Ben Chiraz

Ali

Chiraz Ben

Ali Crefige

Disclosure and minority expropriation: A study of French listed firms

Keywords: corporate governance, minority expropriation, ownership structure Résumé gouvernement d'entreprise, expropriation des minoritaires, structure de propriété

In this paper we examine the influence of minority expropriation on disclosure level in France where shareholders are poorly protected and the main agency problem is the one between controlling and minority shareholders (type II conflict) while prior studies have been undertaken in the United States, in a context of ownership dispersion and high investor protection where the main agency conflict opposes managers to shareholders (type I conflict). Using a sample of 81 French firms on the 2001-2004 period, we find a negative relation between disclosure level and both ownership and control concentration and double voting rights shares. These results confirm that type II conflict exacerbates the disclosure problem. Controlling shareholders benefit from superior information which helps them to profit from private benefits at the expense of minority shareholders particularly when they hold voting rights in excess of their cash flow rights; therefore they are likely to reduce disclosure. Our results show also a positive relation between disclosure level and family control which is similar to Ali, Chen et Radhakrishnan (2007) findings on American firms.

Introduction

Corporate governance has been criticized following the financial crisis of 2001 due to financial scandals (Enron, WorldCom…) that shed light on the importance of the corporate governance issue. Manipulation of accounting numbers and disclosure of false information in Enron showed that minority shareholders rights are not always protected. The main contests that Enron' shareholders have signaled are non disclosure of relevant information, delaying reporting losses, and disclosure of false information. These bankruptcies lead legal authorities to develop new regulations in order to improve shareholders information, for example the Sarbanes Oxley Act (SOX) in the USA, the directive 2004/109/CE of the European Parliament and Council called "transparency directive", and the LSF act in France. Those scandals proved that shareholders are not homogenous. In opposition to individual shareholders who hold limited resources to obtain information about the firm, controlling shareholders often participate in the firms' management, and therefore they can obtain private information which gives them opportunities to expropriate minority shareholders; this situation leads to an agency conflict opposing controlling shareholders to non-controlling shareholders (type II conflict). However, previous studies have focused on the agency conflict between managers and shareholders (type I conflict) [START_REF] Fama | Separation of Ownership and Control[END_REF]. Nevertheless recent studies have shown the prevalence of ownership concentration and the validity of the agency conflict between controlling shareholders and minority shareholders [START_REF] Shleifer | A Survey of Corporate Governance[END_REF]La Porta et al., 1998 ;[START_REF] Porta | Corporate Ownership Around the World[END_REF][START_REF] Porta | Investor protection and corporate governance[END_REF][START_REF] Faccio | The ultimate ownership of Western European corporations[END_REF].

Controlling shareholders have incentives to exploit private benefit at the expense of minority shareholders interest. Because of their proximity to operating activity, they can obtain private information to evaluate their investment return and are therefore reluctant to disclose information to public in order to avoid minority contest and continue to expropriate other shareholders. Controlling shareholders usually have voting rights in excess of their cash flow rights (La Porta, Lopez-de-Silances et [START_REF] Porta | Corporate Ownership Around the World[END_REF]; consequently they don't support all the consequences of their decisions. The separation between ownership and control exacerbate the minority expropriation risk. Family shareholders participate actively in the management; entrenched managers are more likely to execute the family's plan at the expense of other shareholders.

This paper examines the impact of minority expropriation on the disclosure level of French firms. In particular, we study the effect of factors that influence minority expropriation which deals with ownership and control concentration [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF][START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF][START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF], the separation between cash flow and voting rights [START_REF] Grossman | One Share/One Vote and the market for Corporate Contro[END_REF][START_REF] Harris | Corporate control contests and capital structure[END_REF], and family ownership [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF].

We analyze a sample of 81 French firms over the 2001-2004 period. We measure disclosure quality using an indirect measure: analyst earning forecast dispersion and find a negative relation between disclosure level and both ownership and control concentration. First, we conclude that under high ownership concentration, controlling shareholders are less reliant on minority shareholders and are likely to expropriate benefits from them; therefore they have less incentive to disclose information. Second, when the controlling shareholder has high voting percentage, the entrenchment effect is exacerbated because he doesn't need the complicity of other shareholders to increase the chance of approval of one decision submitted to the general meeting.

The results show also a negative relation between the existence of double voting rights shares and disclosure level. When controlling shareholders hold cash flow rights in excess of their control rights, they do not support all the consequences of their decisions which increase their incentives to expropriate minority shareholders. Therefore, they also have little incentive to disclose information to protect themselves.

We find that family controlled firms present better disclosure level than non family firms. This result is similar with Ali et al. (2007) findings that family firms report better quality earnings and are more likely to warn for a given magnitude of bad news. Because of the non separation between management and ownership, family firms face more severe type II conflict and less severe type I conflict. Our results are consistent with the view that the decrease in type I conflict exceed the increase in type II conflict which posits that family firms are likely to present better disclosure quality.

We extend prior studies of voluntary disclosure in four ways. First, our paper adds to the growing literature on the conflict of controlling shareholders with minority shareholders.

Most of the studies that examine disclosure have been conducted in USA or UK where individual shareholders are protected which result in ownership dispersion and focus on the agency conflict between managers and shareholders. Our study extends previous research by examining disclosure transparency in an environment of poor investor protection like France [START_REF] Porta | Investor protection and corporate governance[END_REF] leading to ownership concentration and the raise of an agency conflict between controlling and minority shareholders because of lower law enforcement to protect minority rights.

Second, to the best of our knowledge this is the first study that examines minority expropriation in France. French firms have the possibilities to separate between voting rights and cash flow rights through pyramids, double voting rights shares and crossholding. Due to this complicated ownership structures, controlling have the opportunity to hold more control than their equity ownership indicates, which further amplify the entrenchment effect.

Third, we study the identity of shareholders. Most prior studies [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF][START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF][START_REF] Attig | Effects of large shareholding on information asymmetry and stock liquidity[END_REF] do not distinguish between family controlled firms and nonfamily controlled firms. As suggested before, the intensity of agency conflicts depend on the identity of the major shareholders (family/non family) and influence therefore the corporate disclosure. The French context is worth to study, La [START_REF] Porta | Corporate Ownership Around the World[END_REF] and [START_REF] Faccio | The ultimate ownership of Western European corporations[END_REF] find a prevalence of family controlled groups in France.. Finally, as suggested by [START_REF] Meek | Factors Influencing Voluntary Annual Report Disclosures by U.S., U.K. and Continental European Multinational Corporations[END_REF], understanding why firms invest in disclosure is useful not only for preparers and users of accounting information but also for regulators. Firms with high ownership concentration are less transparent than their counterparts. Should regulatory authorities impose more disclosure requirements on them? Moreover, French firms that want to attract foreign ownership should align their disclosure with international standard and offer higher disclosure transparency.

The remainder of this paper is organized as follows: section 2 reviews the Literature review and hypothesis; section 3 presents the research design and method; section 4 presents the results and their interpretations. Finally, we summarize and conclude in section 5.

Literature review and hypothesis

Our study focus on a new agency conflict that has been less studied in accounting literature and arises when ownership is concentrated, it opposes controlling shareholders to minority ones. This conflict is more relevant in an environment where investors are less protected such in civil law country like France (La [START_REF] Porta | Investor protection and corporate governance[END_REF]. Our research question deals with the way minority expropriation influence corporate disclosure level. We intend to study three aspects that determine minority expropriation intensity: ownership and control concentration [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF][START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF], separation of cash flow and control rights [START_REF] Grossman | One Share/One Vote and the market for Corporate Contro[END_REF][START_REF] Harris | Corporate control contests and capital structure[END_REF] and family ownership [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF]. [START_REF] Healy | Information asymmetry, corporate disclosure, and the capital markets: A review of the empirical disclosure literature[END_REF] present six forces that affect managers' disclosure decisions from a financial perspective : capital market transaction, corporate control contests, stock compensation, litigation costs, proprietary costs, and management talent signaling. Disclosure studies can be classified into two broad categories. The first presents motivations to disclose information (agency costs, capital market transaction, stock compensation, corporate control contests and management talent signaling). The second category of studies examines incentives to withhold and retain information (litigation costs, proprietary costs). Main previous studies focus on traditional agency conflicts and argue that firms suffering from high agency costs between shareholders and managers are likely to offer high disclosure level. In fact, [START_REF] Healy | Information asymmetry, corporate disclosure, and the capital markets: A review of the empirical disclosure literature[END_REF] show that demand for financial reporting and disclosure arises from information asymmetry and agency conflicts between managers and outsiders. Similarly to this study and based on agency theory developments, [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF] finds that disclosure reduces information asymmetry between managers and shareholders and contribute therefore to reduce agency conflicts. [START_REF] Berle | The Modern Corporation and Private Property[END_REF] study the dispersion of the capital and the traditional agency problem resulting from information asymmetry between shareholders and managers. Recent studies show that the validity of the image of the modern corporation drawn by Berle and Means is limited particularly in non American firms [START_REF] Shleifer | Large Shareholders and Corporate Control[END_REF][START_REF] Porta | Corporate Ownership Around the World[END_REF][START_REF] Faccio | The ultimate ownership of Western European corporations[END_REF]. In fact, many studies conducted in other developed countries show more significant concentrations of ownership (La Porta, Lopez-de-Silances et [START_REF] Porta | Corporate Ownership Around the World[END_REF][START_REF] Faccio | The ultimate ownership of Western European corporations[END_REF] and even among the largest American firms, [START_REF] Shleifer | Large Shareholders and Corporate Control[END_REF] find a relative high ownership concentration.

One of the explanations that La [START_REF] Porta | Corporate Ownership Around the World[END_REF] and Roe (2001) suggest for the prevalence of blockholders in continental Europe is that the incapacity of the legal system to protect investors from potential manager's opportunism leads shareholders to obtain large shares blocks to monitor managers themselves. [START_REF] Lee | Structure de l'actionnariat familial et performance des grands groupes coréens[END_REF] argue that controlling shareholders can easily control and influence professional manager and that this situation gives them the possibility to manage the firm in their interest and often at the expense of minority shareholders [START_REF] Shleifer | A Survey of Corporate Governance[END_REF]. [START_REF] Shleifer | Large Shareholders and Corporate Control[END_REF] argue that when the ownership is concentrated, the main agency conflict is between controlling shareholders and minority interest holders rather than between shareholders and managers. [START_REF] Porta R | Tunneling[END_REF] use the term "tunnelling" to refer to "the transfer of resources out of a company to its controlling shareholder (who is typically also a top manager)" (Johnson et al., 2000, p. 3). Controlling shareholders participate to management and have incentives to profit from their proximity to operating activities and make decisions that expropriate minority shareholders' wealth, for example, by paying excessive compensation, acquiring perquisites or making investments that conflict with outsiders' interests.

Following La [START_REF] Porta | Corporate Ownership Around the World[END_REF] and [START_REF] Biebuyck | Les leviers de contrôle des actionnaires majoritaires[END_REF], [START_REF] Villalonga | How do family ownership, control and management affect firm value?[END_REF] and [START_REF] Ho | A study of the relationship between corporate governance structures and the extent of voluntary disclosure[END_REF] notice that agency costs in firms where the capital is hold by few shareholders, are the consequence of interest conflicts between dominant shareholders and minority shareholders. The first have incentives to expropriate private benefits from the second. Moreover, because controlling shareholders hold large blocks of capital and voting rights, they can easily constrain managers to follow them. They can therefore have a total control of the firm in particularly their disclosure strategy. Controlling shareholders have incentives to reduce information disclosure in order to avoid the risk that minority shareholders will be conscious that their interest are in danger and avoid therefore minority contests. Consequently, these firms are likely to offer poor disclosure.

La [START_REF] Porta | Corporate Ownership Around the World[END_REF] find that 75% of controlling shareholders in France are member of the management or the board. [START_REF] Chiang R | Insider Holdings and Perceptions of Information Asymmetry: A Note[END_REF] notice that their position gives them the opportunity to obtain a free access to a wide range of information. This private information help them to make transactions and raise abnormal profit [START_REF] Jaffe | Special Information and Insider Trading[END_REF][START_REF] Finnerty | Insiders and market efficiency[END_REF]. They are therefore reluctant to disclose information to public in order to protect their position. [START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF] examine the relation between ownership concentration, measured by the percentage of voting right of the ultimate shareholder and the quality of accounting information as measured by earning informativeness using a sample of 977 companies in seven East Asian economies. The authors argue that concentrated ownership create agency conflicts between controlling owners and outside investors leading controlling shareholders to report accounting information for self-interested purposes, which result in less credibility in the reported earnings as perceived by outside investors. The authors present the entrenchment effect [START_REF] Morck R | Management Ownership and Market Valuation[END_REF] to explain the relation between ownership concentration and disclosure. They suggest that "as controlling shareholders are entrenched by their effective control of the firms, their decisions that deprive the rights of minority shareholders are often incontestable in the weak legal systems in the region and by ineffective corporate governance mechanisms". [START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF] find that these firms offer poor disclosure quality.

When controlling shareholders hold large voting rights, they can easily control the firm because they don't need the complicity of other shareholders for the approval of their decision in the general meeting. They can influence many of the decision of the firm notably it disclosure strategy. Consequently, firms with a control concentration are likely to present poor disclosure. [START_REF] Attig | Effects of large shareholding on information asymmetry and stock liquidity[END_REF] argue that controlling shareholders have a selfish behavior. Indeed, to increase the chance of executing his plans, the controlling shareholder would limit and delay the disclosure of information in order to avoid the intervention of minority shareholders or to conduct them to take bad decisions based on inadequate information. [START_REF] Lakhal | Les mécanismes de gouvernement d'entreprise et la publication volontaire des résultats en France. (French)[END_REF] examine the association between ownership concentration and voluntary disclosure and find that management earnings forecasts are positively associated to ownership and control concentration.

This discussion leads to the following testable hypotheses:

H1: There is a negative relationship between disclosure level and both ownership and control concentration.

Ben Amar et [START_REF] Ben Amar | Separation of Ownership from Control and Acquiring Firm Performance: The Case of Family Ownership in Canada[END_REF] examine the relation between the separation between voting and cash flow rights and performance in a Canadian context. They suggest that a large proportion of Canadian public companies have controlling shareholders (families) that often exercise control over voting rights while holding a small fraction of the cash flow rights. The authors do not find that separation of ownership and control has a negative impact on performance arguing that Canada offer good investor protection and that "contrary to other jurisdictions offering poor minority shareholder protection or poor corporate governance, separation of control and ownership is not viewed as leading to value destroying mergers and acquisitions, i.e.,market participants do not perceive families as usingM&A to obtain private benefits at the expense ofminority shareholders".

La Porta et al. (1998[START_REF] Porta | Investor protection and corporate governance[END_REF] suggest that countries with a civil law system like France present low investor protection and higher private benefits of control. One example of low investor protection in France is the possibility to separate between voting and cash flow rights which allows one shareholder to control the firm while possessing relatively low ownership percentage. The main mechanism that allows the dissociation between ownership and control is settled by the law of 1966 (du code de commerce) that gives French firms the possibility of issuing shares with double voting rights when they are registered for at least two years. The deviation from "one-share, one-vote" rule increases the risk violating the interest of minority shareholders because controlling shareholders have often voting rights in excess of their cash flow rights, and don't support consequently all the consequence of their decision.

The results of [START_REF] Attig | Effects of large shareholding on information asymmetry and stock liquidity[END_REF] show that stocks with greater deviations between ultimate control and ownership have a larger information asymmetry component of their bid-ask. The authors explain this result using the findings of [START_REF] Claessens | The separation of ownership and control in East Asian Corporations[END_REF], they sustain that the deviation of control from ownership is associated with more selfish behavior by the ultimate shareholder. Following [START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF], the authors suggest that to increase the chance of executing theirs plans, controlling shareholders have incentives to minimize and delay the disclosure to avoid that other shareholders intervene or to base their decisions on inadequate information. [START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF] suggest "Moreover, due to the complicated pyramidal and crossholding ownership structures typical in East Asian companies, a significant number of controlling owners in the region actually possess more control than their equity ownership indicates, which further exacerbates the entrenchment effect". Controlling shareholders can expropriate minority interest benefits and enrich themselves through related party transactions in which profits are transferred to other companies they control. Consequently firms that separate voting rights and cash flow rights are likely to provide poor disclosure level. We thus state the following hypothesis: H2: there is a negative association between disclosure level and the dissociation between voting rights and cash flow rights

In France, family-owned and controlled companies are more in evidence than in Anglo-American stock. La [START_REF] Porta | Corporate Ownership Around the World[END_REF] show that 50% of French firms are controlled by family groups. In family controlled firms, family members actively participate to the management of the firm and staff in their board, therefore, they have access to all the information they need to assess the return of their investment. Moreover, according to Cormier et al. ( 2005) "firms with closely-held ownership are not expected to be responsive to public investors' information costs since the dominant shareholders typically have access to the information they need". These families need less external financing, therefore, external information demand is low. As dominant shareholders, families have less incentive to disclose [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF]. Using a sample of 60 listed firms in Singapore and 62 firms listed in Hong Kong in 1997, [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF] test the relation between ownership structure and voluntary disclosure using the Meek et al. index (1995) to measure disclosure level.

Consistent with the information financing need hypothesis, the authors find a negative relation between family control and voluntary disclosure. [START_REF] Ho | A study of the relationship between corporate governance structures and the extent of voluntary disclosure[END_REF] examine the relationship between corporate governance and voluntary disclosure quality using a sample of 98 listed firms in Hong Kong where family firms are prevalent. The authors find a negative relation between family members in the board and voluntary disclosure. They suggest that boards dominated by family members don't control manager to protect the interest of minority shareholders but their role is to vote decisions that maximize the interest of the family. Therefore, these family shareholders can easily expropriate minority shareholders since they have the support of the board to execute their plan. We suggest then that minority expropriation risk is higher in family firms then their counterparts. Family controlled firm have incentives to limit disclosure in order to continue to easily expropriate private benefit.

Compared to their counterparts, family firms face less severe agency problem due to the separation of ownership and management (type I) but more severe agency problem that arise between controlling and minority shareholders (type II) (Ali, Chen and Radhakrishnan, 2007).

The difference between these two agency conflicts determines the direction of the relation between family control and disclosure. We suppose that the increase of type II conflict exceed the decrease of type I conflict and state the following hypotheses: H3: there is a negative association between disclosure level and family ownership

Research design and method

This section presents the research design.

The sample

The sample was selected from French firms listed on the Paris Stock Exchange and part of the SBF 120 index in 2004 (120 in total). The first step consists in selecting only industrial and commercial firms. We eliminate financial and insurance companies (16 firms) because of their specific disclosure requirements and financial characteristics. Second, 18 companies were excluded from the sample because of a lack of data. Finally, we eliminate 3 outliers. The final sample is composed of 81 companies, as shown in Table 1. A list of the companies included in the sample is reported in the Appendix.

***Insert Table 1 about here***

We selected the period 2001-2004 because of information availability about Analyst earning data from IBES historical database.

Variables and measurement

The disclosure quality is a very difficult variable to measure in a French context as there is no organization that offers a disclosure rating such as the AIMR/CIC (Association of Investment Management Research), and the FAF (Financial Analysts Federation Corporate Information Committee) in USA.

Nelson Sofres Institute conducted a survey for AGEFI and Euronext in 2000 to nominate the prize of the best annual report and the best investor relations. The first step of this contest is to develop a shortlist of SBF120 companies that good annual report/investor relations quality.

The second step consists of electing the prize for the best annual report/investor relations [START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF] use this study to measure disclosure quality. However, this measure offers a binary proxy which limits the number of methodologies that can be used (LOGIT).

Many previous studies have developed disclosure index to measure disclosure level or quality [START_REF] Meek | Factors Influencing Voluntary Annual Report Disclosures by U.S., U.K. and Continental European Multinational Corporations[END_REF][START_REF] Botosan | Disclosure Level and the Cost of Equity Capital[END_REF][START_REF] Michaïlesco | Une étude empirique des déterminants de la qualité de l'information diffusées par les entreprises françaises au cours de la période 1991-1995[END_REF][START_REF] Depoers | L'offre volontaire d'information des sociétés cotées : concept et mesure[END_REF].

Nevertheless, this methodological approach has been recently criticized [START_REF] Chavent | Disclosure and determinants studies: An extension using the divisive clustering method (DIV)[END_REF]. The authors present three limitations to this approach. First, the disclosure index is often determined by summing several items that can be weighted or unweighted. However, there is no dominant practice and the question of who should weight the items remains unanswered. Second, including too many independent variables may create a multicollinearity difficulty. Third, the form of the relationship between dependent and independent variables is not always known [START_REF] Lang | Cross-Sectional Determinants of Analyst Ratings of Corporate Disclosures[END_REF]. [START_REF] Chavent | Disclosure and determinants studies: An extension using the divisive clustering method (DIV)[END_REF] sustain that a 'classical' linear regression is not always the most suitable tool, and propose the use of a rank regression.

Following [START_REF] Lang | Corporate Disclosure Policy and Analyst Behavior[END_REF], [START_REF] Barron | MD&A Quality as Measured by the SEC and Analysts' Earnings Forecasts[END_REF] examine the relationship between analyst earning forecast characteristics and disclosure level. The results of these studies show that disclosure level is negatively associated to earning forecast error and earning forecast dispersion. [START_REF] Dumontier | Adoption volontaire des IFRS, asymmétrie d'information et fourchettes de prix: l'impact du contexte informationnel[END_REF] argue that " the information content of accounting numbers can't be observe directly, they are commonly measured using a proxy of information asymmetry". One of the categories used to proxy information asymmetry is based on analyst earning forecasts : forecast error [START_REF] Labelle R | La gouvernance et la décision d'alerte aux résultats[END_REF], forecast dispersion [START_REF] Daley | Analysts' Forecasts, Earnings, Variability, and Option Pricing: Empirical Evidence[END_REF][START_REF] Brown | Analyst forecasting errors: Additional evidence[END_REF][START_REF] Barron | Using analysts' forecasts to measure properties of analysts' information environment[END_REF]. We choose analyst forecast dispersion to measure disclosure level. Analyst data is taken from the IBES summary tape.

We include 3 proxies for ownership structure relying on [START_REF] Demsetz | The Structure of Corporate Ownership: Causes and Consequences[END_REF] measures. We measure capital concentration by the Herfindhal index, calculated by summing the squared percentages of capital held by each shareholder. We also measure the percentage of voting rights held by the largest and the second largest shareholder. [START_REF] Bebchuk | The "lemons effect" in corporate freeze-outs[END_REF] present mechanisms that allow a controlling shareholder to maintain complete control of the firm even while he owns a relative low percentage of total shares. Indeed, issuing two categories of shares, shares with voting rights and shares without, results in separating voting rights and cash flow rights. We use a second ownership concentration variable denoted VOT1 and VOT2 which take into consideration control and voting rights. We introduce a dummy variable FAM when a firm is controlled by a family. Finally, we introduce four control variables: LNTA, COT, USCOT CAC40 and DEBT. LNTA measures the size of the firm.

COT is a dummy which equals one if the firm is cross listed and 0 otherwise. USCOT is a dummy which equals one if the firm is listed in USA and 0 otherwise CAC40 is a dummy which equals one if the firm is included in the CAC40 index and 0 otherwise. ***Insert Table 2 about here*** Table 2 summarizes the definitions and measurements of all variables used in this study. Data on ownership structure are taken directly from the firm's annual reports or from their official web sites. Earning forecasts are taken from IBES database. Finally, other accounting and financial data were collected from the Compustat database.

The summary descriptive statistics show that the average percentage of voting rights of the major shareholder is 34.9% and the average herfindhal index is 0.14. These high values prove that ownership and control are concentrated in French firms. Theses results are coherent with La Porta, Lopez-de-Silances et [START_REF] Porta | Corporate Ownership Around the World[END_REF] findings. We find that 45.57% of firms in the sample are controlled by families.This result is similar to [START_REF] Faccio | The ultimate ownership of Western European corporations[END_REF] findings. As expected, the descriptive statistics reveal that most of sample firms proceed to a separation between voting and cash flow rights (63% of our firms use double voting rights) and the average voting rights of the dominant shareholders exceed his cash flow rights of 24%. These results proved ownership concentration, the predominance of family controlled firms and the separation between ownership and control in France which leads us to examine type II agency conflicts (controlling vs minority shareholders).

***Insert Table 3 about here***

Method

We use a panel data regression to examine the effects of the independent variables on the disclosure level.

The regression equation is formulated as follows:

Results and discussion

The correlation matrix show many correlations among the variables introduced in the regression, we therefore try to avoid putting them together which results in three models M1 M2 and M3. We discuss hereafter the results of M1, M2 and M3.

We test the presence of specific effects. The chow test reveals the presence of specifics effects. We then use the Hausman test (1978) to determine if they are random effects or fixed effects. The probability of the test is superior to 10% we can't then reject the null hypothesis H0 and decide to use random effect regression. Our time dimension is limited. Probably our result is due to the fact that ownership structure doesn't move a lot in a short period of 4 years. We test for the presence of autocorrelation and heterosedaskicity and find that our data suffer from the two. We correct it using the command xtgls of STATA.

We use the opposite of earning forecast dispersion (STD) which is a proxy of information asymmetry to measure disclosure level (DISCL). Our independent variable DISCL equals the opposite of earning forecast standard deviation (DISCL=-STD that we collect from IBES database.

Table 6 shows the results of the panel regression. Coefficients and Z-statistics are respectively reported. First, it is important to notice that the results of the three models M1, M2 and M3

show a positive and significant relation between STD and HERFI and VOT1. These results suggest a negative association between disclosure level and ownership and control concentration. These results confirm our first hypothesis. Firm with high ownership concentration suffer from more minority expropriation then their counterparts. This risk is higher when the first shareholders possess high voting rights which allow him to vote decision at the expense of minority shareholders without the need to the approval of other shareholders. One explanation of the negative association between disclosure quality and ownership concentration is that controlling shareholders have possibilities to expropriate minority shareholders. They have therefore incentive to reduce disclosure level in order to exploit private benefits of control and avoid minority shareholders contest. Consequently, entrenched managers of controlled firms have little incentives to provide high disclosure level. Our results are in accordance with [START_REF] Ho | A study of the relationship between corporate governance structures and the extent of voluntary disclosure[END_REF], [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF] and Eng and Mak (2003).

Labelle et Schatt (2005) test a non monotonous relation between ownership dispersion and disclosure quality suggestion that when insiders (managers and controlling shareholders) hold large blocks of capital they use to improve disclosure to get advantage as better liquidity and value increase due to disclosure as suggested by [START_REF] Verrecchia | Discretionary disclosure[END_REF][START_REF] Welker | Disclosure Policy, Information Asymmetry, and Liquidity in Equity Markets[END_REF]. On the contrary, in a case of ownership dispersion, managers improve disclosure to decrease type I agency costs (between manager and shareholders). We test this assumption but we find no evidence for this relation.

Second, the regression results show a positive relation between STD and DOUB. Firm issuing double voting right offer low disclosure level. As we conjectured earlier, the separation between cash flow rights and voting rights amplify the agency problem between controlling shareholders and minority shareholders. When controlling shareholders hold voting rights in excess of their control rights, they have more incentives to exploit non-controlling shareholders since they may not pay for all the consequences of their decision. Consequently, they have incentive to retain information inside the company in order to avoid the intervene of non-controlling shareholders. However, we find no significant association between the coefficient of separation between voting rights and cash flows rights of the first shareholders and disclosure level. This variable presents problem of normality, one possible explanation is that our variable is not a good measure of the intensity of the dissociation between voting rights and cash flow.

One explanatory variable related to family ownership is introduced in equation M1, M2 and M3. We suggest earlier that firms controlled by families have little incentive to disclose information to the public for many reasons [START_REF] Chau | Ownership Structure and Corporate Voluntary Disclosure in Hong Kong and Singapore[END_REF]. First, the demand for information in such companies is relatively weak because the major providers of finance already have that information; these families staff many of the senior positions themselves.

Second, these families are controlling shareholders, they have incentives to retain information and expropriate minority shareholders and third these firms suffer from more severe type II conflict.

We find a significant association between disclosure level and family control, nevertheless the coefficient sign is opposite to our prediction. Our hypothesis 3, which predicts that family control influences negatively disclosure level, is not supported. However this result is similar to Ali et al. (2007) findings on a sample of American firms part off S&P 500. The authors argue that "compared to non-family firms, family firms face less severe agency problems due to the separation of ownership and management, but more severe agency problems that arise between controlling and non-controlling shareholders". Consequently, the difference in the quality of disclosure between family and non-family firms would depend on the difference in the severity of their Type I and Type II agency. Ali et al. (2007) find that family firms report better quality earnings and are more likely to warn for a given magnitude of bad news, Our models include firm characteristics as control variables. We find a positive and significant association between US listing and disclosure level. Firms that are listed on the US Stock Exchange are subject to several information requirements than the French ones.

Moreover, the American market is the first financial market and the more active. It requires from firms to comply with more mandatory disclosure. American investors are used to high corporate governance practices particularly high transparency level; therefore they are more demanding in term of information then the French ones. Our result is in accordance with the one found by [START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF] and [START_REF] Lakhal | Les mécanismes de gouvernement d'entreprise et la publication volontaire des résultats en France. (French)[END_REF].

We find that lower leverage is related to greater disclosure. Our result is aligned with Eng and [START_REF] Mak | Corporate Governance and Voluntary Disclosure[END_REF]. These authors suggest that "increased leverage is expected to reduce disclosure because leverage helps control the free cash flow problem, and the agency costs of debt are controlled through restrictive debt covenants in debt agreements rather than increased disclosure of information in annual reports [START_REF] Jensen | Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers[END_REF])".

We find that disclosure level is negatively associated to firm size. We explain this relation using proprietary cost and litigation cost theory frameworks. Larger firms face more severe proprietary costs because of higher competition intensity among them. Consequently, they disclose less information to public in order to avoid competitive disadvantage. Our result can be also explained using the litigation cost theory development [START_REF] Philbrick | Shareholder Litigation and Corporate Disclosures[END_REF]. [START_REF] Healy | Information asymmetry, corporate disclosure, and the capital markets: A review of the empirical disclosure literature[END_REF] suggest that litigation can potentially reduce managers' incentives to provide disclosure, particularly of forward-looking information. The author argue that "this is likely to arise if managers believe that the legal system penalizes forecasts made in good faith because it cannot effectively distinguish between unexpected forecast errors due to chance and those due to deliberate management bias" (Healy et Palepu, 2001, p. 423). Large firms reduce their corporate disclosure in order to reduce the cost of litigation.

Finally, the coefficient of the variable EPSVAR is not significant which suggest no evidence on the relation between disclosure and performance. ***Insert Tables 4-5-6 about here***

Conclusion

In this paper four new determinants of the level of disclosure are identified using agency theory: ownership and control concentration, the existence of double voting share, family ownership. French firms seems to be a suitable setting to test the impact of minority expropriation on disclosure because of the limited academic exploration on the topic and the high level of ownership concentration of French firms increasing controlling private benefits.

Consistent with the predictions of agency theory, we find that managers provide higher disclosures when the capital is less concentrated i.e. the level of public ownership is high. In a context of capital concentration, controlling shareholders retain information in order to expropriate minority shareholders and exploit private benefits.

Many studies show a negative association between disclosure quality and ownership concentration in a US context [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF] characterized by capital dispersion and the prevalence of the agency conflict between shareholders and managers. The study of French firms allows us to examine the topic of corporate disclosures in a different ownership context from that of the US and to focus on the conflict opposing controlling shareholders to noncontrolling shareholders. After controlling for leverage, US listing, performance and size, we find a negative association between disclosure quality and ownership and control concentration, and existence of double voting rights. Our results show a positive association between family control and disclosure level and are consistent with the result found by Ali, Chen et Radhakrishnan ( 2007) on an American firm sample.

This study contributes to the disclosure literature by providing evidence on the impact of minority expropriation on the corporate disclosure practices. Our findings support prior studies [START_REF] Gelb | Managerial Ownership and Accounting Disclosures: An Empirical Study[END_REF][START_REF] Fan | Corporate ownership structure and the informativeness of accounting earnings in East Asia[END_REF][START_REF] Labelle R | Structure de propriété et communication financière des entreprises françaises[END_REF][START_REF] Attig | Effects of large shareholding on information asymmetry and stock liquidity[END_REF] and shows that investor requirements appear to be greater for firms with high levels of individual shareholding. The sample is compose of 81 French listed companies included in SBF120 index in the period 2001-2004.STD measures analyst earning forecast dispersion (we use the standard deviation of the last month preceeding the fiscal year end). HERFI measures ownership concentration and eagal the sum of the square percentage of the capital held by shareholders. VOT1 measures control concentration, it egals the percentage of voting rights of the major shareholders. FAM mesure family control. VA1 and DOUB measure the degree of separation between voting and cash flow rights. VA1 egals the percentage of voting rights to cashflow rights of the first shareholder. au rapport du pourcentage des droits de vote et le pourcentage des droits au capital du premier actionnaire. DOUB is a dummy that equals 1 if the firm issues double voting rights and 0 otherwise. USCOT is a dummy that equals 1 if the firm is listed in USA and 0 otherwise. LNTA and DEBT measure respectively the firm size and its debt ratio.

a We use white method to correct for heteroskedasticity and we use the command robust to correct for first order autocorrelation

Table 1 :

 1 Sampling procedure

	Companies listed on Paris Stock Exchange on	118
	31 December 2004 and part of SBF 120
	Financial companies	16
	Companies with Missing annual report	10
	Companies with Missing data	8
	Outliers	3
	Final sample	81

Table 2 :

 2 Variable definitions

	Variable	Definition	Measurement
	Dependent
	variable:
	DISCL	Disclosure level	The negative absolute value of analyst
			earning forecast standard deviation
	Independent
	variables:
	HERFI	Herfindhal index measuring	The squared sum of shares percentages
		ownership concentration
	VOT1	Large shareholder	Percentage of voting rights held by the
			largest shareholder
	VOT2	Second shareholder	Percentage of voting rights held by the
			second largest shareholder
	FAM	Family ownership	Dummy variable coded as 1 if the firm is
			controlled by a family and 0 otherwise
	DOUBLE	Double voting shares	Dummy variable coded as 1 if there is a
			double voting shares and 0 otherwise
	VA	Separation between property	Voting rights divided by the portion of
		and voting rights	shares of the largest shareholder
	LNTA	Size	Log of total assets
	COT	Foreign quotation	Dummy coded as 1 if the firm is listed on
			foreign market and 0 otherwise
	USCOT	US listing	Dummy coded as 1 if the firm is listed on
			US market and 0 otherwise
	DEBT	Leverage	Total liabilities divided by total assets
	CAC40	CAC40 membership	Dummy coded as 1 if the firm forms part
			of the CAC40 index and 0 otherwise

Table 3 :

 3 Summary statistics of continuous variablesSummary statistics are based on a sample of 84 French listed firms included in the SBF120 index. FAM equals 1 if a large shareholder is a family and 0 otherwise, SO is coded 1 if executives benefit from stock option plans and 0 otherwise, COTATION equals 1 if the firm is cross listed and 0 otherwise, HERFI is concentration index equals to the sum of the squared shares percentages. VOT1 is voting rights of the first large shareholder, VA is the voting right of the first shareholder divided by his shares portion, DEBT is the ratio of total debt per total assets, SIZE is the log of total assets.Descriptive statistics of continuous variables

			Standard
		Mean	deviation Median Maximum Minimum	25%	75%
	STD	0.325	0.291	0.24	0.02	1.84	0.4	0.13
	HERFI	0.14	0.143	0.088	0.544	0.001	0.02	0.223
	VOT1	0.349	0.23	0.323	0.84	0.018	0.139	0.546
	VOT2	0.087	0.079	0.06	0.38	0	0.036	0.121
	VA1	1.241	0.264	1.21	2.009	0.898	1	1.412
	LNTA	8.241	1.677	8.274	11.712	4.836	6.89	9.632
	DEBT	0.257	0.136	0.275	0.69	0	0.162	0.352
		Descriptive statistics of categorical variables
	Variables				Frequency Percentage	Total
		Test variables
	DOUB	No double voting rights		120	36.04	333
		Double voting rights		213	63.96
	FAM	No family control		166	54.43	305
		Family control		139	45.57
		Control variables
	EPSVAR Bad performance		139	42.38	328
		Good performance		189	57.62
	COT	No cross-listing		221	66.37	333
		Cross-listing		112	33.63
	USCOT	No US listing		255	77.51	329
		US listing			74	22.49
	CAC40	No CAC40 membership		218	64.88	336
		CAC40 membership		118	35.12

Table 6 :

 6 Generalized least squared regression a STD is the independent variable that measure the opposite of disclosure level (STD=-

	DISCL)
		1A(1)	1A(2)	1A(3)
	HERFI	0.23014	0.21349
		(0.030)**	(0.061)*
	VOT1			0.11947
				(0.074)*
	DOUB	0.07778		0.06538
		(0.016)**		(0.041)**
	VA1		0.07966
			-0.16
	FAM	-0.09643	-0.07663	-0.0995
		(0.003)***	(0.015)**	(0.003)***
	EPSVAR	-0.03833	-0.0328	-0.0378
		-0.163	-0.245	-0.171
	LNTA	0.03219	0.02991	0.03236
		(0.001)***	(0.002)***	(0.001)***
	USCOT	-0.06231	-0.07347	-0.06727
		(0.089)*	(0.046)**	(0.065)*
	DEBT	0.2791	0.2944	0.27213
		(0.008)***	(0.006)***	(0.010)**
	Constante	-0.03314	-0.0717	-0.03289
		-0.699	-0.502	-0.707
	Number of observations	285	278	285
	Number of companies	81	81	81
	p-values in parentheses
	*, **, *** significant at de 10%, 5% et 1%


DRAFT Politicizing risk assessment on vulnerability conditions

Towards a counter-hegemonic approach of risks in a "world at risk" From the 1970s, hazard-centered approaches have been hugely criticized and challenged by vulnerability-oriented research. Some "critical" English-speaking researchers (like Wisner, Hewitt, Watts or Blaikie), some French sociologist or geographers (Fabiani and Theys, at Mid-1980s, publishing Mary Douglas in French) or Latin-American social scientists, in the La Red network, have contributed to consolidate a social science approach of risks.

Of course, social responsibilities had already been commented a very long time ago. Let"s think about Gilbert White"s work on adaptation in the 1930s as a planner; about the controversy of the Lisboan earthquake between Rousseau and Voltaire, in 1755; or even about colonial archives in Latin America during the late 16 th century.

Though today risk is defined as a combination of hazard and vulnerability, it is worth wondering how important social determination in risk definition and assessment is. Adaptation corresponds to the "adjustment in natural or human systems to a new or changing environment". And I will focus on adjustment of human systems and the consequences on natural systems. But on a theoretical perspective, such an adjustment can be defined either facing an "external" event that societies have to cope with, or as a social contingency which is socially constructed.

Problematic:

In front of such a dilemma I want to ask the question of the social sciences" objectives and functions in risk assessment:

-Are social sciences dedicated to explain any residual irrationality? The reasons why hazard assessment is not completely efficient? -Or are they contributing to a better understanding of the multi-causality of risks? Of a social and performative phenomena? In spite of significant successes in economy for instance, the explanation grounded on psychological frameworks under-estimates collective and structural drivers that are rooted in social relations (in power and culture over time and space).

My hypothesis states that social sciences must highlight political and social implications related with risk and its assessment. Risk and climate change are hegemonic categories that are selflegitimating. They are sufficient to justify actions, initiatives, decisions, evictions, preservations and so on… Nobody can openly talk against climate change mitigation. The fact that the consequences of human actions on physical dynamics must be addressed and the fact that risk assessment must be criticized are different matters.

Social sciences ought to deal with the "how" in addressing such consequences, and with the "what" is really at stake when defining and addressing risks, in our case, to climate change.

My points consist in three main ideas:

-Figure out a theoretical framework for research on risk as a social construction that does not marginalize bio-physical aspects; -Organize such an integrated framework around the notion of territorialization of risks; -Politicize the territorialization of risks regarding climate change so that risks can be analyzed as a product and as a determinant.

As for methods and references, I ground my statements on my PhD research in the field of risk, between 2002 and 2009, mainly in Venezuela (Caracas and Mérida), and in Brazil (Recife). I also base my ideas on the research of French colleagues in Quito, La Paz and Lima. In terms of risks, the critical -and sometimes radical -research of Hewitt, Blaikie, Wisner…, as well as the social scientists" work of La Red have been very helpful.

To support my statements of politicizing adaptation assessment and territorializing risks, my contribution will deal with different scales (Caracas; the European Union and United Nations institutions). And it unfolds in three parts:

-The first one is dedicated to the hardly questionable categories of risk and climate change, and on the different consequences of such hegemony. -I will then focus on what is really at stake when assessing risks, and in which way human adaptation to risks and climate change has consequences on the socio-spatial order. -Finally, I will try to show that the territorial dimension of risks could bring together the traditionally fragmented perspectives on risks (the realistic and the constructivist one). In other words, I will argue that territories of risks could account for materiality and objectivity without ignoring "hidden mechanisms" that are concrete -though intangibledrivers of risk situations.

I. Risk and climate change as hegemonic categories. How practical are fears?

I.1. The emergence of environmental concerns and risks as meta-narratives.

In the western world, in the late 1960s, environmental concerns rise and question productivism and the model of development. Grassroots claims are institutionalizing since early 1970s (Meadows report, 1972;Brundtland report, 1986;Earth Summit in Rio, 1992).

The notion of "risk" rises in the 1970s while the world is being reshaped: crisis of capitalism and the idea of progress, "end of History" and the cold war, the withdrawal of the state… Risks and security become pervasive and define a "dangerous world". "Risk societies" are emerging while risks are turned into a new kind of social interactions.

Nowadays, risk and climate change are among the "buzzwords" of international discourses and political practices. They are showing the hegemonic characteristics (Laclau and Mouffe, 1985):

-They stand as a meta-narrative in public debates.

-They are often presented on a basis of dualism and Manichaeism.

-They appear as a necessity to be addressed. And who could be against risk assessment? Who could rise against the improvement of adaptive capacity and resiliency of society?

An example in Brazil, some colleagues from the State University of São Paolo wonder about the political logics of the distribution of public funds. They say that today, there are plenty of opportunities to lead research on the impacts of climate change on Brazilian megacities whereas other kinds of risks are hardly considered by public authorities, and broader urban concerns, like inequalities, transportation or housing issues, are definitely not assessed in the same way.

What is driving such options? The next point highlights the necessity of political reaction to accidents.

I.2. Events that trigger off public intervention.

Although risks are objective realities that must be addressed, hazard and vulnerability are not the only factors at stake when scientists collect data, or when public authority prioritizes interventions. Here comes an example from the 1999 Vargas Tragedy in Caracas:

The coast of the northern central region in Venezuela has been badly stricken by hard rains, flash floods and landslides in December 1999. The "Vargas Tragedy" triggered off public reactions to improve risk management beyond the crisis period, and prevent further damages and losses of life. Among institutional initiatives that have been adopted at different scales, it has been decided to accept the proposal of the Japanese cooperation to assess risks of landslides, flash-floods and earthquake in Caracas which stands a few kilometers away from the Vargas coast. The research has been done, but it only dealt with part of the agglomeration -the municipalities situated on the left shore of the Guaire river that drains the Caracas valley -, and without questioning the kind of hazards at stake. NGOs have been subcontracted to achieve vulnerability assessment.

The necessity of making decisions and the specific conditions of the Japanese cooperation had much more weight than objective conditions of risks in defining the logics of risk assessment.

I.3. Risk, environment and hegemony. Towards a self-justification for action.

Regarding the hegemony of socio-environmental challenges, I"d like to make two comments:

-On the one hand it appears that in spite of similar objective conditions of risks, responses to risk situations might be different. Risks are not an "absolute" justification for action.

They must be situated. -On the other hand, in spite of a material and realistic identification of risks, individual or collective responses might seem "irrational". After the Vargas Tragedy, people came back quickly to unstable shelters and exposed terrains. Risks must be embodied in social and political commitments, values, principles…

In the case of climate change, the point is that we are dealing with a hegemonic context. Pervasive risks are turned into a discursive justification for decision-making, undermining critical perspectives. Social sciences must give tools to analyze the situatedness and embeddedness of risks to climate change, for instance, by asking two main questions:

-Adaptation to what? And what for? In an instrumental perspective: Is adaptation reproducing structural drivers based on income, education, gender, race, urbanism principles or unequal decision-making features? And in addition to the "what" question, the "how" question also needs to be asked in 2 ways:

-As societies, how do we adapt to climate change? How do we do that? -As scientists, how do we assess adaptation to climate change? How do we measure?

Answers to these questions rely on the kind of theoretical approaches of risks in social sciences.

II. Considering risk as an outcome, or as a driver: what are the implications for adaptation?

On the one hand, risks can be assessed as an objective reality, as a biophysical or technical issue.

On the other hand, they can be addressed as a social construction, embedded in social and power relations, cultural hierarchies, recognition issues, or economic and political interests.

II.1. The realistic approach. The necessity to deal with the drama.

Basically defined, risks are a combination of hazard and vulnerability.

""Hazards relate to the physical processes and the spatial and temporal likelihood of an event occurring. This is the area that physical science addresses" (Leahy, 2006: 83). Hazards are what physical science addresses. In an analytical way, vulnerability is made of exposure and susceptibility of acquired values in front of a hazard. Some authors speak about biophysical vulnerability. But a social approach of vulnerability puts forward multiple factors, from structural to conjectural ones. "Social vulnerability to disasters refers to the inability of people, societies, and organizations to withstand adverse impacts from multiple stressors to which they are exposed. Social vulnerability is due in part to characteristics inherent in social interactions, institutions, and systems of cultural values. The multidisciplinary field of social vulnerability research has emerged to address these complexities" (Warner, 2007: 9). In spite of the materiality of the realistic approach of vulnerability, the weight of culture, representation, faith, relations, hierarchies, status, and so, matters. Research on vulnerability is an opportunity to question the hegemony of exclusively-material, hazard-centred and technical approaches in assessing risk situations.

II.2. The constructivist approach. The ability to take performative dimension of risk into account.

Constructivism states that categories are pre-existing to the apprehension of the world. As social constructions risks are defined by objective conditions of vulnerability and its representations.

Regarding risk components, 2 comments:

-On the one hand, vulnerability is socially defined (exposure is increased). But hazards as well can be human-induced (heat urban islands). -On the other hand, representations of risks are also social constructions. Representations as regards what is socially identified as a risk (it may change according to urban sectors and priorities). But also as regards the way scientific objectivity is constructed, the way risks are monitored, measured and represented.

Risks can be considered as a result of social (re)production. But they are also drivers of sociospatial forms, by influencing territorial practices, rules of urbanism or urban landscapes. They are performative. Risks are both material and intangible constructions. Intangible aspects have concrete and sometimes material consequences.

Adaptation assessment ought to consider these performative / material -intangible aspects.

II. 3. Between realism and constructivism. Some consequences on risk assessment.

The opposition between realism and constructivism is a misleading debate. A social scientist cannot fix weak shelters. As well, a geophysical engineer cannot say why his objective work is not applied on the same way in different places, in spite of similar risk conditions. Power, domination, or equity concerns are fundamental in a social science perspective.

Let"s reconsider some examples and sketch out potential interpretations or key questions through this double theoretical lens which try to conciliate fragmented approaches.

In the case of Caracas and the Japanese cooperation, we should question the choice of risks, and of locations at stake. For instance, why are landslides unevenly addressed by public authorities and the inhabitants, according to urban sectors?

In the case of the European Union, central-European countries had to deal with important floods in Danube and Rhin watersheds in 2002. The German chancellor asked the President of the European Commission for monitoring and assessing risk of flooding at the regional scale.

Since 2002, European Union is developing EFAS (European Flood Alert System), a kind of early warning system. The EFAS is based on probabilistic models whereas deterministic ones were more likely to be used until 2000s. What is such a shift in monitoring corresponding to? What are the practical consequences of such a change? Sébastien Nobert states that probabilistic models are a way to transfer responsibilities from scientists to decision-makers and at last to individuals, accountable of their own decision.

The ways risks are measured are never "neutral". They influence risk representations.

Academic efforts on vulnerability assessment and the International Decade for Natural Disaster Reduction have made possible the constitution of a trans-UN institutional and collaborative platform in 2001: the International Strategy for Disaster Reduction (ISDR). The tsunami in the Indian Ocean in 2004 brought even more attention to environmental risk issues. But it also showed that most of the UN risk programs for more than 20 years have been mainly reactive. UN institutions and initiatives were aiming at improving crisis management and hazard monitoring. The debate between reaction to catastrophes and the conception of risks as development features is still producing critical tensions in (and between) UN institutions. Some academic institutions related with development programs and UN agencies like the World Meteorological Organization are pushing forward to promote early warning systems and technical solutions. In this way, root causes and risk issues to climate change -that have been politicized for more than 30 years (Wisner, 1976;Pelling, 2003) -remain under-considered.

As social scientists, how could we take into account the competition between a reactive and a pro-active perspective? How could we intertwine the two conceptions? How could we consider adaptation as an agency of a socio-spatial order? A critical approach questions limits, categories, boundaries and what appears to be necessary. The notion of territory and its construction processes acknowledges an integrated approach of risks. It helps asking more accurate questions. For all these reasons, adaptation cannot be normative. Risk and adaptation designate things and dynamics, but they also contribute to create them. They have concrete consequences on territories, and they can be instrumentliazed. Because of that, they ought to be politicized in order to get all their implications.

III. Adaptation

III.2. Politicizing risk assessment to achieve accurate adaptation (in both material and intangible ways).

Regarding adaptation assessment, three main objectives should be achieved through politicization:

-The research framework articulates risk situations to a set of multiple causes, to structural, organizational and conjectural ones. -Risks are intertwined with -and sometimes are part of -social dynamics that (re)produce socio-spatial structures. They are a matter of power, control, democracy and justice. Adaptation assessment must account for such political dimensions. -Adaptation to climate change must account for time and space dimensions. At the 2002 Johannesburg conference on sustainable development, Mexican government was ensuring that the Mexican exportation sectors were not contaminating. While southern countries" economies rely on northern localization strategies of production and manufactures units, the definition of environmental problems is not only a local matter. Scaling risk situations is critical to achieve accurate adaptive strategies.

An encompassing approach is required to politicize risk and its assessment. As a geographer, I argue that it leads to consider territorial dimensions that allow integrating scale, power and intangible issues.

III.3. Towards the territorialization of risks. A theoretical sketch.

Methodological aspects are not detailed here.

But as well as risks, territories are both material and intangible social constructions. They are consolidated over time. Territories are identified and characterized by practices and representations.

Territories and risks are narrowly related. Urbanization, for instance, contributes to the construction of risk conditions, and risks -and its representations -shape territorial dimensions, by influencing landscape, practices, laws or economical values. Both categories are interacting and mutually determined.

I define territories of risk as a combination of objective conditions of risks and representations. They correspond to a construction that is grounded on socio-spatial order.

The researcher must be aware of the way risks are handled, and the way they are grounded on social and power relations, values, subjectivity and status. Territories of risk are simultaneously a consequence, a driver, and an instrument of a situated socio-spatial order. They allow taking into account the two main ideas of my statement:

-Risks are social constructions -Adaptation strategies to climate change are performative. It could be a way to organize risk assessment on critical basis:

-By questioning naturalized categories (adaptation to what?) -By considering adaptation to climate change as an agency (territorialization can be considered as an outcome and as a process: the how-question of adaptation).

assessment to climate change through a territorialized approach. III.1. Socio-spatial consequences of adaptation to climate change. Some examples.

  Adaptation is not only a social reaction. It also shapes territories. That is what some territorial consequences of risk assessment are showing. 3 points:-The modification of socio-spatial forms or built environment might resolve some risk situations. But it brings also new exposures (when people feel safer). -In a climate change perspective, risks are hardly calculable. Physical mechanisms of extreme events do not correspond to average mechanisms. Statistical averages based on recurrent events are not accurate if biophysical mechanisms are different. -Finally, social standards of acceptability are shifting over time. Needs, norms, values or risk recognition are not essential categories. They are shifting as society does.


Laurent Poinsot

Boolean Bent Functions in Impossible Cases: Odd and Plane Dimensions

Keywords: cryptography, bent functions, perfect nonlinearity, group actions, fixed-point free involutions

Bent or perfect nonlinear Boolean functions represent the best resistance against the so-called linear and differential cryptanalysis. But this kind of cryptographic relevant functions only exists when the number of input bits m is an even integer and is larger than the double of the number of output bits n.

Unfortunately the non-existence cases, the odd dimension (m is an odd integer) or the plane dimension ( n m = ), are not illegitimate from a cryptographic point of view and even commonly considered. New notions of bentness and perfect nonlinearity are then needed in those impossible cases for the traditional theory. In this paper, by replacing the usual XOR by another kind of bit-strings combination, we explicitly construct new "bent" Boolean functions in traditionally impossible cases.

Introduction

When used as components of a secret-key cryptosystem, Boolean functions must be highly nonlinear in order to avoid vulnerabilities to a differential [START_REF] Biham | Differential cryptanalysis of des-like cryptosystems[END_REF] or linear attack [START_REF] Matsui | Linear cryptanalysis for DES cipher[END_REF]. Informally speaking, linear cryptanalysis relies on the probability of success of a linear approximation of a function. Basically the differential cryptanalysis exploits the differences between plain and ciphered binary strings, i.e. the XOR combination of these bit-strings, which appear more often than at random. The objective of both attacks is to discover the secret key used in the ciphering process. Then Boolean functions must exhibit the optimal resistances against at least these two attacks. Boolean bent functions [START_REF] Dillon | Elementary Hadamard difference sets[END_REF][START_REF] Rothaus | On bent functions[END_REF] are those Boolean functions whose Fourier spectrum contains only two values. In other words, their frequential representation has the simplest form. They ensure the best resistance against linear attacks. Perfect nonlinear Boolean functions [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF] are those Boolean functions whose XORed outputs for XORed inputs are uniformly distributed over the set of all possible values. The best resistance against a differential attack is obtained by this kind of functions. These two notions are actually equivalent by duality using the Fourier transform.

Nevertheless such functions only exist in very restrictive cases: the number of input bits m is an even integer and it is larger than two times the number n of output bits. Unfortunately the impossible cases i.e. when m is an odd integer (call it "odd dimension") or when m is equal to n (call it "plane dimension") occur in many cryptographic applications: for instance, a ciphering function generally maps plain bit-strings to ciphered bit-strings, both with the same length. As linear and differential attacks can obviously be applied in such cases, we need to define a new kind of resistances or in other terms, a new kind of bentness and perfect nonlinearity. In recent contributions [START_REF] Poinsot | Generalized Boolean bent functions[END_REF][START_REF] Poinsot | Group actions based perfect nonlinearity[END_REF], we introduce the concept of G-perfect nonlinearity, which is a natural extension of the traditional one, obtained by replacing the XOR operation by other kinds of binary strings combinations. This combinatorial property has an equivalent characterization by the Fourier transform called G-bentness as in the classical setting. In this paper by using this new approach for nonlinearity, we construct some Boolean G-bent functions in cases impossible for the traditional theory: in odd and plane dimensions. We also present the principle of a G-differential attack and we exhibit a possible weakness of the S-boxes of the famous DES against such new cryptanalysis.

Boolean bent functions: the classical approach

Boolean bent and perfect nonlinear functions

In this paper The zero of m V is denoted m 0 . For any group G, * G is the set of all non-identity elements of G. Using these notations, a Boolean function is simply a function

n m V V f → : (some time called a (m,n)-Boolean function).
The notion of Boolean bent functions originally introduced by [START_REF] Dillon | Elementary Hadamard difference sets[END_REF] and [START_REF] Rothaus | On bent functions[END_REF], is very relevant in cryptography since such functions exhibit the best resistance against the so-called linear cryptanalysis [START_REF] Matsui | Linear cryptanalysis for DES cipher[END_REF]. This notion is closely related to the Hadamard-Walsh transform which is a particular case of the (discrete) Fourier transform.

Definition 1. Let → m V : ϕ R be a function, we denote by → m V : φ R the Walsh-Hadamard transform of ϕ : ∑ ∈ - = ∈ ∀ m V x x m x V . ) 1 )( ( ) ( , α ϕ α ϕ α .
In order to use this transform, we need to identify Boolean functions with real-valued ones. This is done using the notion of characters.

Definition 2. Let

m V ∈ α . The character of α is the map α χ m defined by } 1 { : ± → m m V α χ x x . ) 1 ( α - a . Then a function n m V V f → :
is transformed as a real-valued function defined on m V by considering the maps ) ( .

(-1) : :

x f n x f f β β β χ a o = for n V ∈ β . Definition 3. A Boolean function n m V V f → : is bent if for all * n V ∈ β and for all m V ∈ α , we have 2 2 ) ( ˆm f ± = α β .
Boolean bentness has also a combinatorics characterization called perfect nonlinearity [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF]. Perfect nonlinear functions are those functions that exhibit the best resistance against the famous differential attack [START_REF] Biham | Differential cryptanalysis of des-like cryptosystems[END_REF].

Definition 4.

A Boolean function

n m V V f → : is called perfect nonlinear if for all * m V ∈ α and for all n V ∈ β , n m m x f x f V x - = = ⊕ ⊕ ∈ 2 | } ) ( ) ( | { | β α where | | X is the cardinality of a finite set X . Theorem 1. [7] A Boolean function n m V V f → : is bent if
1 2 GF V V g m = →
be any function and π a permutation over

2 m V . Then the function 1 V : → m V f ) ( ) ( . ) , ( y g y x y x ⊕ π a is bent (we have identified m V with 2 2 m m V V × ).
Bent functions do not exist for each choice of ) , ( n m : there are some important constraints on these exponents. Theorem 2. [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF] Bent functions exist only for n m 2 ≥ and m is even.

In particular bent functions do not exist in odd dimension (m is an odd integer) nor in plane dimension ( n m = ). This is very restrictive since many cryptographic applications used functions that maps bit-strings to bit-stings of the same length. Then we need other objects to define resistances against differential or linear cryptanalysis. In this paper, we present a group action approach to solve this problem; nevertheless some authors have already worked on the subject and used another method.

Almost bent and almost perfect nonlinear functions

If m is even or n m 2 < , bent functions do not exist and we need other bounds to define differential and linear resistant functions. Some results have been found by Chabaud and Vaudenay [START_REF] Chabaud | Links between differential and linear cryptanalysis[END_REF].

Definition 5. Let m m V V f → : be a Boolean function. f is almost perfect nonlinear if for all * m V ∈ α and m V ∈ β , }. 2 , 0 { | } ) ( ) ( | { | ∈ = ⊕ ⊕ ∈ β α x f x f V x m
As differential resistance is measured as the minimum over all maps

m m V V f → : of the maximum over all m m V V × ∈ * ) , ( β α of | } ) ( ) ( | { | β α = ⊕ ⊕ ∈ x f x f V x m
in the plane dimension case ( n m = ), the best possible value is 2. Then almost perfect nonlinear functions have the optimum resistance to differential cryptanalysis.

In a similar way, we can define the optimum resistance against linear cryptanalysis as follows.

Definition 6. Let

m m V V f → : . The Boolean function f is called almost bent if for all * ) , ( m m V V × ∈ β α , } 2 , 0 { ) ( ˆ21 + ± = m f α β .
Such functions only exist when m is an odd integer.

Finally we have

Theorem 3. [2] If the function

m m V V f → : is almost bent then it is also almost perfect nonlinear.
The reciprocal assertion of the theorem above is false in general since the map

) 2 ( ) 2 ( : m m GF GF → σ defined by 1 ) ( - = x x σ if * ) 2 ( m GF x ∈ and m m 0 ) 0 ( = σ is almost
perfect nonlinear (when m is an odd integer [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF]) but it is not almost bent (it a consequence of [START_REF] Lachaud | The weights of the orthogonal of the extended quadratic binary Goppa codes[END_REF]).

In this paper we present another way to treat a similar problem using group actions.

Boolean bent functions: the group action approach

Basics on the theory of characters and group actions

In the remainder of this paper, the letter "G " stands for a finite abelian group in multiplicative representation. Our approach of nonlinearity and bentness is described using the Fourier transform and the notion of group actions. So in this subsection are recalled some basics on them.

Theory of characters

A character of a group G is a group homomorphism from G to the unit circle of the complex field C. The set of all characters, denoted by , Ĝ when equipped with the point-wise multiplication of functions, is a group isomorphic to G itself. We always suppose that an

→ G : φ C . ) ( ) ( ∑ ∈G x x x α α χ ϕ a
The Walsh-Hadamard transform is a particular instance of the discrete Fourier transform for elementary abelian 2-groups.

Group actions

A group action of a group G on a nonempty set X is a group homomorphism ϕ from G so ) ( X S the symmetric group of X (the group of all permutations over X). For instance we can define the regular action by translation of a group G on itself as follows

S(G) : → G ϕ ) : ) ( ( x x α α ϕ α a a . Instead of writing " ) )( ( x α ϕ " for X G x × ∈ ) , (α , we
use the convenient notation " x .

α " similar to the regular action by translation. The orbital of an element

X x ∈ under the action of G is . } | . { ) ( G x x O G ∈ = α α
The orbitals define a partition of the set X.

An action is called faithful if the corresponding homomorphismϕ is injective or in other terms there is no α (in particular a regular action is also faithful). For instance, the regular action by translation is regular.

* G ∈ α , such that x x = .
There is a particular kind of group actions on Boolean vector spaces which will be useful in the sequel. A

fixed-point free involution σ of m V is an element of ) ( m V S such that 1. σ σ = -1 , 2. for all m V x ∈ , . ) ( x x ≠ σ For instance the translations over m V , x x ⊕ α σ α a : , are fixed-point free involutions. Let identify k V as a subvector space of m V of dimension k (with m k ≤ ≤ 1 ). A group action of k V on m
V can be naturally defined as follows: for

m k V V x × ∈ ) , (α , we define x x ⊕ = α α.
. Moreover it is obvious that k V acts faithfully (and even

regularly if k = m) on m V .

G-perfect nonlinearity and G-bentness

Embedded in the definition of perfect nonlinearity is the regular action by translation of m V . So we can naturally extend this notion of nonlinearity by replacing translations by another kind of group actions. In [START_REF] Poinsot | Generalized Boolean bent functions[END_REF][START_REF] Poinsot | Group actions based perfect nonlinearity[END_REF], we have introduced the concept of G-perfect nonlinearity that we briefly recall in the Boolean setting.

Definition 8. Let G be a finite abelian group thats acts faithfully over

m V . A Boolean function n m V V f → : is called G-perfect nonlinear if for all * G ∈ α and all n V ∈ β , we have n m m x f x f V x - = = ⊕ ∈ 2 | } ) ( ) . ( | { | β α .
This notion is similar to the classical one except that we have replaced the translations by the action of G on m V .

On an other hand, we can define a group action based notion of bentness.

Definition 9.

Let G be a finite abelian group thats acts faithfully over

m V . A Boolean function n m V V f → : is called G-bent if for all * n V ∈ β and all G ∈ α , we have | G | | ) ( | 2 2 1 = ∑ ∈ m m V x x f α β where for each X x ∈ , x f is defined as n V : → G f x ) . ( x f α α a and |z| is the complex modulus of ∈ z C.
These two concepts are actually identical as it is the case in the traditional setting.

Theorem 4. [START_REF] Poinsot | Group actions based perfect nonlinearity[END_REF] Let G be a finite abelian group thats acts faithfully over

. m V A Boolean function n m V V f → : is G-perfect nonlinear if and only if it is G-bent.
This definition of G-bentness permits us to explicitly construct (see Section 4) "bent" functions in cases otherwise impossible with the standard notion of bentness. These constructions rely on the characterization of G.

G-difference sets

In the paper [START_REF] Poinsot | Group actions based perfect nonlinearity[END_REF] we gave a combinatorics characterization of G-bentness and G-perfect nonlinearity for 1 V -valued functions in terms of some special objects called G-difference sets.

Definition 9.

Let G be a finite abelian group thats acts faithfully over The indicator function of a subset S of a set X is the map

. m V A subset D of m V is called a G- ) , ( λ K -difference set of m V if 1. K = |D|,
} 1 , 0 { : 1 = → V X i S defined by 1 ) ( = x i S if S x ∈ and 0 ) ( = x i S if .

S x ∉

Theorem 5. [START_REF] Poinsot | Group actions based perfect nonlinearity[END_REF] Let G be a finite abelian group that acts faithfully over

. m V Let . m V D ⊂ Then D is a G- ) , ( λ K -difference set of m V such that ) ( 4 2 λ - = K m if and only if } 1 , 0 { : 1 = → V V i m D is G-bent.
In the following section, we use this particular characterization to construct a G-bent function in odd dimension.

Boolean bent functions in odd and plane dimensions

In this section we construct some G-bent Boolean functions

n m V V f → :
in some cases impossible for the traditional theory: m is an odd integer or m=n. As we want these constructions to be relevant for cryptographic applications we restrict our choices for G:

either k V or * ) 2 ( m GF .
With these particular choices, we remain in the Boolean setting which is important for implementations of cryptosystems.

Boolean bent functions in odd dimension

In this subsection, we present two constructions of

k V -bent functions n m V V f → :
where m is an odd parameter. Note that the action of k V over m V has been described at the end of paragraph 3.1.2.

Theorem 6.

Let

n k V V f → : be a (classical) bent function and n l V V g → : be any function. Let m=k+l and m V is identified with l k V V × . Then the map h defined by n m V V h → : ) ( ) ( ) , ( y g x f y x ⊕ a is a k V -bent Boolean function. Proof. Let 2 ) , ( m V y x ∈ and * n V ∈ β . We must compute ) ( ˆ) , ( α β y x h i.e. the Fourier transform of ) , ( y x n h o β χ in k V ∈ α . ' . ' ) ' ( . ) , ( ) 1 ( ) 1 ( ) ( ˆ) , ( x V x x h y x k y x h α β α β ∑ ∈ - - = ∑ ∈ ⊕ ⊕ - = k V x x y x x h ' ' . ) , ' ( . ) 1 ( α β ∑ ∈ ⊕ ⊕ ⊕ - = k V x x y g x x f ' ' . )) ( ) ' ( .( ) 1 ( α β ∑ ∈ ⊕ ⊕ - - = k V x x x x f y g ' ' . ) ' ( . ) ( . ) 1 ( ) 1 ( α β β ∑ ∈ ⊕ ⊕ - - = k V x x x x f y g ' ' ) ' ' .( ) ' ' ( . ) ( . ) 1 ( ) 1 ( α β β ). ( ) 1 ( . ) ( . α β α β f x y g ⊕ - = Then we have for all k V ∈ α and for all * n V ∈ β , ∑ ∑ ∑ ∈ ∈ ∈ = = m k m k m k V y x k V y x y x V y x f h ) , ( 2 1 2 ) , ( 2 1 2 ) , ( ) , ( 2 1 2 | ) ( | | ) ( | α α β β (because f is bent). Then the sum is equal to m 2 and therefore h is k V -bent.

QED

In particular if l is chosen so that m=k+l is odd then we have constructed a "bent" function in the odd dimension. However this construction is not satisfactory since it is based on a classical bent function. We now give another construction which is independent of the traditional notion of bentness.

Lemma 1.

Let m be a non-zero positive integer. Each

* m V ∈ α is contained in 1 2 1 - - m subvector spaces of m V of dimension m-1 over ) 2 ( GF .
Proof. We will build up a basis that contains .

α We can not use m 0 orα for the second vector in the basis, so we Continuing in this manner, we will have

2 )( 4 2 )( 2 2 ( ) 2 2 )...( 8 2 )( 4 2 )( 2 2 ( 2 1 1 1 1 2 - - - - - - - - - - - - - - m m m m m m m m m m 8
By pulling out all of the powers of 2 and then canceling all the terms, this reduces to

1 2 1 - - m . QED Theorem 7.
Let k and l be any positive integers, k ≠ 0 and l can be equal to zero. There exists a

k V - ) 2) - 1)(2 - (2 2 1), 1) - 1)(2 - ((2 2 ( 1 - 1 - 1 - k k l k k l + -difference set of l k V + 2 .
In particular its parameters satisfy the equation

). ( 4 2 2 λ - = + K l k Proof. Since k V is a k-dimensional vector space, it contains 1 2 - k subvector spaces of dimension k -1 denoted i W for 1 2 1 - ≤ ≤ k i . We observe that each orbital of l k V + 2 under the action of k V has exactly
1} - {0,...,2 } 2 ,..., 1 { j) (i, l k × ∈ j i x , is the representative of the k j i 2 + -th orbital }. | { ) ( 2 , , k l k j i j i V V V x x O k ∈ ∈ ⊕ = + α α In particular if ' ', , j i j i x x ≠ then ø ) ( ) ( ' ', , = ∩ j i V j i V x O x O k k . For each 1}, - 2 ,..., 0 { j l ∈ we associate the subvector space i W to the orbital ) ( , j i V x O k (so for i from 1 to 1 2 - k ) and we build-up the set } 0 , W | { 2 i 2 , , l k l k j i j i V x D + + ≠ ∈ ∈ ⊕ = α α α which is a subset of . ) ( , j i V x O k Finally we construct }. { ) ( , 2 1 2 1 , j i j i j k k x D D U - = ∪ = By construction, with a fixed j, ø ', , = ∩ j i j i D D for all '. i i ≠ In particular . | | 1 | | 1 2 1 , ∑ - = + = k i j i j D D But 1 2 1 | | | | 1 , - = - = - k i j i W D and then 1. 1) - 1)(2 - (2 | | 1 - k k + = j D
Then we construct the following set:

.

1 2 0 U - = = l j j D D We note easily that ø ' = ∩ j j D D for all ' j j ≠ and then D is a disjoint union. Its cardinality is then equal to ). 1 ) 1 2 )( 1 2 (( 2 | | 1 1 - 2 0 j + - - = - = ∑ k k l j l D
We have already proved that the parameter

| | D K = is equal to the value given in the statement of the theorem. It is now sufficient to prove that D is a k V - ) , ( λ K -difference set of l k V + 2 with ). 2 2 )( 1 2 ( 2 1 1 - - = - - k k l λ Now we show that for all nonzero , k V ∈ α they are exactly λ solutions in 2 D to the equation . y x ⊕ = α
Note that if x and y are not in the same orbital, we have no solution.

So let 1}. - {0,...,2 } 2 ,..., 1 { ) , ( l k j i × ∈ If k i 2 = then there is no solution in 2 D for a nonzero k V ∈ α (because D only contains j k x , 2
and α acts as a fixed-point free involution). Let suppose that

1}. - 2 ,..., 1 { k i ∈ If 2 , ) ( ) , ( j i V x O y x k ∈ is a solution in 2 D of the equation y x ⊕ = α for , i W ∈ α then j i x x , ⊕ = β and j i x y , '⊕ = β for 2 ) ' , ( i W ∈ β β implies that ). ).( ' ( , , j i j i x y x x β α α β ⊕ = ⊕ = = ⊕
Since the action of k V on each its orbital is regular, we have '

β α β ⊕ = and then i W ∈ ⊕ = ' β β α
, which is a contradiction and there is no solution of this form.

Boolean bent functions in plane dimension

In

β α = ⊕ ) ( ) . ( x f x f β α = ⊕ ⇔ ) ( x x f ) ( ) 1 ( 1 β α - = ⊕ ⇔ f x ) 1 ( ) ( 1 ⊕ = ⇔ - α β f x
2 ( m GF ∈ β , | ) 2 ( | | ) 2 ( | | } ) ( ) ( | ) 2 ( { | m m GF GF m x f x f GF x = = ⊕ ∈ β α =1. Then f i s * ) 2 ( m GF -perfect

G-differential attack

G-differential attack algorithm

Bentness and perfect nonlinearity are closely related to differential and linear cryptanalysis. By analogy we can interpret G-bentness and G-perfect nonlinearity as resistances against G-linear and G-differential attacks. group action based differential attack would be similar to the classical one except that we replace the XOR differences by their group actions based counterparts: the

G-difference of , ) , ( m V y x ∈ G x ∈ = Δ α (if there exists) such that y x . α = (with G acting faithfully on m V ).
The sketch of a G-differential attack, when applied on a cryptosystem with round function f that maps an encrypted message 1 i C (that consists in a m bit-string) of the (i-1)-th round on a encrypted message ) ( ) , ( 1

i K i i i C f K C f C i = = - ( ) (x f x K a is a
permutation on m-bit strings) where i K is the i-th round key, could be represented by a four steps algorithm:

A possible weakness in the S-boxes of the DES

In this last subsection, we present an amazing structure in the S-boxes of the famous DES, based on fixed-point free involutions, that could be used in a G-differential attack of this system. However this raises a question not answered in this paper: how to use this structure in a G-differential attack? But since the DES cryptosystem has been widely used since its introduction in the 1970s, any theoretical advances could lead to new generation of DES-like cryptosystem.

The DES has eight S-boxes i S (i = 1,…,8) from [START_REF] Matsui | Linear cryptanalysis for DES cipher[END_REF] V to 4

V and its solidity is based on these particular nonlinear functions. Nevertheless it appears that each S-box has a remarkable structure that seems to be a weakness regarding to the G-differential attack. In other words, there exists a fixed-point free involution σ such that the difference in output of i S of all input x and x', that differ from the application of σ , is constant equals to β .

Proof. In each S-box i S , each output occurs exactly four times (because they are equidistributed). So if we arrange each input by its output value, we obtain 16 sets y I with 4 elements

) 2 (

 2 GF denotes the finite field of modulo-2 integers and, as usually, it is considered as the subset field R and the modulo-2 sum (or XOR) is designed by the symbol " ⊕ ".A m-dimensional vector space over

  spaces of the same (finite) dimension are isomorphic, we use the same symbol "." which stands for a dot-product for m V (which is the ideal m-dimensional ) 2 ( GF -Hilbert space) and can take one of the previous forms. We also use the symbol " ⊕ " to denote at the same time the component-wise modulo 2 sum of symbol is used for the addition in m V .

C

  isomorphism from G to G ˆis fixed and the image of G ∈ α by such isomorphism (called the character of α ) already introduced. Using the theory of characters we can define the (discrete) Fourier transform on G. be a function. The Fourier transform of ϕ is the map ϕ ˆdefined by

  the action is the regular action by translation, this notion is equivalent to the traditional concept of difference sets in m V .

  to fill the second basis vector (call it 2 v ). Once we have chosen 2 v , we can not take any linear combination of α and 2 v , so we have 4 2m choices for the third vector (call it 3 v ).

  counted, so we need to divide by the number of basis for a given (m-1)-dimensional subspace (that containsα )

QED

  Since the parameters of the G-difference set satisfy the assumption of theorem 5, the indicator function of this set is k V -bent. Moreover this construction is much more relevant than the previous one. Indeed we can choose the parameters k and l that make impossible the existence of any traditional bent function in this setting. If k is an odd integer, classical) bent and if l is also an odd integer, there is no bent function from

)

)

  is as far away as possible from the equidistribution, where M is the clear message to encrypt.2. Choose at random a clear text M and submit M and M . α to ciphering. We obtain two pairs clear-encrypted )

4 .

 4 Iterate steps 2. and 3. until one of the values ' i K occurs more than others. We will consider this value as the i-th round key.

  only if m is odd whereas with our construction we always have bijective functions no matter if m is an odd or even integer.

							nonlinear and by
	equivalence it is also	GF	2 ( m	)	*	-bent.
							QED
	The interest of such construction seems to be less clear than
	the previous one if we remain in the traditional setting with
	a DES-like cryptosystem equipped with XOR as internal
	operation since the automorphisms are in particular linear
	and do not offer great resistance against cryptanalysis.
	Nevertheless if we think about a DES-like system in which
	the key is combined with the message by a multiplication
	rather than an addition, this construction gives the
	best-resistant function against a differential attack where
	the difference is taken in the sense of multiplication
	whereas such functions in the traditional setting are
	impossible: the best ones are almost perfect nonlinear.
	Moreover it is widely conjectured that almost perfect
	nonlinear permutations on	GF	2 ( m	)

exist

Acknowledgments

The author thanks Professor James Davis, from the University of Richmond, for his help on constructions of G-difference sets [4] and Assistant Professor Cyril Prissette, from the University of South Toulon-Var, for the anatomical study of the S-boxes of the DES [11].