopoly or oligopoly) and consequently their profits in the telecommunications industry has been variable. There are Many European operators active in the Arab markets. That's why we will begin this work by exposing cases of detection of collusion in European countries. We'll then study the degree of competition in the Tunisian market and similar Arab countries.

Studies conducted by The "Arab Advisor Group" show that the Cellular Competition Intensity Index results for April 2011 revealed that Saudi Arabia tops the score as the most competitive Arab market with a 76.01% mark followed by Jordan (75.37%), in the 6 th rank Morocco (64.72%), Tunisia (63.23%) ranked 8 th , and Algeria (61.17%) who was ranked 9 th . The Cellular Competition Intensity Index is relative in nature as it compares the state of every market relatively to other markets. Consequently, even if a market's absolute level of competition improved, its score in this relative index will also depend on at which level other markets are developed.

In this work, we will first present microeconomic models dealing with market competition and the preference for collusion of operators on interconnected markets. Our goal is to study the state of mobile phone market in Tunisia and to compare the preference for collusion through the threshold value with three Arabian markets that are Morocco, Algeria and Jordan, in duopoly and oligopoly structures, where the actors are private, mixed or public. The market power will be estimated, based on the work of [START_REF] Parker | Collusive conduct in duopolies: multimarket contact and cross-ownership in the mobile telephone industry[END_REF] and on the assumption of "Balanced Calling Pattern". The comparison of the critical threshold of preference for collusion will be based on the model of [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF].and the results of [START_REF] Cortade | Règlementation, Structures de Marché et Comportements Stratégique Sur le Marché de l'Internet[END_REF] and Debbichi and Hichri (2013a). Our findings can be used by the decision makers to control collusion, by acting on the level of interconnection fees for each market structure and by implementing the suitable market liberalization policies in this sector.

Interconnection is a key factor for competition (Laffont et al. (1996)). Each operator must pay an interconnection charge to its competitor for routing the call on its network (twoway interconnection) [START_REF] Baranes | Cost-based access regulation and collusion in a differentiated duopoly[END_REF], [START_REF] Bulatovic | Les Enjeux Economiques de l'Interconnexion des Réseaux de Télécommunications[END_REF]). The stability of the interconnection rate leads us to ask about the state of competition, firstly, on the interconnection market and, secondly, on the retail market (Parsons (2002)). In this context, collusion is one of the possibilities and strategies [START_REF] Colombier | Une analyse empirique des stratégies d'interconnexion des opérateurs internet[END_REF]) that actors may adopt to control the market. Collusion is a strategic behavior chosen by economic agents when it allows better results in comparison to competition. Several studies have already highlighted the determinants of the choice of colluding [START_REF] Parker | Collusive conduct in duopolies: multimarket contact and cross-ownership in the mobile telephone industry[END_REF] and [START_REF] Hoffler | Mobile Termination and collusion, Revisited[END_REF]), and especially its relationship with the level of interconnection fees. Also, telecommunications operators may even use a high access charge as an instrument of collusion [START_REF] Dessein | Network competition in nonlinear pricing[END_REF]). [START_REF] Laffont | Competition in Telecommunications[END_REF] present a study of competition in telecommunications. In the same context, a view of this competition from the United States was presented by Parsons (2002).

The paper is organized as follows: Section 1 presents the European cases detecting collusive behavior. In Section 2, we extend the work of [START_REF] Debbichi | Market Conduct, interconnection costs and benchmarking in mobile phone industry: the Tunisian case[END_REF] to study the structure and state of the mobile phone market in Tunisia. The results of Debbichi and Hichri (2013a) related to the model of [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF] are presented in Section 3. These theoretical results will be applied to different countries and cases following several market structures and presented in section 4. Finally, we present concluding remarks.

The European cases:

On a practical level and starting international experiences, several regulatory frameworks were able to detect these collusive behaviors and their natures, enhancing the damage incurred by the economy and to impose sanctions on operators due to these injuries. The cases of French and Czechs operators and even other operators of the European Union have the best-known cases worldwide. Detection instruments, the nature, scope and level of damage varies from one country to another.

On the mobile phone market, factors that facilitate the implementation of collusive behavior are: exchange of information, convergence of market shares, the monetary transfer between operators and communication between them through regular meetings [START_REF] Pénard | Competition and Strategy on the Mobile Telephony Market : a Look at the GSM Business Model in France[END_REF]). These factors are risky for possible collusion, as it is the case of operators of mobile phone in France. Indeed, they have implemented two types of cartel practices to restrict competition: exchanges of strategic information (these practices may facilitate collusion in the first row) and agreement between 2000 and 2002 to stabilize their market shares. This information exchange reduces the intensity of competition in the mobile market for a main reason: on a market where the entry is very difficult, the exchange of information of this type is likely to affect competition. Consultations in order to stabilize their market share has been established through the intersection of several major indexes, accurate and consistent, such as the existence of handwritten documents explicitly mentioning an "agreement" (explicit collusion), as well as similarities identified during this period in the trade policies of operators, particularly in terms of acquisition costs and pricing of communications. The Competition Council has therefore fined the three mobile operators, Orange France, SFR and Bouygues Telecom respectively amounting to € 256 millions, € 220 millions and € 58 millions.

Collusion may be related to price, quality of service and technical standardization networks. The establishment of such a practice is done through regular meetings of leaders. This is the case between "Deutsche Telekom", "France Telecom", "Telecom Italia", "Telefonica" and "Vodafone" convicted of monopolistic agreement after secret meetings, four meetings in fifteen months, October 2010 in Paris, February 2011 in Barcelona, July 2011 in Venice and on January in London. This is the subject that could be investigated by the European Commission concerning a possible collusion between them, including the standards for future mobile communication services.

The mobile phone market in Tunisia: Structure and Market Power:

Tunisia has one of the more developed mobile markets in Africa as indicated by the high level of mobile penetration. There are three mobile phone operators in the country. The market structure of the mobile phone in Tunisia has gone through several stages, from the monopolistic structure (1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001), to the duopolistic one (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009) until reaching a three operators structure (from 2010 until today). In addition to the market structure, there have been changes in the market shares of the public and the private operators. Indeed, the privatization of Tunisia Telecom (T.T.) (the historical national operator) in 2006 transformed (theoretically) the market from a mixed (Private-Public) duopoly to a private one. With the entry of Orange Tunisia on the market in 2010, one should talk about a three private operators market. Obviously, the preference for collusion is certainly not the same in these different market structures. At the same time, some changing in the market structure was the result of a strategic behavior adopted by the several economic operators. In this context, collusion is one of the possibilities and strategies [START_REF] Colombier | Une analyse empirique des stratégies d'interconnexion des opérateurs internet[END_REF]) that actors may adopt to control the market and practice market power. Latter is the ability to profitably alter prices away from the competitive price. It is measured often by the Lerner index.

We suppose that an operator has a real market power if it sets non-competitive prices above marginal cost. In absence of cost accounting, the measurement of the marginal cost will be more difficult and assessment of market power will become impossible. For this reason, [START_REF] Parker | Collusive conduct in duopolies: multimarket contact and cross-ownership in the mobile telephone industry[END_REF] consider "the conduct parameter" , defined by:

+ ́ q q = θ (1) -θ = -́ q q ⇔ = -1 ́ q q -θ (2) Finally, = ́ ⇔ = (3)
where q is the quantity of interconnection exchanged between two networks, θ is the marginal cost, is the termination price, and represents the demand elasticity of interconnection. The price elasticity of demand is assumed constant (8%) during the period 2002-2011 for both operators, is calculated from the following formula and based on the hypothesis "Balanced Calling Pattern"1 [START_REF] Debbichi | Market Conduct, interconnection costs and benchmarking in mobile phone industry: the Tunisian case[END_REF]):

\ = " -# # "$ " -"$ # "$ # (4)
The Lerner index (margin) L of one operator is equal to its market share divided by demand elasticity (request to interconnection) and is given by the following expression2 :

-

& = ∝ = (5)
The Herfindahl-Hirschman Index (HHI) is defined as the sum of the squared market shares:

))* = + , $ (6)
.

/"

and 0 = + , . /" ⇔ 0 = + , -θ = -θ 1 0 (7)
.

/"

The average index is equal to 0 = 3

θ 4 0 5 where & ̅ = ∑ , $ /"
θ is the weighted average unit cost of interconnection service.

Moreover, as = ∝ 8 this average value is also given by:

0 = + , $ . /" = ))* (8)
The average Lerner index is proportional to the HHI on the interconnection market. We are faced with two alternatives; → 0 (perfect competition of interconnection market) and → 1 (the market is monopolistic). Generally, in the case of Cournot competition between < symmetric operators, → " .

. The parameter measures then the degree of collusion. In this case it's possible to construct an econometric test to reject or to accept the assumption according to which the industry is (or not) competitive, and to compare the theoretical values to the estimated ones.

If

= 0, then = & and prices are equal to marginal costs, which means that the industry is perfectly competitive.

If > 0, then the prices are above marginal costs and the industry is in a collusive situation.

Table 1 [START_REF] Debbichi | Market Conduct, interconnection costs and benchmarking in mobile phone industry: the Tunisian case[END_REF]) presents values of market power (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) during the duopoly era. We will retain these values and complete the study of two eras: monopoly (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001) and oligopolistic in 2011. In the table above, values marked by asterisks are not estimated by the conduct parameter, but using the equation = "

. with < = 1 (Number of operators) in (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001) and < = 3 after 2010. We deduce that → " . = " $ > 0 and that the interconnection price is above marginal cost. This result is valid for the three structures of the Tunisian market (monopoly, duopoly and triopoly). However, the intensity of market power decreases when the number of operators on the market increases. An econometric analysis of market power on the Tunisian mobile industry is presented in [START_REF] Debbichi | Market Power and Competition Effect on Termination Rate: Econometric Analysis of the Tunisian Case[END_REF]. Debbichi and Hichri (2013a) present a Cournot model that compares the intensity of market power using the critical threshold of collusion in duopoly and oligopoly Markets, where the actors are private, mixed or public. Their findings can be used by the decision makers to control collusion by acting on the level of interconnection fees for each market structure and by implementing the suitable market liberalization policies in this sector. An example of the market power applied to the case of The U.S Airline Industry can be found in [START_REF] Murakami | Multimarket contact and Market Power: A case of The U.S Airline Industry[END_REF].

The Results of Debbichi and Hichri (2013a) based on the model of Friedman (1971):

In this section, we present the theoretical results of Debbichi and Hichri (2013a), based on the model of [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF]. The authors assume that the market is composed of two to three operators which are in Cournot competition. Each operator ? = 1,2,3 is characterized by an interconnection fee [START_REF] Flochel | Interconnexion de Réseaux et Charges daccès : Une Analyse stratégique[END_REF], [START_REF] Harbord | Network-Based Price Discrimination and 'Bill-and-Keep' vs. 'Cost-Based' Regulation of Mobile Termination Rates[END_REF]).

The operators agree on a common interconnection tariff " = $ = A = . The authors also assume that the two operators charge retail rates that are very close ). Let B = 1 -C = 1 -( "$ + $" ) the inverse demand function and the Q total amount of exchanged traffic between the two networks.

In this model, there are two industrial configurations in the market, a private and a public operator. The first one maximizes his profit:

D " = (1 -"$ -$" -) "$ + $" (9)

and the second operator maximizes the following function:

D $ = " $ ( "$ + $" ) $ + (1 -"$ -$" -) $" + "$ ( 10
)
If there is privatization of the incumbent operator, the duopoly market structure becomes then private.

The model assumes that there are N operators on the Phone Market, who have the choice between colluding and competing. The incitation to collude will depend on the critical threshold of preference for collusion (C.T.P.C.) that is related to the discount factor F of each operator. Each operator has to choose between two strategic behaviors: either competing or colluding, regarding to the comparison between short-term gains to deviate and long-term losses after deviation, in a repeated game. In such a context, collusion is possible (Debbichi and Hichri (2013a)) when the preference for the present, reflected by the discount rate G, (with F = " "HI and 0 ≤ F ≤ 1) is very low [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF]).

Players are concerned with an indefinitely repeated sequential game where in the first stage, at period K = 0 , they decide to collude. If they cooperate in period K = 1, player ?, where 1, 2 i = (N = 2) realizes a profit equal to D LMNN . A unilateral deviation from collusion will change this profit to D O8P , with D O8P > D LMNN . Debbichi and Hichri (2013a) suppose that a deviation of one player in period t will be followed by a change in the cooperative behavior of the other operator in period K + 1, such that the profit of each operator becomes equal to π QRST , as both operators deviate from collusion. Calculations of the updated value of profit after Deviation U O8P and the updated value of profit after Collusion U LMNN show that:

U O8P = ∑ δ V ∞ V/" π QRWX = δ " δ π QRWX (11) and U LMNN = + δ V ∞ V/" π QRYY = δ 1 -δ π QRYY (12)
Then, collusion is a better strategy if the profit resulting from Deviation, in a repeated game, is lower than the difference between the updated value of profit after Collusion and the updated value of profit after Deviation: 13) From this inequality, Debbichi and Hichri (2013a) calculate the threshold of the discount factor F ̅ from which collusion becomes possible:

Z " Z π QRYY -π QRWX > D ? [ \ -D ? ]^__ (
δ > δ `= π abc -π QRY π abc -π QRde (14)
Consequently, if the value of δ for one operator is higher than F 0 , collusion will be the best strategy to choose. Table 2 presents the different theoretical values of the critical threshold of preference for collusion, as calculated in Debbichi and Hichri (2013a), in different market structures. Duopoly with private operators Oligopoly with three private operators

δ = 1 64 + 1 4 a 3a - 1 2 5 9 64 + 1 4 a 3a - 1 2 5 - 1 9 (1 + 2a)(1 -a) = 9 17 F = 3 4 $ - 2 3 + 3/54 3 4 $ - 2 3 + 1 9 + 1 16 (1 -). (1 -3 )
Duopoly with mixed operators Mixed oligopoly with a public operator and two private

F ̅ = a $ + a - 3 4 5a $ -3a + 1 4 δ `= 3a $ + 2a - 5 3 9a $ -4a + 1/3
As shown in Table 1, the critical threshold of preference for collusion is constant in a private duopoly and is equal to

j "k
. This result is similar to those found by [START_REF] Cortade | Règlementation, Structures de Marché et Comportements Stratégique Sur le Marché de l'Internet[END_REF] in the case of internet operators.

The critical threshold of preference for collusion depends on the interconnection fees . That's why Debbichi and Hichri (2013a) discuss then the variation of F ̅ , the critical threshold of preference for collusion (C.T.P.C.) when the interconnection fees vary. A study of the regulated interconnect rates paid by entrants to incumbents can be found in [START_REF] Edwards | The Effects of Public Ownership and Regulatory Independence on Regulatory Outcomes[END_REF]. Another study related to Internet Interconnection is presented by [START_REF] Laffont | Internet Interconnection and the Off-net Cost Pay Principle[END_REF].

Application to different Arabian Market Structure:

We propose to discuss in this section the variation of F ̅ , when the interconnection fees vary, for each Arabian market structure between 1999 and 2008.

Structure of some Arabian mobile phone Markets:

Algeria

The Algerian mobile market structure was historically a public monopoly with one operator (Algeria Telecom). The first offer "GSM" (Global System for Mobile) was launched in 1999. Since 2001, a private operator, "Orascom Telecom Algeria," has entered on the mobile phone market. The first foreign private operator has commercially launched its brand "Djezzy" in February 2002.Finally, last arrived on the market, "Kuwait Wataniya Telecom" took its license in December 2003 and, six months later, created his brand "Nedjma." Since 2004, three operators are competing in the market for mobile phone in Algeria (see Table 3).

Morocco

The Moroccan mobile market structure was historically a public monopoly with one operator (Maroc Telecom). With the opening of postal and telecommunications competition a second mobile license of type "GSM" is authorized to (Médi Telecom) in 1999. Indeed, the privatization of (Maroc Telecom) (the historical national operator) in 2001 transformed (theoretically) the market from a mixed (Private-Public) duopoly to a private one. Finally, in 2010, a third private operator (Wana) joined the market to transform the duopoly private structure into a three private operators market (see Table 3).

Jordan

The Jordanian market structure has also experienced a profound change. Indeed, a second mobile license type "GSM" was launched in 1999. The privatization of the historical national operator in 2001 transformed (theoretically) the market from a mixed (Private-Public) duopoly to a private one. In 2009, a third private operator joined the market to transform the duopoly private structure into a three private operators market (see Table 3).

5.2) Results and Discussion:

In this section, and based on calculations made by Debbichi and Hichri (2013 critical threshold of preference for collusion for several market structures, we propose to calculate this threshold in the case of Tunisia, Algeria, Morocco and Jordan. (C.T.P.C.) depends on the interconnection along the values taken by the threshold function by changing the value of As shown in Figure 1 (C.T.P.C) in Tunisia is increasing but negative during the period (2002 the year following the privatization remains constant. This is due to a transition from the mixed structure to the private market structure when the public opera Tunisian mobile market, we had and after 2006:

π abc -
As shown in Figure 2 preference for collusion (C.T.P.C year of entry of a third operator, t near to zero). This is due to a transition from the market structure.

In Jordanian and Moroccan market the critical threshold of preference for collusion ( decreasing, but remains positive are substantially the same. taken by the threshold function by changing the value of "a" each year.

1, the value of the critical threshold of preference for collusion increasing but negative during the period (2002ng the privatization of "Tunisia Telecom," the threshold becomes constant. This is due to a transition from the mixed structure to the private market structure when the public operator "Tunisia Telecom" (T.T.) became private. In Tunisian mobile market, we had between 2002and 2006: In this section, and based on calculations made by Debbichi and Hichri (2013a) for the critical threshold of preference for collusion for several market structures, we propose to calculate this threshold in the case of Tunisia, Algeria, Morocco and Jordan. Since the the curves of Figures (1 to 5) are plotted "a" each year.

D LM.l m D O8P m D LMN -π QRY = k(π abc -π QRWX ) with k = 2,
of the critical threshold of preference for collusion 2006). From 2007, becomes positive but constant. This is due to a transition from the mixed structure to the private market private. In fact, in the If we compare the Tunisian case to the Algerian one, the value of the critical threshold of preference for collusion is higher during the period 1999-2007, in the Algerian market than its value in the Tunisian one (see Figure 5). After 2007, the two values are very similar, with a relatively higher value on the Tunisian market. The comparison of the value of the critical threshold of preference for collusion for the four studied countries allows us to conclude that collusion is easier on the Tunisian market, than on the Algerian, Jordanian, or Moroccan one. This result is logical, as the market for mobile phone in Algeria is more competitive (oligopolistic) during the period of study, in comparison to its Tunisian counterpart. The entry of a third operator on the Algerian market was in 2004, whereas it was the case in Tunisia only in (2010).

Obviously, there is a relationship between the number of competitors on one market and collusion, as shown in [START_REF] Selten | A simple model of imperfect competition, where 4 are few and 6 are many[END_REF] who presents a theory that investigates "the connection between the number of competitors and the tendency to cooperate."

Concluding remarks:

We characterized in this paper the state of mobile phone market in Tunisia, Algeria, Morocco and Jordan through the comparison of the level of the critical threshold of preference for collusion. This comparison is based on the market power of operators on interconnection markets, using the Lerner index. This index is a relevant indicator available for the regulator to judge the nature of competition. To keep a certain degree of competition, the regulator, as in [START_REF] Flacher | Réguler le secteur des Télécommunications ? Enjeux et perspectives[END_REF], can set the level of interconnection rate at a level that minimizes collusion. The regulator can control market structure to minimize prices. The "conduct parameter" measuring the intensity of competition is not null during the period (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011), which means that interconnection prices are not oriented to marginal cost and that mobile phone operators practice market power. In fact, operators can maintain these high interconnection charges to inflate prices paid by consumers and reduce the probability of detecting collusion retail prices. Results show that collusion is easier on the Tunisian market that on the Algerian, Jordanian, and Moroccan one. A possible extension of our work could lead us to estimate the "conduct parameter" depending on variables related to market structures and prices.

Figure1

  Figure1:Evolution of the Critical Threshold of Preference for Collusion in the Tunisian Market

Figure1

  Figure1:Evolution of the Critical Threshold of Preference for Collusion in the Tunisian Market

Figure2

  Figure2:Evolution of the Critical Threshold of Preference for Collusion in the Algerian Market

Figure 5 :

 5 Figure 5 : Comparing the critical threshold of preference for collusion in the Tunisian and Algerian Market

Table 1 : Tunisian Market Power value (1993-2011)

 1

Table 2 : Value of the critical threshold of preference for collusion in different Market structures (Debbichi and Hichri (2013a))

 2

Table 3 : The history of some Arabian mobile phone market structures from 1998 to 2010 Years Tunisia Alegria Jordan Morocco 1998 Monopoly Monopoly Monopoly Monopoly 1999 Duo poly Duo poly 2000 Duo poly (priv) 2001 Duo poly(Priv)

 3

	2002	Duo poly	Duo poly
	2003
	2004		Trio poly
	2005
	2006	Duo poly(Priv)
	2007
	2008
	2009		Trio poly
	2010	Trio poly	Trio poly

[START_REF] Laffont | Competition in Telecommunications[END_REF] define this hypothesis as the fact that fraction of calls that is generated on one network and that ends on the other competing network is proportional to the market share of the latter. In other words, the flow of incoming and outgoing calls is balanced even if market shares are not.

For More details see[START_REF] Debbichi | Market Conduct, interconnection costs and benchmarking in mobile phone industry: the Tunisian case[END_REF].


Amirali Habibi

Eugen Dedu

Julien Bourgeois

Guillaume J Laurent

Nadine Le Fort-Piat

Distributed Pneumatic MEMS for Fast Conveyance of Fragile Objects

Keywords: pneumatic MEMS, distributed MEMS, real-time system

In this paper we present a distributed and modular system to convey small and fragile objects. This is done by attaching similar modular blocks together to form a larger conveyance surface. Similar to other networked control systems, each block is composed of several sensors, actuators and communication infrastructure. Control of a levitating object should be done distributed and real-time. We emphasize on realistic simulations in multiple domains such as asynchronous control and communication. Simulations with two strategies on object motion show that we can meet all the real-time requirements for a successful conveyance.

I. INTRODUCTION

Moving very small and fragile objects has been studied through the capabilities of Micro-Electro-Mechanical System (MEMS) devices. In order to achieve it, two main categories of MEMS conveyors were introduced. The first category is grippers and ciliary actuators which have physical contact with the object. In [START_REF] Sardan | Topology optimized microgrippers for nanomanipulation of carbon nanotubes[END_REF], microgripping is presented as a way to grip and move objects. The second category is contactless, like pneumatic actuation and electromagnetic forces. Both also allow to move, sort and settle objects. In particular, [START_REF] Nakazawa | Electromagnetic micro-parts conveyer with coil-diode modules[END_REF], [START_REF] Liu | Parts manipulation on an intelligent motion surface[END_REF] study contactless micro-conveyors made of magnetic alloys and magnetic fields. In [START_REF] Fukuta | Design fabrication and control of MEMS-based actuator arrays for air-flow distributed micromanipulation[END_REF], [START_REF] Nakazawa | Electromagnetic micro-parts conveyer with coil-diode modules[END_REF], [START_REF] Liu | Parts manipulation on an intelligent motion surface[END_REF], [START_REF] Laurent | A new aerodynamic traction principle for handling products on an air cushion[END_REF], the design and fabrication of one conveyor are studied, whereas in [START_REF] Möbes | Toward a 2D modular and self-reconfigurable robot for conveying microparts[END_REF] the application and control of ensemble of MEMS devices is studied.

Basically, our idea is to attach similar conveyors together and make a bigger conveyance surface. For that, using monolithic conveyors is not a good solution, because it needs to be rebuilt for each application. We therefore use modular conveyors. They are scalable, fault-tolerant and could be optimized to use a minimum number of conveyor modules. As a consequence, network delay, asynchronous control and nonprecise sensing will be the tradeoff of our system. Fig. 1 shows the basic idea of how modular blocks can be used in order to move small object over them [START_REF] Möbes | Toward a 2D modular and self-reconfigurable robot for conveying microparts[END_REF]. In order to study distributed algorithms, communication infrastructure and control of our system, we design a simulator. This simulator is event-driven and can simulate group of modules, communication among them and the physics laws applied to move objects with microsecond precision.

The remainder of this paper is organized as follows. In Section II, we briefly go over the previous works. In Section III, we present our design and architecture of the blocks, the aerodynamics of the system and communication infrastructure. In Section IV and V, we briefly present the distributed object recognition and strategies for object motion respectively. In Section VI, we show our simulation results, and finally Section VII concludes the paper.

II. RELATED WORKS

Previous works regarding this research is in two domains. First, MEMS-based and contactless systems for applying forces to move an object. Second, modular robotic systems which use modular robots attached together to fulfill a goal.

In [START_REF] Liu | Parts manipulation on an intelligent motion surface[END_REF] an Intelligent Motion Surface has been introduced as a fast manipulation surface. Authors used arrays of micro-robots programmed as an ensemble of robots for manipulating the objects. Using air force for conveying object as a contactless moving paradigm has been addressed for fragile objects or medicinal products.

For moving small objects, different paradigms have been introduced and implemented. In [START_REF] Becker | Automated manipulation of spherical objects in three dimensions using a gimbaled air jet[END_REF], a robotic air-jet was presented for moving objects with 2-DOF (Degree of Freedom). In [START_REF] Reed | High-speed conveyor junction based on an air-jet floatation technique[END_REF], a high-speed conveyor was introduced and tested for moving fragile objects. Another type of pneumatic microconveyor was presented in [START_REF] Zeggari | Design and fabrication of a new two-dimensional pneumatic micro-conveyor[END_REF] for moving micro objects. Another surface manipulator used for moving and sorting objects, SmartSurface, has been widely studied through modeling of air jets, open-loop vs closed-loop, distributed algorithms etc. Fig. 2 shows the SmartSurface which we use it hardware for our tests. In all previous works on pneumatic systems, the whole system is based on one solid part which makes the object levitate using air force.

Modular robots has been studied in a variety of projects like Kilobots [START_REF] Rubenstein | Kilobot: A low cost scalable robot system for collective behaviors[END_REF], M-TRAN [START_REF] Kurokawa | Distributed self-reconfiguration of M-TRAN III modular robotic system[END_REF], Superbot [START_REF] Salemi | Superbot: A deployable, multifunctional, and modular self-reconfigurable robotic system[END_REF], Molecube [START_REF] Zykov | Evolved and designed self-reproducing modular robotics[END_REF] and roombots [START_REF] Spröwitz | Roombots: Reconfigurable robots for adaptive furniture[END_REF]. Superbot, Molecube and roombot are self-reconfigurable robots similar to our research except that most of those researches are 3-D whereas we use 2-D. In [START_REF] Möbes | Toward a 2D modular and self-reconfigurable robot for conveying microparts[END_REF] a self-reconfigurable robot with array-based MEMS actuators on top was introduced. This concept which is the foundation of our implemented system used Electro-Permanent Magnets to join and disjoin the modules.

III. ARCHITECTURE OF DISTRIBUTED PNEUMATIC BLOCKS

In this section we will introduce the distributed pneumatic system for moving objects. This system uses air forces to lift an object on an air-cushion, afterwards to move it.

In order to move an object over a distributed pneumatic MEMS, the blocks consist of three layers, as shown in Fig. 3. The top layer is formed by the pneumatic MEMS and the sensors, whose role is to blow the air to the above area. The air flow is used for accomplishing two tasks. First, to levitate an object over the whole surface; second, to make a large air flow in front of the object to pull it. Besides, in the first layer we have installed some proximity (on/off) sensors to detect the presence of the object over the block. The middle layer is the electric layer, the printed circuit board (PCB), used for controlling the valves and sensors. Its design is not relevant to the current topic. The bottom layer is the controller and the communication infrastructure. In order to make a big conveyance surface we should stick the blocks together. Fig. 3 shows that blocks could stick together and make a surface to convey the object.

A. Aerodynamic Effect of Airjets over the Object

In this section we briefly introduce the impact of air flows over the object. In [START_REF] Delettre | 2-DOF contactless distributed manipulation using superposition of induced air flows[END_REF], the basics of fluid mechanics have been introduced. In [START_REF] Delettre | 3-DOF potential air flow manipulation by inverse modeling control[END_REF], it has been suggested that any vertical airjet could be considered as a sink which attracts the air through itself. Equation ( 1) shows that the velocity vector of the air U i has direct relation with the strength of i th sink which is Λ i and inverse relationship with distance of the sink which is r i . e r,i is the unit direction vector from sink to the center of the object.

U i = - k i=1 Λ i 2πr i e r,i (1)
From [START_REF] Delettre | 3-DOF potential air flow manipulation by inverse modeling control[END_REF] the given forces in each dimension are as follows:

F air,x = N i=1 -b 2π [f 1 cos(φ + θ) + f 2 sin(φ -θ)]Λ i F air,y = N i=1 -b 2π [f 1 sin(φ + θ) + f 2 cos(φ -θ)]Λ i (2)
The details of aerodynamics are beyond the scope of the current paper, but according to (2) the applied forces at any time to the object is the superposition of the forces applied by all sinks.

In our simulator and models we use these aerodynamic equations to implement the effect of airjets over the object.

B. Communication and Constraints

Before we present our distributed system, we will state some rules and constraints among the whole system. According to (1) the applied force to an object has a reverse relation with the distance of the object to the position of the open valve. So, we define a parameter called effective radius which means the maximum distance from the object as which the open valve could have a noticeable effect on it. In the current paper, the effective radius is considered to be ten centimeters. As a result, we consider in our basic design that any given intelligent block just has information about the position and status of the blocks within the effective radius. The status of a block consists of In order to exchange status of the blocks and objects in the whole system, we propose a Serial Peripheral Interface (SPI) for peer-to-peer communication within the neighboring blocks. Each block sends and receives messages from the four neighbouring blocks.

IV. OBJECT RECOGNITION

In [START_REF] Boutoustous | Distributed control architecture for smart surfaces[END_REF], object recognition in the context of distributed movement of the object has been introduced and experimented. We implement a similar strategy which has differences comparing with that system. The first difference is that the blocks in our experiment are asynchronous, which means different blocks could be in different states. Secondly, the communication infrastructure for message passing is distributed, nondeterministic and has real delays.

When the object is placed over the blocks the binary representation of the object is extracted by proximity sensors. In real pneumatic object levitation, the object does not stand still over the surface. Actually the object is vibrating randomly because of the air turbulence, effect of the airjets over the object and imperfect flat surface underneath the object. So, while the blocks are trying to discover the shape of the object with message passing, there is a probability for the object to move, consequently change the shape of the object.

Object recognition and reconstruction should be done when the object moves over the surface. All the blocks in the trajectory path should know the position and shape of the object. We have approached this problem in two different ways. First, we have implemented a leader-based algorithm, where the leader is the block considered as the right-most and top-most block under the object considered as the responsible for the shape reconfiguration. The matter which let us implement such technique is because the total message passing for object recognition will be less. Second approach, called information forwarding, is that each block sends the status of its sensors to the next blocks in the route, then each block construct the shape of object and decides whether to open its valve or not. Fig. 4 shows the operation of two different strategies.

V. OBJECT MOTION

In this section we present the movement of object over the surface built by modular blocks.

A. Verification of the Object Motion

Before discussing our distributed control algorithm, first we will explain how we have verified our model and simulator by using real data captured from SmartSurface. In this procedure we have conducted experiments which move the object over the surface. We capture the position of the object, status of each airjet from the real system. After gathering the data from the real system we apply the same scenario to our system simulator. In addition, we have added random noises caused by air flows, parasitic noise of levitation airjets that make the formation of the object movement. Finally, the model and simulator came to work the same as the real system with the maximum error of 2 millimeters on object position.

B. Object Movement Strategy

In this part, we propose two different strategies to apply forces to the object. These strategies are based on the object recognition techniques presented in the previous section. In the first strategy (leader-based strategy), the leader reconstructs the shape by exchanging messages with the other blocks under the object, then sends the position and status to next blocks in the path. Second strategy is that the blocks under the object send object information to all the blocks in the path, so each block according to its pre-programmed control law will make a decision for opening the valve or not.

C. Scheduling Policy

Different messages for different tasks such as leader election, status exchange etc. might appear in the middle of other tasks execution such as control algorithm. If a task started (any message can be implied as a new task), it would be queued and processed according to its deadline. Some tasks, such as control to decide for opening or closing a valve and object recognition, are considered as hard real-time, so their deadline should not be missed. The missing of such deadline may lead the object to either running over an open airjet or getting out of the trajectory. In either cases the object may hit a solid place and damaged.

In our implementation we have made the following assumptions. All the tasks are nonpreemptive and all the messages are categorized in different levels of priority. The categorization of the messages are performed off-line by the designer and according to the design goals. We used dynamic priority scheduling for incoming tasks; therefore, we decided to use the Earliest Deadline First (EDF) as scheduler in our implementation [START_REF] Carpenter | A categorization of real-time multiprocessor scheduling problems and algorithms[END_REF].

VI. SIMULATION RESULTS

In order to express the results we use 12 × 12 blocks that shape a conveyance surface. The goal is to move an object from left to right. In each experiment, the velocity of the same object for the two strategies (leader-based and information forwarding) is shown in one axis. Fig. 5 and6 show the velocity when only one row or three rows of airjets in front of the object are opened, respectively. Fig. 5 shows similarity between the two strategies. On the other hand, Fig. 6 shows that the leader-based strategy performs better. This is mainly caused by the high traffic imposed by many messages of the second strategy to the network. As a result, the delays in message reception accumulate and cause the next blocks in the trajectory to open their valves later than in the leader-based strategy.

VII. CONCLUSION

In this paper we presented a distributed real-time system for conveyance of small and fragile objects. We have implemented a precise model of airflows and its physical impact over levitating objects.

We have successfully designed a surface where objects levitate and move. We implemented the reconstruction of an object using proximity sensors over blocks by a completely distributed algorithm. We also proposed two strategies for object reconstruction and control; faster conveyance is obtained when using a leader block which gathers all the information then dispatches it to next blocks in the trajectory.

Fig. 1 :

 1 Fig. 1: Basic Concept of Modular Conveyor.

Fig. 2 :

 2 Fig. 2: Hardware set-up of SmartSurface.

Fig. 3 :

 3 Fig. 3: Schematic view of Smart Blocks, (a) a block, (b) top layers of a block consisting of sensors and an actuator, electronic and microcontroller, (c) group of attached blocks making a bigger conveyance surface and an object over it.

Fig. 4 :

 4 Fig. 4: (a) Leader-based strategy, (b) Information forwarding strategy.

Fig. 4 .

 4 a. and Fig. 4.b. show the command message dispatch from the leader-based and information forwarding strategies respectively.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Velocity of the object for the two strategies with one row of open airjets.


Pierre Toulemonde
email: pierre.toulemonde@grenoble.cnrs.fr

Céline Goujon

Laetitia Laversenne

Pierre Bordet

Rémy Bruyère

Murielle Legendre

Olivier Leynaud

Alain Prat

Mohamed Mezouar

High pressure and high temperature in situ X-ray diffraction studies in the Paris-Edinburgh cell using a laboratory X-ray source

Keywords: X-ray diffraction, laboratory setup, Paris-Edinburgh cell, high pressure crystallography, phase diagram, phase transitions

Introduction

The recent development of synchrotron radiation sources has made easier the study of pressure-temperature phase diagrams of pure elements or compounds by in-situ x-ray diffraction. In addition to this development of brilliant x-ray sources, the use of diamond anvil cells (DAC) in such experiments has permitted, for example, the measurement of melting curves of pure elements at very high temperature and pressure. DAC are also sometimes used in setups using laboratory x-ray sources ( [1] and references therein).

In synchrotron and neutron sources large volume cells are also used, i.e. multi-anvil presses and Paris-Edinburgh cells. One obvious advantage of using a PE cell compared to a DAC is the sample volume probed, typically 1-5 mm 3 : phase transitions observed in PE cells really characterize the main phase and not minor phases or impurities. In addition, considering the acquisition time with laboratory x-ray sources, the thermodynamic equilibrium is more likely to be achieved. PE cells are well adapted for moderate temperature range (700-1200˚C) experiments at HP where setups based on DAC and conventional laser heating do not provide a constant and homogeneous heating of the sample. The most significant application using a large volume cell is the study of HP-HT in-situ reactions between different products. For instance, HP in-situ hydrogenation has been carried out using an internal hydrogen source (anthracen for example) that decomposes upon heating in a Conac setup [3]. Such experiments require that the gasket contains the sample plus the hydrogen source and diffusion barriers (BN). Such assembly is only achievable with a large volume cell.

The use of a large volume press coupled with a laboratory x-ray source is rare. Such development has been made for example by the group of Di Cicco et al. in Italy [2]. Their setup is made up of a 4 columns Paris-Edinburgh cell coupled to a rotating Mo anode generator and operates in an energy dispersive x-ray diffraction mode. In this article, we describe a new setup developed at Institut Néel, CNRS, Grenoble, in France.

Experimental

Our new setup operates a Paris-Edinburgh cell (PE-cell), of the VX type with two columns [4]. It works in the angular dispersive diffraction mode. This new equipment is inspired by the setup developed earlier at the id30 beamline of the ESRF and now improved at the id27 beamline (see drawing fig. 1 in [5]). The figure 1 (a) shows a picture of our setup comprising three main components aligned together on a common track: the x-ray source, the PE-cell and the detector. Our VX PE-cell is mounted on two translation stages (a vertical and an horizontal one, both perpendicular to the x-ray beam) to allow the alignment of the sample in the x-ray beam. The detector, a photosensitive Fuji image plate, can be easily removed from its holder for reading on an off-line reader. After being erased, the image plate can be reproductively repositioned for a new acquisition. We have chosen an x-ray source which produces a quasi-monochromatic micro-beam source from Incoatec operating at the 22.2 keV silver radiation (λ(Ag) Kα1,2 ≈ 0.56 Å). With Ag radiation, the x-ray beam transmission through the HP assembly and sample is higher than with Mo radiation (17.5 keV), 2 times better for a silicon (Z=14) sample for instance and higher than 4 orders of magnitude for samples with high atomic number (germanium for example, Z=32). The huge absorption at the copper Kα radiation (8.05 keV), i.e. far above 4 orders of magnitude compared to the one for Ag radiation, makes definitively impossible to run such an experiment with this low energy source. The Incoatec 30 W air cooled Ag radiation source is equipped with Montel focusing optics. The image plate is placed at the focal distance of 600 mm. The x-ray beam, collimated by vertical and horizontal slits positioned few centimeters next to the optics, shows a divergence of 2.7 mrad and a size of 90 µm at the focus point (on the image plate). Then, the beam size is typically around 200 -250 µm at the sample position. The sample itself is placed in compression, in its gasket, between the anvils of the PE-cell which can be placed at a variable position between the x-ray source and the image plate.

As preliminary check of the setup, we have acquired images on different samples (diameter: 1.5 mm) loaded in the standard 5 mm (in diameter) B-epoxy gasket. For diluted or low to moderate absorbing materials we obtained good quality diffraction images after only 0.5-2 h of counting time. Figure 2(a) and (b) show respectively the image obtained from mixtures of BN + Si and BN + LaB 6 powders and the corresponding intensity versus 2-theta pattern. For samples with large absorption, no usable diffraction signal is obtained through the gasket: this was the case for a mixture Bi 2 O 3 :Mn 2 O 3 1:1 (% mol) for example (XRD image not shown). Note the width of the Bragg peaks, typically around 0.1-0.2deg.(2-theta FWHM). The x-ray beam is not fully monochromatized, then at ambient pressure it is possible to distinguish the XRD signal coming from both K α1,2 radations at large angle.

We have used two types of anvils: tungsten carbide anvils (2 different pairs compatible with 2 different diameters of B-epoxy gaskets: the 5 mm and the 7 mm ones) and sintered diamond (SD) anvils designed for 5 mm gaskets. The high pressure assembly used in all experiments was the standard one [6]. Figure 1(b) shows the main components of the HP assembly. Each sample (diameter: 0.8 mm and 1.2 mm, then typically 1 mm 3 and 3-4 mm 3 for respectively 5 mm and 7 mm gasket), composed of a mixture of powders, was placed in a crucible made of hexagonal BN. After pressurization it was heated at HP by Joule effect induced by the injection of a current in the cylindrical graphite heater. The alignment of the sample in the x-ray beam was performed by measuring the transmission profile of the beam in the vertical plane and in the horizontal plane, in the direction perpendicular to the beam. The acquired images were azimuthally integrated with respect to the beam center using the Fit2D software. Lattice parameters of each phase were determined by adjusting the theoretical XRD pattern of each compound, calculated with PowderCell software (or Fullprof software), with the observed pattern. The precision of these values was better than 0.1%.

Results

The experimental setup was first calibrated to adjust for the applied temperature and pressure on samples before conducting XRD measurements. On one hand, we established the relationship between the oil pressure applied to the anvils and the pressure obtained on the sample. On the other hand, we also investigated the dependence between the power injected in the graphite heater and the temperature obtained at the sample position for several applied pressures. For this purpose we used the diffracted signal of h-BN, MgO, NaCl and gold calibrants (i.e. their lattice volume obtained from the lattice parameters calculated from the position of Bragg peaks) and their associated equations of state (EOS). The typical lattice parameters uncertainty of our calibrants was in the order of 0.001-0.005 Å depending on the P,T conditions (± 0.002 Å for MgO at 7-8GPa for example).

Pressure calibration

Figure 3(a) and (b) shows the pressure calibration curves obtained for WC anvils and related to 5 mm and 7 mm gaskets. SD anvils calibration curve up to 12 GPa is not shown but is in accordance with the one previously obtained in a standard 5 mm gasket (see fig. 1 in [7]). Our WC anvils calibration are also in agreement with previous ones obtained in similar HP assemblies [7]. Both graphs have been drawn with the same vertical and horizontal scales to better show the higher conversion of applied oil pressure in the case of smaller 5 mm gasket. A pressure around 6.5 GPa is reached for an applied nominal oil pressure of 200 bar in the 5 mm gasket while only 3 GPa is obtained for the larger 7 mm gasket. We have observed that old B-epoxy gasket stored in air are less efficient to properly transmit the pressure to the sample than new ones. To have always the same efficiency, gaskets have to be stored under inert gas or vacuum.

Temperature calibration

Figure 4 shows one example of a temperature calibration curve obtained with 5 mm gaskets compressed in WC anvils at two different nominal pressures: 5 GPa and 6.5 GPa (at room temperature (RT)). The four calibrants used in a set of three experiments are indicated: BN, NaCl, Cu and MgO. The P and T cross-calibration was obtained from the EOS of a couple of calibrants (NaCl and Cu in experiment number 4 (labeled exp4 in fig. 4) and MgO and BN in exp9 respectively). For this purpose, we have considering that the pressure and the initial (ambient) slight difference between calculated pressures for both calibrants do not change significantly in this range of heating. We observe that the higher the applied pressure is, the lower the heating efficiency. For instance, a power of 125 W applied to the carbon resistive element allows to reach a temperature around 600˚C and 1100˚C at 6.5 GPa and 5 GPa, respectively. We believe that this large difference is more related to a large difference in thermal conductivity of the calibrants constituting the sample rather than the pressure-induced change of the geometry and resistivity of the graphite heater, or a consequence of the approximation made in the calibration procedure (pressure is considered as constant).

First application: detection of the melting points of pure elements

In this section we show two examples of melting point determination under high pressure: magnesium (at 6.6 GPa) and germanium (at 5 GPa). For both cases, a mixture of fine Mg or Ge with NaCl and BN powders in a weight ratio 1:1:1 was introduced in the BN crucible, then pressurized and heated at different temperatures using progressive power steps. XRD patterns were successively acquired during 1 h to 2 h at each power plateau.

Figure 5 shows the integrated patterns of the experiment performed on Mg upon heating after pressurization at 6.6 GPa. As expected, we clearly see the shift of the Bragg peaks towards lower angles due to the increase of lattice parameters associated with thermal expansion. The step heating experiment was stopped when a drastic decrease of the intensity of the Mg Bragg peaks occurred which is the signature of melting (arrows in fig. 5). The estimated temperature was 1200K ± 100K, in quite good agreement with the melting point reported around 1300K at 6.6 GPa in the literature (see fig. 3 in [8]).

Germanium has a melting point which decreases under pressure with an initial value of 1210 K at ambient pressure and a rate of 35 ± 3K/GPa (determined experimentally, see related references in ref. [9]). In our experiment performed at 5 GPa (not shown here), the Ge Bragg peaks disappeared at around 1100 ± 100K which is in agreement with literature, in particular with our previous experiment of differential thermal analysis performed under pressure in a Conac large volume cell (see fig. 5 in ref. [9]).

Second application: P,T phase diagram of MgH 2

Magnesium hydride and related phases are interesting materials for potential hydrogen storage applications. Nevertheless the P,T diagram of MgH 2 is not established and even at room temperature pressure induced phase transformation are subject of controversy.

We have started the MgH 2 P,T phase diagram study using our new setup. XRD experiments performed during the compression at RT have allowed us to determine that the ambient tetragonal (rutile type) form of MgH 2 transforms into the orthorhombic form at 2.15 ± 0.25 GPa. An incomplete conversion was observed up to 10 GPa. The third MgH 2 listed polymorph, the cubic one [10,11], predicted to appear at 3.9 GPa [12], 7 GPa [13] and 9.7 GPa [14] from calculations is not found. A second set of in-situ XRD experiments was performed upon heating at different nominal pressures to establish the P,T phase diagram up to 8 GPa and 1400 K. Fig. 6 illustrates one of these experiments performed at 7 GPa. In this case BN and MgO were used to determine P and T. The shift of the Bragg peaks and their broadening during the compression at 7 GPa is visible by comparison of the two first XRD patterns. At 7 GPa, as previously observed, the tetragonal towards orthorhombic transformation is not fully completed at RT. However, we observe that a temperature increase of 100˚C is sufficient to fully transform the remaining rutile phase into the orthorhombic phase (see the indicated main 110, 111 reflections). By further heating above 745K a new phase transition is observed towards a another phase which is still under characterization. A detailed study of the P-T phase diagram will be published elsewhere.

Third application: HP-HT synthesis of MgB 2 and C-diamond

As another example, we have followed the in-situ synthesis of MgB 2 and carbon diamond, i.e. the chemical reactions between the different starting precursors.

For magnesium diboride we have compressed at 5PGa and heated up to 800˚C a mixture of 1:2 Mg:B powders placed in a BN crucible. A slight amount of NaCl powder was added to the mixture as a P,T calibrant. The corresponding XRD patterns are shown in fig. 7(a). We observe that Mg starts to react with B from 350˚C (see the (101) main Bragg peak of MgB 2 in fig. 7(a)), far below its melting point at 5 GPa (above 950˚C [8]). The reaction between Mg and B starts in a solid state under pressure, as already observed in an energy dispersive diffraction study using synchrotron radiation [15,16]. This experiment shows that this new XRD setup is well adapted to follow in-situ the synthesis of low Z materials, for instance borides.

The diamond synthesis has already been studied in the past, for example by energy dispersive XRD starting from a HP-HT treatment of a Fe 60 Ni 40 C 20 alloy mixed with graphite [17]. In our case, we have conducted experiments at two different pressures, starting from carbon graphite and using nickel as a catalyst. P and T were determined from the EOS of NaCl and BN calibrants. The melting of nickel (1455˚C at ambient pressure) was reached during the heating in the 5-7 GPa range, marked by the loss of the diffraction peaks of Ni, but no diffraction peaks from C-diamond were detected at "CrystalPress" No. ANR-07-JCJC-0030 (CSD4). Part of this work was supported by the Pole SMINGUE of Université Joseph Fourier Grenoble.

  Overview of the angular dispersive x-ray diffraction setup. From left to right: the image plate for acquisition of diffraction patterns, the VX Paris-Edinburgh cell mounted on motorized translation stages, and the microfocus x-ray beam source. The main components of the standard high pressure cell based on a 5 mm B-epoxy gasket.

Figure 1 .

 1 Figure 1. Pictures of the setup that consists on a Paris-Edinburgh cell and a laboratory x-ray source for in-situ x-ray diffraction at high pressure and high temperature.

Figure 2 .

 2 Figure 2. Transmission XRD images and corresponding integrated patterns for two 5 mm B-epoxy gasket filled with two different mixtures.

  Pressure calibration curve for 5 mm gasket (from 5 experiments); the different calibrants (h-BN, NaCl and Cu) are indicated. Pressure calibration curve for 7 mm gasket; the two calibrants (NaCl and BN) are indicated. Inset: typical XRD image acquired at 1.5GPa.

Figure 3 .Figure 4 .

 34 Figure 3. Calibration curves of the sample pressure versus the oil pressure applied to the piston for both sizes of WC anvils (respectively adapted for 5 mm and 7 mm B-epoxy gaskets).

Figure 5 .

 5 Figure 5. XRD patterns of Mg mixed with BN and NaCl, compressed at 6.6GPa, and heated successively at different temperatures.

2 Figure 6 .

 26 Figure 6. XRD patterns of MgH 2 mixed with BN and NaCl and compressed at 7GPa versus temperature.

Acknowledgements

The authors thank the master students V. Valenti (2011), Le T.N. Huynh (2012) and A. Vauthelin (2013) who participated actively to the related experiments during their internship. This work was supported by Agence Nationale de la Recherche (ANR), project

high temperature, due to the overlapping of those potential peaks with the ones of the sample environment (BN, graphite furnace, NaCl). After quenching, we observe Bragg spots which could be assigned to diamond in the XRD image acquired at a slightly smaller pressure of 4.5 GPa (see fig.

Conclusion

The phase transformations under extreme conditions of pressure and temperature are an original way for obtaining new compounds with unique physical properties like ultrahardness. Synchrotron x-ray diffraction is now become a precious tool to explore the pressure phase diagram of a pure element or a compound, in particular its melting curve under pressure, or to determine the optimal P,T conditions for the synthesis of new metastable (HP) phases. To cope with the limited beam time access in large facilities we have developed a similar in-situ XRD experimental setup based on a laboratory microbeam monochromatic x-ray source working in an angular dispersive mode.

In this article we describe in details this new scientific instrument which is now fully operating up to 12 GPa and 1500 K. It is also a complementary tool to differential thermal analysis under HP-HT setup recently developed in our laboratory. The feasibility of in situ HP-HT XRD experiments by use of this laboratory setup is demonstrated through three types of applications. This field of materials science remains still largely unexplored because the HP-HT study of a combination of elements or compounds are infinite. In the future, such kind of new laboratory setup will help the prospective search and discovery of new materials with interesting properties recovered after HP-HT treatment.


Samir Chouali
email: schouali@lifc.univ-fcomte.fr

Sebti Mouelhi
email: smouelhi@lifc.univ-fcomte.fr

Hassan Mountassir
email: hmountassir@lifc.univ-fcomte.fr

Adapting Component Behaviours using Interface Automata

One of the principal goal of

is to allow the reuse of components in diverse situations without affecting their codes. To reach this goal, it is necessary to propose approaches to adapt a component with its environment when behavioural mismatches occur during their interactions. In this paper, we present a formal approach based on interface automata to adapt components in order to eliminate possible behavioural mismatches, and then insure more flexible interoperability between components.

Introduction

The idea of component based software engineering [START_REF] Szyperski | Component software: Beyond object oriented programming[END_REF][START_REF] Heineman | An evaluation of component adaptation techniques[END_REF] is to develop software applications not from scratch but by assembling various library components. This development approach allows software reuse without changing components code. A component is a unit of composition with contractually specified interfaces and explicit dependencies [START_REF] Szyperski | Component software: Beyond object oriented programming[END_REF]. An interface may describe component information at the level of signature (method names and their types), behavior or protocol (scheduling of method calls), semantic (method semantics) without disclosing the component implementation. The success of applying the component based approach depends on the interoperability which holds between two components when their interfaces are compatible. Usually interoperability is not guaranteed when we reuse and assemble components. This is due to possible mismatches that may occur, between components, at different levels cited above [START_REF] Canal | Software adaptation[END_REF]. In this case, components adaptation should be performed in order to eliminate the resulting mismatches.

In this paper 1 , we focus on adapting components, described by interfaces which are specified by interface au-tomata [START_REF] Alfaro | Interface automata[END_REF]. This formalism has the ability to model a temporal order of, both the input requirements (input actions), the output behavior (output actions) of a component, and the local events of a component. The composition of two interfaces is achieved by synchronizing their shared output and input actions. An interesting verification approach was also proposed in [START_REF] Alfaro | Interface automata[END_REF], to detect interface incompatibilities that may occur when, from some states in the synchronized product, one automaton issues a shared action as output which is not accepted as input in the other. We say that those states are illegal. Our purpose is to generate automatically an adaptor (interface in the middle) for exactly two component interface automata according to a mapping that establishes a number of rules relating the component actions. It allows the elimination of mismatches at the signature and the protocol levels.

One of the aims of this paper is to bring together the notion of adaptability of interface automata and their optimistic approach of composition. The latter makes the two interface automata A 1 and A 2 compatible if there is an environment that prevents their synchronization to enter illegal states. The composition approach of the other automatabased interface formalisms are considered pessimistic.

Interface Automata

Interface automata have been introduced by L. Alfaro and T. Henzinger [START_REF] Alfaro | Interface automata[END_REF], to model the temporal behavior of software component interfaces. Every component interface is described by one interface automaton. In this article, we use the formalism of interface automata (IAs) defined by L. Alfaro and T. Henzinger without any extension, except that the set of hidden actions of an interface may contain the special action epsilon ǫ that symbolizes the no-operation event. We redefine, as consequence, the composability condition of interface automata. Two interface automata A 1 and A 2 are composable if

Definition 1 (Interface Automata). An interface automaton

A = S A , I A , Σ I A , Σ O A , Σ H A ,
Σ I A1 ∩ Σ I A2 = Σ O A1 ∩ Σ O A2 = Σ H A1 \ {ǫ} ∩ Σ A2 = Σ H A2 \ {ǫ} ∩ Σ A1 = ∅.
All the notations used and the specificities of the optimistic view of interface automata composition are detailed in the reference article of the formalism [START_REF] Alfaro | Interface automata[END_REF].

Behavioural Mismatch in IAs

The behavioural mismatch of IAs cannot be detected by applying the synchronized product between two composable interface automata as it was defined in [START_REF] Alfaro | Interface automata[END_REF], because the case where the action names do not correspond leads to make them absent in the set of shared actions. Thus, all of mismatched actions do not synchronize and they are interleaved asynchronously in the product. To solve this constraint, we define a set of rules called a mapping which provides the minimal specification of the adaptor of two adaptable components. These rules define correspondences between the mismatched actions.

Definition 2 (Rules and Mappings). A rule α for two composable interface automata

A 1 and A 2 , is a tuple L 1 , L 2 ∈ (2 Σ O A 1 × 2 Σ I A 2 ) ∪ (2 Σ I A 1 × 2 Σ O A 2 ) such that (L 1 ∪ L 2 ) ∩ Shared(A 1 , A 2 ) = ∅ and if | L 1 | > 1 (or | L 2 | > 1) then | L 2 |= 1 (or | L 1 |= 1). A mapping Φ(A 1 , A 2 ) for two composable interface automata A 1 and A 2 is a set of rules α i , for 1 ≤ i ≤ | Φ(A 1 , A 2 ) | 2 .
According to the Definition 2, a rule in our approach deals with one-to-one and one-to-many correspondences between actions. More clearly, the adaptation may in general relate either an action or a group of actions of one automaton with one action in the other. The adaptability of interface automata permits to make sense to the synchronization of some non-shared actions between two composable interface automata A 1 and A 2 according to a fixed mapping Φ(A 1 , A 2 ). We denote the set of the mismatched actions by

Mismatch Φ (A 1 , A 2 ) = {a ∈ Σ ext A1 ∪ Σ ext A2 | ∃ α ∈ Φ(A 1 , A 2 ) . a ∈ Π 1 (α) ∨ a ∈ Π 2 (α)} 3 .
We call that two interface automata A 1 and A 2 are adaptable if they are composable, their mapping Φ(A 1 , A 2 ) is not empty, and there exists an adaptor for them which is compatible with both A 1 and A 2 . We distinguish two principle cases: (i) if the generated adaptor is empty then A 1 and A 2 cannot be adapted (ii) Otherwise, if A 1 Ad A 2 is non empty, then A 1 and A 2 are compatible after their adaptation. 2 | S | is the cardinality of some set S. 3

Adaptor Specification

We present the adaptor specification of two composable interface automata A 1 and A 2 . Some preliminaries have to be introduced before. Given an interface automata A, we denote by Θ S A (s) ⊆ S * A the set of successor finite runs θ = s 1 a 1 s 2 a 2 ...s n such that s 1 = s, s n is the initial state or a state that have no outgoing transitions, and for all 1 ≤ i < n, there is a transition (s i , a i , s i+1 ) ∈ δ A . We denote by Θ P A (s) ⊆ S * A the set of predecessor finite runs θ = s 1 a 1 s 2 a 2 ...s n is defined exactly as Θ S A (s) except s 1 = i where i ∈ I A and s n = s. The set Θ A of all finite runs of A equals to Θ S A (i) where i ∈ I A . We say that a succession of transitions s 1 a 1 s 2 a 2 ...s n (for n 2) is included in a run σ in Θ S A (s) or Θ P A (s) (denoted by the operator ⊑), if all transitions of s 1 a 1 s 2 a 2 ...s n are transitions of σ.

Definition 3 (Adaptor). Given two composable interface automata A 1 , A 2 , and a mapping Φ(A 1 , A 2 ), an adaptor for A 1 and A 2 according to the mapping

Φ(A 1 , A 2 ) is an interface automata Ad = S Ad , I Ad , Σ I Ad , Σ O Ad , Σ H Ad , δ Ad such that • Σ I Ad = {a ∈ Σ O A1 ∪ Σ O A2 | a ∈ Mismatch Φ (A 1 , A 2 )}; Σ O Ad = {a ∈ Σ I A1 ∪ Σ I A2 | a ∈ Mismatch Φ (A 1 , A 2 )}; Σ H Ad ⊆ {ǫ}; δ Ad ⊆ S Ad × Σ I Ad ∪ Σ O Ad ∪ {ǫ} × S Ad ; • Shared(Ad, A 1 ) = α∈Φ(A1,A2) Π 1 (α); Shared(Ad, A 2 ) = α∈Φ(A1,A2) Π 2 (α);
• For all s ∈ S Ad and σ ∈ Θ P Ad (s), if there exist r 1 a 1 ...r n a n s ⊑ σ and α ∈ Φ(A 1 , A 2 ) such that Π i (α) ⊆ Σ O Ai for i ∈ {1, 2} and Π i (α) ⊆ k∈1..n {a k }, then for all ρ ∈ Θ S Ad (s), there exists

sb 1 ...b m t m ⊑ ρ such that Π 3-i (α) ⊆ Σ I A3-i and Π 3-i (α) ⊆ l∈1..m {b l }.
Property 1 An adaptor Ad for two composable interface automata A 1 and A 2 according to a mapping Φ(A 1 , A 2 ) is composable with A 1 and A 2 .

The property can be easily verified according to Definition 3 and the definition of the interface automata composability.

Example 1. Given the two composable interface automata Client and Server shown in Figure 1 and a mapping Φ(Client,Server) = { {login!}, {usr?,pass?} , {req!}, {query?} , {arg!}, {value?} , {ack?}, {service!} {ok?}, {connected!} }. As the reader can conclude, Client and Server are adaptable since their adaptor Ad (cf. Figure 1) is composable and compatible with both of them and it satisfies all the items of Definition 3. The composite interface automaton (Client Adaptor) Server is non empty.

Adaptor Construction

The adaptor construction is splited into two algorithms. The result of the first one (Algorithm 1) is illustrated in Figure 2. It traverses the two interface automata A 1 and A 2 to be adapted, according to a mapping Φ(A 1 , A 2 ), by exploring alternatively the states and the transitions as far as possible along the branches of each of them. It updates as one goes along the set of states S and the set of transitions T initially set to the empty set. Red states represents states that from where the output actions of a given rule α in the mapping are not followed by all their corresponding input ones. These states are called blocking.

To satisfy the last condition of Definition 3, Algorithm 2 removes all the fragments of runs leading to the blocking states. It traverses the resulting graph of Algorithm 1 using a loopback starting from blocking states until states which have at least one outgoing transition leading to a non marked state by the loop. If there is no remaining state then the adaptor is not defined between A 1 and A 2 . We introduce the set

PRED d T (B) (B ⊆ S) by { λ ∈ S\B. λ ∈ Pred T (µ) ∧ µ ∈ B ∧ (deg o T (λ) 4 = 1 ∨ (deg o T (λ) > 1 ∧ Succ T (λ)⊆ B))}
where Pred T (λ) and Succ T (λ) are respectively the sets of predecessor and successor of λ ∈ S in T .

Algorithm 2 Adaptor Constructor Input: The resulting sets S and T of Algorithm 1

Initialisation: B 0 ={ν ∈ S . ν est bloquant}.

1. For all k 0, let 4 For a state s, we define by deg o T (s) the number of transitions which have s as origin in a set of transitions T. By applying Algorithm 2 to the transition system presented in Figure 2, we obtain our desired adaptor shown in Figure 1. It performs a loop back from the blocking states 11, {1ogin!}, {pass?} and 14, {connection!}, {ok?} until states which have more than one outgoing transitions. Given two interface automata A

B k+1 = B k ∪ PRED d T (B k ); 2. Back to the step 1 while B k ⊆ B k+1 ; Output: The adaptor Ad of A 1 and A 2 according to Φ(A 1 , A 2 ) tel que -S Ad = { ν ∈ S \ B k } and -δ Ad = {(ν, a, ν ′ ) ∈ S Ad × Σ Ad × S Ad such that ν, ν ′ ∈ T \ { (µ, b, µ ′ ) ∈ T | µ, µ ′ ∈ S \ B k }} .

Related Works

Many papers about component adaptation are based on the work presented in [START_REF] Yellin | Protocol specifications and component adaptors[END_REF], where the authors proposed an interesting approach, based on finite state machine, to adapt components specified by interfaces describing component protocol and action signatures. This approach deals with one-to-one relations between actions. In [START_REF] Min | Using smart connectors to resolve partial matching problems in cots component acquisition[END_REF], the authors propose the Smart Connectors approach which allows the construction of adaptors using the provided and required interfaces of the components in order to resolve partial matching problems in COTS component acquisition. In [START_REF] Vanderperren | Towards a new component composition process[END_REF], the authors have developed the tool PaCoSuite to edit visually components, and to generate adaptors using signature and protocols interfaces. In [START_REF] Schmidt | Generating adapters for concurrent component protocol synchronisation[END_REF], Schmidt and Reussner presented a particular adaptation approach as a solution to synchronisation problems between concurrent components. The approach addresses for instance situations where one component is accessed simultaneously by two other components. The proposed method was implemented in the CoConut/J tool suite [START_REF] Reussner | Automatic component protocol adaptation with the coconut/j tool suite[END_REF], where the authors introduce the concept of parameterized contracts and a model for component interfaces. They also present algorithms and tools for specifying and analyzing component interfaces in order to check interoperability and to generate adapted component interfaces.

Others approaches for the generation of adaptor protocols from component behavioral interfaces and composition contracts based on process algebra were proposed in [START_REF] Bracciali | A formal approach to component adaptation[END_REF][START_REF] Mateescu | Behavioral adaptation of component compositions based on process algebra encodings[END_REF]. In [START_REF] Passerone | Convertibility verification and converter synthesis: Two faces of the same coin[END_REF], the authors developed a game-theoretical approach to find out whether incompatible component interfaces can be made compatible by inserting a converter between them which satisfies specified requirements. In [START_REF] Autili | Automatic adaptor synthesis for protocol transformation[END_REF], the authors proposed an approach based on transition systems and message sequence charts, for automatic synthesis of failure-free coordinators (adaptor) for COTS component based systems. In [START_REF] Mouakher | Component adaptation: Specification and verification[END_REF], the authors proposed a model of adaptors expressed in the B formal method, allowing to define the interoperability between components.

The approaches described above propose solutions for the component adaptation based on different specification formalisms of component interfaces. In our approach, we propose a solution to adapt particular components that are specified by interface automata. This formalism is more general and based on rich notation, allowing to deal with more complex adaptation scenarios.

Conclusion and Future Works

In this paper, we propose a methodology for the automatic development of component adaptors, allowing the elimination of behavioural mismatches between interacting components, described by interface automata. We propose an algorithm that generates the adaptor for two adaptable interface automata according to a fixed mapping. The generated adaptor allows to eliminate mismatches at the signature and the protocol levels. The proposed algorithms were implemented in Java in order to validate them, and we plan to propose a complete tool in the future works. Actually, we are developing a tool that implements a framework checking the compatibility between interface automata at the protocol and the semantic levels [START_REF] Mouelhi | Refinement of interface automata strengthened by action semantics[END_REF]. We plan also to implement the proposed adaptation approach in our framework.

  Π 1 ( a, b ) = a and Π 2 ( a, b ) = b are respectively the projection on the first element and the second element of the tuple a, b .

Figure 1 .

 1 Figure 1. The adaptor Ad for a variant of a client/server system

Figure 2 .

 2 Figure 2. The result of Algorithm 1 for the client/server system of Example 1

1 and A 2 ,

 2 Algorithm 1 has the complexity O(|S A1 × S A2 |.(|δ A1 | + |δ A2 |)). The complexity of Algorithm 2 is in linear time on the number of states in S.

  A ⊆ S A × Σ A × S A of transitions between states.

	δ A consists of: a finite set
	S A of states; a subset of initial states I A ⊆ S A . Its cardi-
	nality card(I A ) ≤ 1 and A is called empty if I A = ∅; three
	disjoint sets Σ I A , Σ O A and Σ H A of inputs, output, and hidden
	actions; a set δ

 [START_REF] Alfaro | Interface automata[END_REF]This work has received support from the French National Agency for Research , ANR-06-SETI-017 (TACOS).


Christophe Guyeux
email: christophe.guyeux@univ-fcomte.fr

Jacques M Bahi
email: jacques.bahi@univ-fcomte.fr

A NEW CHAOS-BASED WATERMARKING ALGORITHM

Keywords: Watermarking, Chaotic iterations, Topological chaos, Information hiding

This paper introduces a new watermarking algorithm based on discrete chaotic iterations. After defining some coefficients deduced from the description of the carrier medium, chaotic discrete iterations are used to mix the watermark and to embed it in the carrier medium. It can be proved that this procedure generates topological chaos, which ensures that desired properties of a watermarking algorithm are satisfied.

INTRODUCTION

Information hiding has recently become a major security technology, especially with the increasing importance and widespread distribution of digital media through the Internet. It includes several techniques, among which is digital watermarking. The aim of digital watermarking is to embed a piece of information into digital documents, like pictures or movies for example. This is for a large panel of reasons, such as: copyright protection, control utilization, data description, integrity checking, or content authentication. Digital watermarking must have essential characteristics including imperceptibility and robustness against attacks. Many watermarking schemes have been proposed in recent years, which can be classified into two categories: spatial domain [START_REF] Wu | A chaos based robust spatial domain watermarking algorithm[END_REF] and frequency domain watermarking [START_REF] Cong | A wavelet packets watermarking algorithm based on chaos encryption[END_REF], [START_REF] Dawei | A chaos-based robust wavelet-domain watermarking algorithm[END_REF]. In spatial domain watermarking, a great number of bits can be embedded without inducing too clearly visible artifacts, while frequency domain watermarking has been shown to be quite robust against JPEG compression, filtering, noise pollution, and so on. More recently, chaotic methods have been proposed to encrypt the watermark, or embed it into the carrier image for security reasons.

In this paper, a new watermarking algorithm is given. It is based on the commonly named chaotic iterations and on the choice of relevant coefficients deduced from the description of the carrier medium. This new algorithm consists of two basic stages: a mixture stage and an embedding stage. At each of these two stages, the proposed algorithm offers additional steps that allow the authentication of relevant information carried by the medium or the extraction of the watermark without knowledge about the original image.

This paper is organized as follows: firstly, some basic definitions concerning chaotic iterations is recalled. Then, the new chaos-based watermarking algorithm is introduced in Section 3. Section 4 is constituted by the evaluation of our algorithm: a case study is presented, some classical attacks are executed and the results are presented and commented on. The paper ends by a conclusion section where our contribution is summarized, and planned future work is discussed.

BASIC RECALLS: CHAOTIC ITERATIONS

In the sequel S n denotes the n th term of a sequence S, V i denotes the i th component of a vector V and f k = f • ... • f denotes the k th composition of a function f . Finally, the following notation is used: 1; N = {1, 2, . . . , N}.

Let us consider a system of a finite number N of cells, so that each cell has a boolean state. Then a se-quence of length N of boolean states of the cells corresponds to a particular state of the system. A sequence which elements belong in 1; N is called a strategy. The set of all strategies is denoted by S.

Definition 1 Let S ∈ S. The shift function is defined by σ : (S n ) n∈N ∈ S -→ (S n+1 ) n∈N ∈ S and the initial function i is the map which associates to a sequence, its first term: i :

(S n ) n∈N ∈ S -→ S 0 ∈ 1; N .
Definition 2 The set B denoting {0, 1}, let f : B N -→ B N be a function and S ∈ S be a strategy. Then, the so-called chaotic iterations are defined by x 0 ∈ B N and ∀n ∈ N * , ∀i ∈ 1; N ,

x n i = x n-1 i if S n = i f (x n-1 ) S n if S n = i.
(1)

A NEW CHAOS-BASED WATERMARKING ALGORITHM

Most and Least Significant Coefficients

Let us first introduce the definitions of most and least significant coefficients of an image.

Definition 3 For a given image, the most significant coefficients (in short MSCs), are coefficients that allow the description of the relevant part of the image, i.e. its most rich part (in terms of embedding information), through a sequence of bits.

For example, in a spatial description of a grayscale image, a definition of MSCs can be the sequence constituted by the first three bits of each pixel. Definition 4 By least significant coefficients (LSCs), we mean a translation of some insignificant parts of a medium in a sequence of bits (insignificant can be understand as: "which can be altered without sensitive damages").

The LSCs are used during the embedding stage: some of the least significant coefficients of the carrier image will be chaotically chosen and replaced by the bits of the (possibly mixed) watermark.

The MSCs are only useful in case of authentication, mixture and embedding stages will then depend on them. Hence, a coefficient should not be defined at the same time both as a MSC and a LSC: the LSC can be altered, while the MSC is needed to extract the watermark (in case of authentication).

Stages of the Algorithm

Our watermarking scheme consists of two classical stages: the mixture of the watermark and its embedding into a cover image.

Watermark mixture

For security reasons, the watermark can be mixed before its embedding. A common way to achieve this stage is to use the bitwise exclusive or (XOR), for example, between the watermark and a logistic map. In this paper, we will introduce a mixture scheme based on chaotic iterations. Its chaotic strategy will be highly sensitive to the MSCs, in case of an authenticated watermark [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF]. For the details of this stage see the Paragraph 4.1.2 in Section 4.

Watermark Embedding

This stage can be done either by applying the logical negation of some LSCs, or by replacing them by the bits of the possibly mixed watermark.

To choose the sequence of LSCs to be changed, a number of integers, less than or equals to the number N of LSCs, corresponding to a chaotic sequence U k k , is generated from the chaotic strategy used in the mixture stage and possibly the watermark. Thus, the U kth least significant coefficient of the carrier image is either switched, or substituted by the k th bit of the possibly mixed watermark. In case of authentication, such a procedure leads to a choice of the LSCs which are highly dependent on the MSCs.

On the one hand, when the switch is chosen, the watermarked image is obtained from the original image, whose LSCs L = B N are replaced by the result of some chaotic iterations. Here, the iterate function is the vectorial boolean negation, defined by f 0 : B N -→ B N , (x 1 , . . . , x N ) -→ (x 1 , . . . , x N ), the initial state is L and strategy is equal to U k k . In this case, it is possible to prove that the whole embedding stage satisfies topological chaos properties [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF], but the original medium is needed to extract the watermark.

On the other hand, when the selected LSCs are substituted by the watermark, its extraction can be done without the original cover. In this case, the selection of LSCs still remains chaotic, because of the use of a chaotic map, but the whole process does not satisfy topological chaos [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF]: the use of chaotic iterations is reduced to the mixture of the watermark. See the Paragraph 4.1.3 in Section 4 for more details.

Extraction

The chaotic sequence U k can be regenerated, even in the case of an authenticated watermarking: the MSCs have not been changed during the stage of embedding watermark. Thus, the altered LSCs can be found. So, in case of substitution, the mixed watermark can be rebuilt and "decrypted". In case of negation, the result of the previous chaotic iterations on the watermarked image, is the original image.

If the watermarked image is attacked, then the MSCs will change. Consequently, in case of authentication and due to the high sensitivity of the embedding sequence, the LSCs designed to receive the watermark will be completely different. Hence, the result of the decrypting stage of the extracted bits will have no similarity with the original watermark.

A CASE STUDY

Stages and Details

Images Description

Carrier image is the famous Lena, which is a 256 grayscale image and the watermark is the 64 × 64 pixels binary image depicted in Fig. 1a. The embedding domain will be the spatial domain. The selected MSCs are the four most significant bits of each pixel and the LSCs are the three following bits (a given pixel will at most be modified by four levels of gray by an iteration). The last bit is then not used. Lastly, LSCs of Lena are substituted by the bits of the mixed watermark.

(a) Watermark.

(b) Watermarked Lena. Figure 1: Watermark and watermarked Lena.

Mixture of the Watermark

The initial state x 0 of the system is constituted by the watermark, considered as a boolean vector. The iteration function is the vectorial logical negation f 0 and the chaotic strategy (S k ) k∈N will depend on whether an authenticated watermarking method is desired or not, as follows. A chaotic boolean vector is generated by a number T of iterations of a logistic map ((µ,U 0 ) parameters will constitute the private key). Then, in case of unauthenticated watermarking, the bits of the chaotic boolean vector are grouped six by six, to obtain a sequence of integers lower than 64, which will constitute the chaotic strategy. In case of authentication, the bitwise exclusive or (XOR) is made between the chaotic boolean vector and the MSCs and the result is converted into a chaotic strategy by joining its bits as above. Thus, the mixed watermark is the last boolean vector generated by the chaotic iterations.

Embedding of the Watermark

To embed the watermark, the sequence (U k ) k∈N of altered bits taken from the M LSCs must be defined. To do so, the strategy (S k ) k∈N of the mixture stage is used as follows

U 0 = S 0 U n+1 = S n+1 + 2 × U n + n (mod M).
(2)

To obtain the result depicted in Fig. 1b.

Remark that the map θ → 2θ of the torus, which is a famous example of topological Devaney's chaos (?), has been chosen to make (U k ) k∈N highly sensitive to the chaotic strategy. As a consequence, (U k ) k∈N is highly sensitive to the alteration of the MSCs: in case of authentication, any significant modification of the watermarked image will lead to a completely different extracted watermark.

Simulation Results

To prove the efficiency and the robustness of the proposed algorithm, some attacks are applied to our chaotic watermarked image. For each attack, a similarity percentage with the watermark is computed, this percentage is the number of equal bits between the original and the extracted watermark.

Zeroing Attack

In this kind of attack, some pixels of the image are put to 0. In this case, the results in Table 1 have been obtained. We can conclude that in case of unauthentication, the watermark still remains after a cropping attack: the desired robustness is reached. In case of authentication, even a small change of the carrier image leads to a very different extracted watermark. In this case, any attempt to alter the carrier image will be signaled.

Rotation Attack

Let r θ be the rotation of angle θ around the center (128, 128) of the carrier image. So, the transformation r -θ • r θ is applied to the watermarked image. The good results in Table 2 are obtained.

JPEG Compression

A JPEG compression is applied to the watermarked image, depending on a compression level. Let us notice that this attack leads to a change of the representation domain (from spatial to DCT domain). In this case, the results in Table 3 have been found. A good authentication through JPEG attack is obtained. As for the unauthentication case, the watermark still remains after a compression level equal to 10. This is a good result if we take into account the fact that we use spatial embedding.

Gaussian Noise

Watermarked image can be also attacked by the addition of a Gaussian noise, depending on a standard deviation. In this case, the results in

DISCUSSION AND FUTURE WORK

In this paper, a new way to generate watermarking methods is proposed. The new procedure depends on a general description of the carrier medium to watermark, in terms of the significance of some coefficients we called MSC and LSC. Its mixture and also the selection of coefficients to alter are based on chaotic iterations, which generate topological chaos in the sense of Devaney. Thus, the proposed algorithm possesses expected desirable properties for a watermarking algorithm. For example, strong authentication of the carrier image, security, resistance to attacks, and discretion.

The algorithm has been evaluated through attacks and the results expected by our study have been experimentally obtained. The aim was not to find the best watermarking method generated by our general algorithm, but to give a simple illustrated example to prove its feasibility. In future work, other choices of iteration functions and chaotic strategies will be explored. They will be compared in order to increase authentication and resistance to attacks. Lastly, frequency domain representations will be used to select the MSCs and LSCs.

Table 2 :

 2 Rotation attacks.

	UNAUTHENTICATION	AUTHENTICATION
	Size (pixels) Similarity	Size (pixels) Similarity
	10	99.08%	10	89.81%
	50	97.31%	50	54.54%
	100	92.43%	100	52.24%
		Table 1: Zeroing attacks.
	UNAUTHENTICATION	AUTHENTICATION
	Angle	Similarity	Angle	Similarity
	5	94.67%	5	59.47%
	10	91.30%	10	54.51%
	25	80.85%	25	50.21%

Table 4

 4

				have been
	found.
	UNAUTHENTICATION	AUTHENTICATION
	Ratio	Similarity	Ratio	Similarity
	2	82.95%	2	54.39%
	5	65.23%	5	53.46%
	10	60.22%	10	50.14%
		Table 3: JPEG compression attacks.

Table 4 :

 4 Gaussian noise attacks.