Camille Salesse
email: camille.salesse@umontpellier.fr

Inequality in exposure to air pollution in France: bringing pollutant cocktails into the picture

Keywords: air pollution, cocktail, inequality, environmental justice JEL Codes: D63, Q53, I14

I estimate the relationship between income, the number of days of exposure to the four main air pollutants and the proportion of "cocktail days" with French municipal data over the period 2012-2018. I find contrasting results between rural and urban areas.

The most affluent urban municipalities have on average a lower number of pollution days compared to the poorest urban municipalities. In urban areas, the pollution days are composed of an equal proportion of cocktail days between the poorest and the most affluent municipalities. On the other hand, in the rural areas the better-off municipalities have on average a higher number of days of pollution, composed of more toxic mixtures, compared to the poorer municipalities. I also show that the pollution levels and the difference in the number of pollution days between the better-off and poorer municipalities are higher in urban areas.

Introduction

Exposure to air pollution has negative effects on health, productivity, cognitive performance and educational outcomes1 . The overexposure of a part of the population to pollution thus reduces its earning potential, challenges the principle of equal opportunity and increases health inequalities. Quantifying environmental inequalities is important for determining optimal environmental policies that take into account these equity concerns. In this paper, I have two main objectives. First, to quantify the difference in air pollution exposure days between the richest and poorest municipalities. Second, to assess inequalities in exposure to air pollutant cocktails.

This study focuses on the four main pollutants identified by the World Health Organization (WHO) and regulated in France: fine particulate matter (PM2.5), PM10, nitrogen dioxide (NO2) and ozone (O3). The health effects of these pollutants are widely documented [START_REF] Deryugina | The Mortality and Medical Costs of Air Pollution: Evidence from Changes in Wind Direction[END_REF][START_REF] Knittel | Caution, Drivers! Children Present: Traffic, Pollution, and Infant Health[END_REF][START_REF] Anderson | As the Wind Blows: The Effects of Long-Term Exposure to Air Pollution on Mortality[END_REF][START_REF] Moretti | Pollution, Health, and Avoidance Behavior: Evidence from the Ports of Los Angeles[END_REF] and each of them has a recommended concentration threshold defined by the WHO. I evaluate the exposure to these pollutants because of their different origins and spatial dispersion patterns, some of which are more urban and others rural (Figure 1). In addition, exposure to these different pollutants can accumulate and form an even more toxic mixture2 . For these reasons, it is necessary to assess inequalities in exposure to the different main pollutants, taking into account their simultaneous presence when measuring exposure.

I use French daily concentration data for these pollutants as well as socio-economic data at the municipal level over the period 2012-2018. I exploit the daily availability of our pollution data to construct a new exposure indicator measuring the number of days in the year for which the concentration thresholds defined by the WHO for these pollutants are exceeded. Exposure to pollutant cocktails is measured by the share of cocktail days 3 in the total number of days for which at least one pollutant is above the thresholds. I use these indicators to examine the relationship between municipal income 4 , the number of pollution days and the share of cocktail days among these pollution days. I examine these relationships in rural and urban areas separately, on the one hand because these areas concentrate different types of pollutants and on the other hand because the literature has already shown a heterogeneity of results between these two areas. In all our main specifications, I control for the year fixed effect and the regional fixed effect (département level).

I find contrasting results between rural and urban areas. I show that fine particulate matter (PM2.5) and nitrogen dioxide (NO2) account a greater number of pollution days compared to the other pollutants (PM10 and ozone) and they are highly concentrated in urban areas. In urban areas, our results suggest that disadvantaged municipalities accumulate more days of exposure to fine particulate matter (PM2.5) and nitrogen dioxide (NO2), all else being equal. In this area, a one standard deviation increase in income is associated with a decrease of 8.57 nitrogen dioxide (NO2) pollution days and 2.3 fine particulate matter (PM2.5) days on average over the year. In urban areas, the pollution days are composed of an equal proportion of cocktail days on average between the poorest and richest municipalities. In rural areas, the results are different. I show that the richest rural municipalities accumulate more fine particulate matter (PM2.5), particulates PM10 and nitrogen dioxide (NO2) pollution days compared to the least rich rural municipalities. However, the exposure gap between rich and poor municipalities, other things being equal, is smaller compared to urban areas. Pollution days have on average a significantly higher proportion of cocktail days for affluent rural municipalities compared to disadvantaged municipalities.

This paper relates to the literature that measures environmental inequalities in exposure to air pollution5 . Many studies have shown the overexposure of the poorest and minorities to air pollution in the US, there is less evidence in France and Europe. While most studies consider only a limited number of pollutants for which exposure is measured in most cases through the annual average concentration of the pollutant, I estimate for the first time the unequal simultaneous exposure to air pollutants over the whole territory. I propose a new method for assessing exposure to air pollution cocktails that can be applied in studies evaluating the effects of air pollution exposure on multiple outcomes.

The rest of the paper is organized as follows. The following section presents the related literature. Section III describes the data. Section IV presents our empirical strategy. Section V presents the results, Section VI the robustness tests and the last section concludes.

Related Literature

In the US, areas with a higher proportion of minorities and disadvantaged people are exposed to higher concentrations of fine particulate matter (PM2.5) [START_REF] Jbaily | Air pollution exposure disparities across US population and income groups[END_REF] and industrial air pollutants [START_REF] Ard | Trends in exposure to industrial air toxins for different racial and socioeconomic groups: A spatial and temporal examination of environmental inequality in the U.S. from 1995 to 2004[END_REF]. [START_REF] Mikati | Disparities in Distribution of Particulate Matter Emission Sources by Race and Poverty Status[END_REF] also find these results using fine particulate matter emissions data6 . The results differ depending on the level of data aggregation7 and the area considered. [START_REF] Hsiang | The Distribution of Environmental Damages[END_REF] show that the relationship between income and nitrogen dioxide (NO2) exposure is positive at the Metropolitan statistical area (MSA) level while it takes a U-shape at the census block level, the relationship becomes negative when estimated within the MSA. Rosofsky et al. ( 2018) use socio-demographic data at the block group unit and concentrations data for fine particulate matter (PM2.5) and nitrogen dioxide (NO2) in Massachusetts to assess environmental inequalities in rural and urban areas. They show descriptive evidence that the poorest and minorities are more exposed to pollution in urban areas, while in rural areas inequalities are lower. Finally, the extent of exposure inequalities within regions varies across the US (Zwickl, Ash, and Boyce, 2014). Several studies have looked specifically at inequalities in exposure between ethnic groups in the US. [START_REF] Christensen | Housing Discrimination and the Toxics Exposure Gap in the United States: Evidence from the Rental Market[END_REF] show that African American or Hispanic tenants have a lower probability of response from homeowners in areas further away from emission sources where the concentration of pollutants is lower. [START_REF] Gillingham | Racial Disparities in the Health Effects from Air Pollution: Evidence from Ports[END_REF] show that air pollution from port activity8 disproportionately increases the number of hospitalizations of African Americans. This is due to an overexposure of African Americans but also a potential greater vulnerability. More generally, African Americans are exposed to higher concentrations of fine particulate matter (PM2.5), PM10 and nitrogen dioxide (NO2) compared to white Americans but this gap has narrowed in recent years [START_REF] Currie | What Caused Racial Disparities in Particulate Exposure to Fall? New Evidence from the Clean Air Act and Satellite-Based Measures of Air Quality[END_REF][START_REF] Kravitz-Wirtz | The Long-Term Dynamics of Racial/Ethnic Inequality in Neighborhood Air Pollution Exposure, 1990-2009[END_REF].

Several studies have also highlighted inequalities in exposure to air pollution in Europe. [START_REF] Germani | Environmental justice and air pollution: A case study on Italian provinces[END_REF] show an inverted U-shaped relationship between income and industrial pollutant emissions at the level of Italian provinces. [START_REF] Neier | Austrian Air -Just Clean for Locals: A Nationwide Analysis of Environmental Inequality[END_REF] uses socioeconomic data from a fine grid of the Austrian territory combined with industrial pollutant emission data. He shows that a higher proportion of foreigners in the area is associated with a higher exposure to pollution. Moreover, in urban areas the better-off are less exposed to pollution while in rural areas the relationship is reversed.

In France, several evidence suggest the overexposure of the poorest and minorities to air pollution and more particularly in urban areas. Our study is most closely related to [START_REF] Champalaune | Inequality in Exposure to Air Pollution in France: Evidence from a City-Level Public Policy[END_REF]. She uses administrative data at the neighborhood level and annual average estimates of fine particulate matter concentrations at a fine scale over the whole French territory. She shows through an individual fixed effect model that within urban and peri-urban areas, the poorest neighborhoods and those with a higher proportion of migrants are more exposed to fine particulate matter. On the other hand, in rural areas, areas with a lower share of migrants are more exposed and the relationship between income and exposure is not significant. These estimates are robust to spatial autocorrelation (using a model close to the SAR model, see empirical strategy section). [START_REF] Laurian | Environmental justice in France? A spatiotemporal analysis of incinerator location[END_REF] show that towns with a higher share of migrants are more exposed to incinerators in France. Ouidir et al. (2017) assess the exposure of pregnant women in France to three air pollutants: fine particulate matter (PM2.5), particles PM10 and nitrogen dioxide (NO2).

The study shows that in urban areas the most disadvantaged women are more exposed to the three pollutants, while in rural areas it is the most affluent and disadvantaged women who are most exposed. Padilla et al. (2014) use socio-economic data at the neighbourhood level in 4 large French cities and show that the strength and direction of the link between wealth, share of migrants and NO2 exposure can vary between cities9 . [START_REF] Fosse | Inégalités environnementales et sociales se superposent-elles ?[END_REF] show descriptive evidence of a U-shaped relationship between fine particulate matter (PM2.5) emissions and income of municipalities in France and an inverted U-shaped relationship between ammonia (NH3) emissions and income.

Overall, in the US and Europe, there is an overexposure of the poorest and minorities to air pollution. However, inequalities are more pronounced in urban areas. They are also greater in some regions/cities. Most studies consider only a limited number of pollutants and exposure is measured in most cases by the annual average concentration of the pollutant. I estimate for the first time the inequality in exposure to pollutant cocktails, analysing urban and rural areas separately. I also explicitly take into account the problem of spatial autocorrelation in a series of robustness tests.

Data

I build a database at the communal level including about 30 000 French municipalities over the period 2012-2018 10 . I use PREV'AIR (INERIS) background pollution data as well as localized socio-economic data from the FILOSOFI (INSEE) database and the census (INSEE) that I describe below.

Pollution data

I use the PREV'AIR air pollution data produced by INERIS for the four main regulated pollutants: ozone (O3), nitrogen dioxide (NO2), fine particulate matter PM2.5 and PM10. For each day of the year, I have the average concentration in micrograms per cubic meter of air (µg/m 3 ) of each of these pollutants in each of the 34,000 French municipalities over the 2012-2018 period.

These data are constructed using the CHIMERE pollutant transport simulation model. From pollutant emission inventory data, sampling data, metrological data and other types of input data, the model is able to predict pollution concentrations over the entire territory at a fine scale. An additional statistical treatment based on real observations of concentrations is then applied to the predictive outputs of the CHIMERE model allowing to further increase the reliability of the concentration estimates. PREV'air data are very reliable data available on the whole French territory with a fine resolution of about 4km2.

Based on the reliability of the pollution concentration estimates and the daily availability of our data, I calculate for each municipality the number of days in the year for which the threshold defined by the WHO was exceeded for each of the 4 pollutants. I use the most recent WHO thresholds (2021).

Fine particulate matter (PM2,5) : 15 µg/m 3 24-hour average value Coarse particles (PM10) : 45 µg/m 3 24-hour average value Nitrogen dioxide (NO2) : 25 µg/m 3 24-hour average value Concerning ozone (O3), the threshold recommended by the WHO is 100 µg/m 3 , maximum daily value of 8-hour rolling averages. I do not have measurements in 8-hour rolling averages, so I define our threshold at 100 µg/m 3 24-hour average value.

I choose to use the number of days for which the concentration of the pollutant is above the WHO threshold as an exposure indicator. To our knowledge, this indicator has never been used before and I argue that it improves the quality of the analysis. Indeed, I argue that the overexposure to pollution of a part of the population is a problem insofar as this population will suffer significant negative health consequences. Indeed, environmental inequalities measured from the annual average concentration of the pollutant are not systematically associated with a significant increase in health risk for the most exposed population. If one population is exposed to an average of 3 µg/m 3 of PM2.5 particles over the year and another the data of the municipalities forming the future merger by averaging them in order to have in the database only the municipalities newly formed by the merger over the entire period studied.

to 1 µg/m 3 of PM2.5 particles, in this case there is indeed an environmental inequality but the health consequences for the overexposed population remain limited since neither group exceeds the WHO threshold of 5 µg/m 3 annual average. I argue that environmental inequalities must be considered in terms of the health consequences they cause, and our indicator of the number of days above the thresholds allows us to identify exposure gaps with significant health risks.

Cocktails

I exploit the daily availability of pollution data and the very high reliability of the estimates to define several cocktails of pollutants. The objective is to identify the unequal exposure of the municipalities to various mixtures of pollutants. For this purpose, for each municipality, each day in the year for which the threshold of at least one pollutant is exceeded is classified in one of the 15 possible "pure" mixtures. This method allows us to define 11 pure mixtures without interaction with other pollutants (the pollutants of the considered mixture are above the WHO thresholds and the other pollutants are below the thresholds) and 4 pure monopollutant mixtures (only the considered pollutant is above the thresholds). I sum for each year for each municipality the number of days that each mixture represents. From these data I build an exposure indicator which measures the total number of pollution days of the municipality in the year by adding all the pollution days of the 15 possible mixtures. I also construct an exposure indicator to pollutant cocktails by calculating for each municipality each year the share that each mixture represents in the total number of pollution days. I also calculate the share of the "cocktail" mixtures in the total number of pollution days.

Figure 2 shows the distribution of all pollution days in each of the possible groups. 62% of the days above the WHO thresholds, for at least one pollutant, are days where only the PM2.5 threshold is exceeded. 19% of the pollution days are a pure mixture of PM2.5 and nitrogen dioxide (NO2), 6% pure nitrogen dioxide (NO2), 5% NO2 PM2.5 and PM10, 4% O3, 3% PM2.5 PM10 (see Table 1 descriptive statistics).

Socio-economic data

I match pollution data with several types of socio-economic data. First, I use the FILOSOFI database produced by INSEE, which contains a wide variety of indicators on available income at the municipal level. The data are available for the years 2012 to 2018. Our main indicator for measuring the wealth of a municipality is the median standard of living of the municipality (the standard of living is the disposable household income divided by the number of consumption units). Data on this indicator are available for almost all the mu-nicipalities. I also mobilize other variables such as the poverty rate, the first decile and the ninth decile of standard of living. However, data for these indicators are only available for municipalities with more than 1000 households or more than 2000 inhabitants. I also use many socio-economic indicators from INSEE census data such as the share of each socioprofessional category in the municipality or the share of women and men in the municipality, the composition by age and type of household in the municipality (see Table 1 descriptive statistics).

Descriptive statistics

The richness of this study lies in the consideration of various forms of air pollution that have different origins. The correlations Table 2 show that the different pollutants are not perfectly correlated with each other, so it is relevant to study these different forms of pollution separately. The origin of the emissions of these four pollutants partly explains the imperfect correlation between them. About half of the fine particulate matter come from residential heating (wood heating is largely responsible for these emissions) and about a quarter from road transport. The concentration of nitrogen dioxide in the air comes mainly from the road sector and, to a lesser extent, from industry. PM10 emissions come from residential heating, industry but also from construction sites which produce large particles. Ozone is not emitted directly by human activities, it is formed by a chemical reaction between nitrogen oxides and other components in interaction with solar radiation11 .

The maps Figure 3 show the dispersion of the 4 pollutants over the whole French territory in 2012 and 2018 (the same scale was used from one year to another). First, on average over the period, pollution levels decreased for PM10, PM2.5 and nitrogen dioxide (NO2) but not for ozone (see Figure 3 andFigure A.3). Secondly, the geographical distribution of the four pollutants at the national level and in rural and urban areas is not similar (which confirms the relevance of studying exposure to these different pollutants without being limited to fine particulate matter).

Aggregation level & ecological fallacy

The problem of the ecological fallacy is to erroneously infer a correlation at the individual or higher level from a correlation obtained from more aggregated data. The way to avoid this problem is to use the least aggregated data possible, but also to infer conclusions only at the level of data aggregation. The PREV'AIR pollution data are background pollution data, which means that although the resolution of the data is very fine, areas of high pollution that are very localized, such as areas with heavy road traffic, have very smoothed concentration levels.

I argue that the level of aggregation of our data at the municipality level is the most relevant for our study, whose objective is to investigate unequal exposure to national background pollution at the municipality level. The municipality level of our data allows to take into account part of the exposure to pollution when individuals move close to their home (to work for example). The municipal level is suitable for background pollution data, as the concentration of background pollutants is homogeneous over most municipalities. There is a finer level of administrative division (IRIS) adapted to very localized pollution data but which is not however adapted to our study devoted to background pollution. The French municipalities are very numerous (more than 30 000) and form a fine grid of the French territory allowing to avoid the problem of ecological fallacy 12 .

4 Empirical strategy 4.1 Specification I estimate the following fixed-effects linear regression model for municipality c, in département (region) d in year t, in each of the sub-samples of urban and rural municipalities.

y cdt = βx cdt + Θz cdt + λ d + α t + ε cdt (1)
y cdt is the number of days in year t in municipality c of department d for which the concentration levels of the pollution considered exceeded the WHO health threshold. x cdt is the median standard of living in the municipality. The vector z cdt includes a set of control variables such as the share of the population belonging to each socio-professional category, the share of women and men, the share of each age group, the share of each type of household. The regressions include département α d and year λ t fixed effects. I cluster standard errors at the département level 13 . I estimate this model in each of the 6 types of rural or urban communes defined by INSEE on the basis of the density of the municipality and its attraction to a pole. I also estimate this model at the national level in the appendix.

12 Many studies on environmental inequality in the United States have used more aggregated data at the county level, which is equivalent to the French département.

13 The traditional justification for clustering standard errors is to account for the correlation of residuals within département. [START_REF] Abadie | When Should You Adjust Standard Errors for Clustering?[END_REF] propose a new framework in which clustering is justified by two parameters: sampling and treatment design. In our case, only the treatment design is important since we have the full set of available data. Since income is partially correlated within departments, the robust standard errors (Eicker-Huber-White) are too small and the clustered standard errors at the department level too large (conservative). We prefer clustering to minimize the risk of bias.

I estimate this fixed-effect model to identify the correlation between the wealth of municipalities and their level of exposure to pollution, holding socio-demographic characteristics, department and year constant. This specification deliberately does not resolve the potential reverse causality of the income-pollution relationship. Indeed, environmental inequalities can be formed either by a variation in the wealth of the municipalities inducing a variation in pollution levels, or by the opposite phenomenon. The objective of this study is to identify environmental inequalities as they currently exist and not to identify the mechanism of formation. Identifying the causal effect of wealth on pollution is therefore not the objective of this paper.

To estimate the relationship between income and pollution, I exploit the variation within different geographical areas at different time periods in pollution levels and income of the municipalities while controlling for their socio-economic characteristics. I do not include municipality fixed effects because the objective is not to study the mechanism by which income variations influence pollution or vice versa or both at the same time within municipalities.

Spatial autocorrelation

I argue in this section that our fixed-effects model must account for the spatial dependence of our observations. Air pollution propagates continuously through space in a non-random manner (Figure 3). As a result of this spatial dependence or spatial autocorrelation, OLS estimation can be either inefficient or biased. To gauge the importance of this problem, I perform a series of tests on our OLS estimates.

There are 3 main specifications to account for different types of spatial interactions with panel data, the Spatial Error Model (SEM), the spatial autoregressive model (SAR) and the Durbin model which is a combination [START_REF] Agha | Chapter 7. Spatial econometrics on panel data[END_REF].

To determine which of these models to use, I perform a Lagrange multiplier test on the residuals of the OLS model for error dependence (LM-error) and for the presence of the spatially lagged dependent variable in the model (LM-lag) [START_REF] Anselin | Simple diagnostic tests for spatial dependence[END_REF]. I also perform the robust version of these tests to the presence of the other type of spatial relationship.

Depending on the results of the tests I estimate the SEM or SAR models (see appendix details). The SAR model reduces the bias of the OLS estimates while the SEM model has an impact on the efficiency of the model. These models are estimated by the maximum likelihood (ML) method with fixed effects14 .

Results

Income and number of days of exposure to pollutants

I estimate the relationship between income and the number of days of exposure to the pollutants in the rural and urban areas in Table 3. The poorest dense urban municipalities accumulate on average significantly more days of fine particulate matter (PM2.5) and nitrogen dioxide (NO2) pollution compared to the better-off municipalities, all else being equal. The increase of one standard deviation15 in income is associated with a decrease of 8.57 days of pollution for nitrogen dioxide (NO2) and 2.3 days for fine particulates (PM2.5) over the year. It seems that income is not significantly related to the level of PM10 and ozone (O3) pollution in this area. In urban municipalities with intermediate densities, no coefficient of income is significant for the 4 pollutants. In rural areas, income is significantly positively related to nitrogen dioxide (NO2), fine particulates (PM2.5) and PM10 pollution. Ozone level seems to be unrelated to income.

One additional standard deviation of income in rural areas under weak influence of a pole is associated with an increase of 3.68 days of nitrogen dioxide (NO2) pollution and 0.89 days of PM2.5 pollution and 0.32 days for PM10 on average. In sparsely populated rural areas, the number of additional days for nitrogen dioxide (NO2), fine particulate matter (PM2.5), and PM10 per standard deviation of income is on the order of 2.29 , 0.97 , and 0.29 days on average.

Finally, I also evaluate the link between the wealth of the municipalities and the number of days for which the threshold for at least one of the four pollutants is exceeded (ALL column). The results are similar to the previous results. One additional standard deviation of income is associated with a decrease of on average 8.25 days with at least one pollutant above the thresholds in dense urban areas and an increase of 1.57 days in sparsely populated rural areas.

Figure 1 shows that fine particulate matter (PM2.5) and nitrogen dioxide (NO2) are the pollutants for which the number of days above the thresholds is the highest and these pollutants are concentrated in urban areas (very largely for NO2). I also note that among rural municipalities, the most affluent municipalities are more exposed to PM2.5, PM10 and nitrogen dioxide (NO2). The concentration of these 3 pollutants in rural areas is on average lower compared to urban areas (Figure 1).

It should also be noted that the magnitude of exposure inequalities in terms of additional days of pollution per standard deviation of income is higher in urban municipalities. Not only is pollution much higher in terms of level in urban municipalities, but the exposure gap is also larger in urban areas.

The literature developed mainly in the United States has identified mechanisms that can explain the inequalities in exposure to nitrogen dioxides (NO2) and fine particulate matter (PM2.5) that are observed in urban areas. Environmental inequalities can be explained both by the location choice of inhabitants, the location choice of polluting firms and a combination of both (Banzhaf, Ma, and Timmins, 2019a). Wealthier people are more willing to pay the price of not living in polluted municipalities [START_REF] Banzhaf | Do People Vote with Their Feet? An Empirical Test of Tiebout[END_REF]. Highly imperfect household information on pollution also promotes exposure inequalities [START_REF] Hausman | Inequality, information failures, and air pollution[END_REF]. Polluting firms may also choose to locate disproportionately in or near less affluent municipalities (Banzhaf, Ma, and Timmins, 2019b).

The relatively small difference in pollution levels that I find between the better-off and poorer municipalities in rural areas can theoretically be explained by both the level of economic activity and the specialization of the municipalities. The rural population is divided into two types of municipalities, the densely populated municipalities that concentrate the most economic activity and the less densely populated municipalities that concentrate less economic activity. Dense municipalities have higher incomes than less dense municipalities, but also higher levels of pollution, both of which are partly explained by the higher economic activity (see annex for correlations between municipality density, income, economic activity and pollution, Figure A.5) Like rural municipalities, the densest urban municipalities have the highest economic activity and the highest pollution levels. However, the densest urban municipalities have lower incomes than the less dense urban municipalities (see figure Figure A.4). In urban municipalities, the better-off are located in the less dense and less polluted municipalities, while in rural municipalities they are located in the dense municipalities where economic activity and pollution are higher.

The origin of the pollutants may partly explain the non-significant results for ozone. Indeed, ozone, unlike other pollutants, is a so-called "secondary" pollutant, i.e. it is not directly emitted into the atmosphere but results from chemical transformations influenced in particular by solar radiation. This may explain the absence of a significant relationship with income, as ozone is determined more by meteorological than socio-economic variables (the relationship has the lowest R 2 compared to other pollutants).

In this section I have highlighted several results. Firstly, the 4 pollutants studied do not represent the same amount of days above their respective thresholds. Fine particulate matter (PM2.5) and nitrogen dioxide (NO2) have very high average levels in urban areas of around 100 and 117 days respectively over the period. PM10 and ozone (O3) have a maximum average of 13 and 5 days in the areas with the highest values. I have highlighted the significant negative relationship between income and exposure to PM2.5 and nitrogen dioxide (NO2) in urban areas (these areas being the most polluted for these pollutants). Conversely, I find a significant positive relationship between income and PM2.5, PM10 and nitrogen dioxide (NO2) pollution in rural areas. However, the average difference in exposure days according to income is greater in urban areas than in rural areas. The high level of pollution and the large exposure gap to the detriment of the less well-off in urban areas should lead to the implementation of policies to reduce pollutant emissions in these areas.

Unequal exposure to pollutant cocktails

The toxic effect of a pollutant can be multiplied in the presence of other pollutants, which may explain the differences in toxicity between different mixtures. Nevertheless, the interaction effects of pollutants on health are still poorly understood. To our knowledge, no study has assessed the toxicity associated with each mixture of air pollutants.

In this section, I define for the first time the share of multi-pollutant days compared to single-pollutant days. The impact of air pollution on health depends both on the number of days of exposure to pollutants and on the composition of these pollution days. Some mixtures of pollutants are more harmful than others, what share does each mixture of pollutants represent in the pollution days? In the first part of the analysis, I assessed the link between income and the number of exposure days. In this part, I will assess the link between income and the composition of pollution days.

As described in the data section, each pollution day is associated with one of 15 possible mixtures. The variables I used previously to estimate the pollution-income relationship are a simple measure of the number of days above the threshold for the pollutant of interest, across all mixtures. However, cocktails represent a significant number of days (Figure 2).

Figure 4 shows the average share of each mixture in the total number of days with at least one pollutant above the threshold at the national level and in urban and rural areas. According to these estimates, at the national level, on average 25% of the days with one pollutant above the threshold, the threshold of at least one other pollutant is also exceeded in the municipalities (Figure 4). The cocktail that represents the largest share is the mixture of PM2.5 and NO2. It represents on average 17% of the total pollution days. The composition of the pollution days varies between rural and urban areas. In dense urban areas 43% of pollution days are cocktail days compared to 34.6% in intermediate density urban municipalities and 23.5% in rural areas on average. Nitrogen dioxide, a very urban pollutant has a higher share in pure and mixed form in the pollution days of urban municipalities. The fine particulate matter PM2.5 are also a very present pollutant in the urban municipalities which favor the mixture with nitrogen dioxide. Fine particulate matter are also present in rural municipalities but not nitrogen dioxide which is much less present.

I have just shown that our measure of exposure to a pollutant in terms of the number of days above the threshold is composed of a significant proportion of "cocktail days". The shares of each mixture in the total number of pollution days that I have estimated in Figure 4 are averages. Some municipalities may therefore have a higher or lower share of cocktail days than the average.

Table 4 presents the results in each of the rural16 and urban areas. I use Z-score on the income variable to facilitate interpretation of the coefficients17 . The composition of pollution days can be influenced by the number of pollution days in the municipality. Since income is related to the number of pollution days, I add it as a control variable to avoid an omitted variable bias. I also include all control variables and fixed effects from previous regressions. In dense and intermediate density urban areas, the share of cocktail days is on average similar between poor and more affluent municipalities. In contrast, among rural municipalities, wealthier municipalities have a significantly higher share of cocktail days on average. One additional standard deviation of income is associated with a 1.75 percentage point increase in the cocktail share.

To investigate these results further, I now evaluate the relationship between income and the share of each of the major mixtures (those with a significant share, see Figure 4) represents. I calculate the share that each mixture represents in the total number of pollution days. Table 5 shows a non-significant relationship between the richness and the share of each mixture in urban areas. This result is consistent with our previous results. The pollution composition between rich and poor municipalities is on average similar in urban areas. Table 5 shows which mixes are related to income in rural areas. On average in this area, the pollution days in the richer municipalities are composed of a significantly lower share of pure fine particulate matter (PM2.5) and a higher share of a mixture of nitrogen dioxide (NO2) and fine particulate matter (PM2.5) , nitrogen dioxide (NO2), fine particulate matter (PM2.5) and PM10 but also pure nitrogen dioxide (NO2). One additional standard deviation of income is associated with an average decrease of 2.8 percentage points in the share of pure fine particulate matter and an increase of 1.5 points in the share of NO2 PM2.5, 0.2 points in NO2 PM2.5 PM10 and 1 point in pure NO2.

In the wealthy rural areas the nitrogen dioxide (NO2) takes a more important place in pure and mixed form compared to the poor areas. As I have shown, in this area the betteroff accumulate more nitrogen dioxide (NO2) pollution days and the coefficient is the most important compared to the other pollutants (Table 3). This surplus of NO2 is associated with a change in the composition of pollution days. As NO2 is a pollutant from urban emissions, this result is consistent with our theoretical explanation of the overexposure of the better-off in rural areas. Affluent rural municipalities are denser and more urban, concentrating more economic activity favoring the formation of NO2, in a proportion nevertheless much lower than urban areas (see appendix the correlations between density of the municipalities, income, economic activity and composition of the pollution Figure A.5).

Overall, among urban municipalities, pollution days have on average the same composition in the poorest and most affluent municipalities. On the other hand, in rural areas, pollution days are composed of a larger share of cocktail days in the better-off municipalities.

Robustness

Measures of wealth

Income is more heterogeneous in urban municipalities compared to rural municipalities. The median standard of living is a good socio-economic indicator, but information about the top and bottom of the standard of living distribution is lost, particularly in urban municipalities. I use a set of data on the poverty rate, the first and ninth decile of the standard of living, the share of social benefits in income and the share of income from assets in the total income of the municipality. These data from the FILOSOFI database are only available for municipalities with more than 1,000 households or more than 2,000 people, a condition met by almost 100% of dense urban municipalities and 66% of intermediate density urban municipalities.

Table 6 reports the results for the subsample of dense urban municipalities. Each coefficient is associated with a regression. The coefficients for the poverty rate, the first decile and the share of assets in income are highly significant for nitrogen dioxide (NO2), particulate matter (PM2.5) and for the measurement of days with at least one pollutant above the threshold. The direction and significance of these coefficients point to an inequality of exposure to the disadvantage of the poorest for nitrogen dioxide (NO2) and PM2.5 particles, which confirms our previous results. PM10 and ozone (O3) have non-significant coefficients which further affirms our previous results.

Table 7 presents the results for the subsample of intermediate density urban municipali-ties. The vast majority of the coefficients are insignificant in accordance with our previous estimates, with the exception of ozone (O3). Ozone (O3) appears to be more concentrated in the more affluent intermediate density urban municipalities. Estimation by new wealth indicators generally supports our results with the exception of ozone (O3).

Table 8 shows the results for the cocktail share in urban municipalities. No coefficients are significant for all wealth measures which confirms our results.

Spatial autocorrelation

In this section I explicitly take into account spatial autocorrelation in our estimates. First, as described in the empirical strategy section, I perform a series of tests on the residuals of our OLS estimates. For each regression performed on each subsample the results are Table 9. For computational reasons I perform the tests on the error term of the national regression taking data only for the year 2018, (the number of municipalities at the national level is very large which requires many computational resources). The test results for each year are similar to the year 2018 (results not shown). For the same reason of computational limitation, I do not perform the tests on the errors of the subsample of sparsely populated rural municipalities (which has the most municipalities among the subsamples).

I begin by performing a Moran test on the error terms of the national estimates (these regressions are weighted by the population of each municipality, see appendix). For all specifications I reject the hypothesis of no spatial autocorrelation of the error terms. The Moran test allows us to formally confirm the presence of spatial autocorrelation, the following tests allow us to identify the spatial model best suited to our specifications. The results of the LM-error and LM-lag tests on the different fixed effects models do not allow us to decide in favor of a particular model, I reject the null hypothesis of both tests. On the other hand, the robust version of these tests leads us to favor the implementation of a spatially autocorrelated error model (SEM). This model consists in estimating a spatial autocorrelation parameter in the error term (see appendix E). I can already conclude from the results of these tests that the majority of our specifications estimated by OLS are not biased because of autocorrelation, but they are less efficient.

By specifying the SEM model I model the spatial correlation of errors as being caused by the spatial correlation of variables affecting pollution levels but which are not correlated with the independent variables in our model. Meteorological variables are a plausible explanation for these spatial relationships in our error term. The results of the SEM model estimates are presented in Table 10. The spatial autocorrelation coefficients ρ are positive and highly significant. The SEM model estimates support our previous results.

Conclusion

The health risk associated with exposure to air pollution depends on the time of exposure but also on the toxicity of the pollutants. In this study, I evaluate the relationship between income, the number of days of exposure to air pollutants and the relationship between income and the composition of these pollution days.

First, I showed that urban areas concentrate the most days of fine particulate matter (PM2.5) and nitrogen dioxide (NO2) pollution. Among the dense urban municipalities, the poorest are significantly overexposed to these two pollutants in terms of number of days, all other things being equal. In urban area, income is not related to the number of PM10 and ozone (O3) pollution days. The share of cocktail days is on average similar between the poorest and the most affluent municipalities in the urban area.

In rural areas, the most affluent municipalities accumulate more PM2.5, PM10 and nitrogen dioxide (NO2) pollution days, all else being equal. However, rural areas are less polluted and the exposure gap between affluent and disadvantaged municipalities is smaller compared to urban areas. Among rural municipalities, the most affluent municipalities have on average a significantly higher share of cocktail days.

I showed in this paper the high concentration of pollutants in urban areas and the important overexposure of the poorest within these areas. Several studies have shown the vulnerability of the poorest to air pollution [START_REF] Deguen | Neighbourhood Characteristics and Long-Term Air Pollution Levels Modify the Association between the Short-Term Nitrogen Dioxide Concentrations and All-Cause Mortality in Paris[END_REF][START_REF] Morelli | Air pollution, health and social deprivation: A fine-scale risk assessment[END_REF][START_REF] Hsiang | The Distribution of Environmental Damages[END_REF][START_REF] Deryugina | Geographic and Socioeconomic Heterogeneity in the Benefits of Reducing Air Pollution in the United States[END_REF] which reinforces the need to prioritize the implementation of public policies to reduce pollution in urban areas and particularly in the poorest urban areas. It is also necessary to take environmental justice into account in public policies aimed at reducing pollution so as not to further reinforce inequalities in exposure. For example, recently in Paris, the closure of roads has created a phenomenon of displacement of road traffic on roads close to the homes of the poorest and therefore increased the pollution exposure of this population [START_REF] Sleiman | Displacing Congestion: Evidence from Paris[END_REF]. Thus, although pollution levels have tended to decrease in France in recent years, public policies aimed at reducing pollution must also take into account the distributional aspect. 1) in each of the sub-samples of urban and rural municipalities. The dependent variable is the number of days in the year for which the health threshold concentration (WHO) of the considered pollutant is exceeded. The dependent variable ALL is the number of days in the year when at least one of the pollutants exceeds its respective threshold. Robust standard errors are clustered at the local département level. * p<0.1; * * p<0.05; * * * p<0.01 Note: Table reports maximum likelihood estimates of the Spatial Error Model (SEM) described in the empirical strategy section and in the appendix section, in urban and rural sub-samples. The dependent variables are the number of days in the year for which the health threshold concentration (WHO) of the considered pollutant is exceeded (columns 1 to 4) and the share of cocktail days (days with at least two pollutants above the health thresholds) among pollution days (at least one pollutant above its health threshold). ρ is the spatial autocorrelation parameter in the error term as described in the spatial aurocorrelation section in the appendix. I use the k nearest neighbors spatial weight matrix, using k=5. We use the Z-score on the income variable for the cocktail share regressions for ease of interpretation of the coefficients. * p<0.1; * * p<0.05; * * * p<0.01

C Estimation at the national level

We estimate a linear regression model with fixed effects at the national level. France has a large number of small municipalities with very small populations, which gives them more weight in the analysis. We weight the regressions at the national level according to the population in each municipality at each period. This weighting allows us to identify a link that is more representative of the situation of the largest number of inhabitants. We cluster the standard errors at the department level.

The results are presented in Table A1. The estimates suggest a significant negative relationship between municipality income and the exposure to PM2.5 and NO2 pollution. Ozone (O3), on the other hand, is concentrated in affluent and rural communes although the effect is not significant. Income is negatively associated with the level of PM10 pollution although the coefficient is not significant.

A one standard deviation increase in income in the municipality is associated with a reduction of 3 days above the thresholds in the year for nitrogen dioxide (NO2) and of 1.2 days for PM2.5 particles on average. The coefficients for PM10 and ozone (O3) are lower. We slightly underestimate the number of days above the threshold for ozone because the daily average ozone concentration is not measured as a rolling 8-hour average as measured by the WHO but as a 24-hour average, which may slightly influence the estimated coefficient for ozone (O3). For ozone (O3), a one standard deviation increase in income is associated with a 0.3 increase in the number of days above the threshold, and a 0.04 decrease in the number of days for PM10.

The coefficients of the dummy years indicate for nitrogen dioxide (NO2), PM2.5 and PM10 a continuous and significant decrease in pollution levels between 2012 and 2018. For ozone (O3) on the other hand, the overall levels of ozone (O3) pollution above the threshold seem to evolve randomly over the period.

Our national results reflect relationships in urban areas. Indeed, since a large proportion of the population lives in urban municipalities, the population weighting of municipalities in the national regressions gives more weight to this type of municipality.

FiguresFigure 1 :Figure 4 :Figure

 14 Figures

  Figure A.3: Evolution of the average number of pollution days in urban and rural areas

Figure A. 4 :

 4 Figure A.4: correlations between municipal density, income and pollution (ALL) in urban area (2018)

Figure A. 5 :

 5 Figure A.5: correlations between municipal density, income and pollution (ALL) in rural area (2018)

Table 1 :

 1 manuel Rivière, Ludivine Launay, Cécile Zaros, Marie Cheminat, Marie-Aline Charles, and Rémy Slama. 2017. "Is atmospheric pollution exposure during pregnancy associated with individual and contextual characteristics? A nationwide study in France." Journal of Epidemiology and Community Health 71 (10):1026. Padilla, Cindy M., Wahida Kihal-Talantikite, Verónica M. Vieira, Philippe Rossello, Geraldine Le Nir, Denis Zmirou-Navier, and Severine Deguen. 2014. "Air quality and social deprivation in four French metropolitan areas-A localized spatio-temporal environmental inequality analysis." Environmental Research 134:315-324. Rosofsky, Anna, Jonathan I. Levy, Antonella Zanobetti, Patricia Janulewicz, and M. Patricia Fabian. 2018. "Temporal trends in air pollution exposure inequality in Massachusetts." Environmental Research 161:76-86. Zwickl, Klara, Michael Ash, and James K. Boyce. 2014. "Regional variation in environmental inequality: Industrial air toxics exposure in U.S. cities." Ecological Economics 107:494-509. Descriptive statistics

	Tables
	Statistic	N	Mean	St. Dev.	Min	Max
	Pollutions
	O3	217,889	4.375	5.324	0	53
	NO2	217,889 26.554	29.645	0	325
	PM25	217,889 77.409	31.645	0	279
	PM10	217,889	6.854	7.019	0	61
	ALL	217,889 85.941	35.314	5	331
	Cocktails
	Pure PM2.5	217,889 52.623	20.848	0	194
	Pure NO2	217,889	5.357	12.308	0	210
	Pure O3	217,889	3.162	4.746	0	50
	NO2 PM2.5	217,889 16.736	16.528	0	167
	NO2 PM2.5 PM10	217,889	4.436	5.472	0	55
	PM2.5 PM10	217,889	2.407	2.959	0	20
	PM2.5 O3	217,889	1.184	1.923	0	23
	Social variables
	Median standard of living	217,681 20,770 3,016.255	9,958	53,500
	poverty rate (60 % threshold)	30,667	13.219	6.417	1.000	54.000
	share of social benefits	36,145	5.137	2.567	0.400	27.800
	first decile	36,145	12,235	1,975	6,153	21,800
	ninth decile	36,145	36,497	8,370	21,863 145,730
	share of income from assets	36,145	10.235	4.685	2.100	74

Table 3 :

 3 Income and number of pollution days in rural and urban areas

	ALL	NO2	PM25	PM10	O3
	(1)	(2)	(3)	(4)	(5)

Note: Table reports OLS estimates of equation (

Table 4 :

 4 Income and share of cocktail days in rural and urban areasNote: Table reports OLS estimates in each of the sub-samples of urban and rural municipalities. The dependent variable is the the share of cocktail days (days with at least two pollutants above the health thresholds) among pollution days (at least one pollutant above its health threshold). I use Z-score on the income variable to facilitate the interpretation of the coefficients. Robust standard errors are clustered at the local departement level.

			Share of cocktail days
		DENSE URBAN URBAN INTERMEDIATE DENSITY RURAL
		(1)	(2)	(3)
	Income	0.002	0.005	0.019 * * *
		(0.002)	(0.005)	(0.003)
	Number of pollution days	0.0001	0.001 * * *	0.001 * * *
		(0.0002)	(0.0002)	(0.0002)
	Département dummies	Yes	Yes	Yes
	year dummies	Yes	Yes	Yes
	controls	Yes	Yes	Yes
	Observations	5,362	23,471	188,484
	R 2	0.742	0.744	0.752

* p<0.1; * * p<0.05; * * * p<0.01

Table 5 :

 5 Income and share of mixture in total pollution days Table reports OLS estimates in each of the sub-samples of urban and rural municipalities. The dependent variable is the the share of the mixture (days when the pollutant(s) in the mixture are above the thresholds but no other pollutant exceeds its threshold) among pollution days (at least one pollutant above its health threshold). I use Z-score on the income variable to facilitate the interpretation of the coefficients. Robust standard errors are clustered at the local departement level.

				Share of mixture in total pollution days
		PM2.5	NO2 PM2.5 PM2.5 PM10 PM2.5 O3 NO2 PM2.5 PM10	NO2	O3
		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	DENSE URBAN
	Income	-0.005	0.004	-0.0005 *	-0.001	-0.001	0.003	-0.00005
		(0.004)	(0.003)	(0.0003)	(0.0005)	(0.001)	(0.004)	(0.001)
	Observations	5,362	5,362	5,362	5,362	5,362	5,362	5,362
	R 2	0.809	0.637	0.524	0.346	0.760	0.850	0.651
	URBAN INTERMEDIATE
	Income	-0.011	0.005	0.0003	0.0001	-0.001	0.006	0.001
		(0.007)	(0.005)	(0.0002)	(0.001)	(0.001)	(0.005)	(0.002)
	Observations	23,471	23,471	23,471	23,471	23,471	23,471	23,471
	R 2	0.687	0.701	0.539	0.380	0.650	0.601	0.648
	RURAL
	Income	-0.031 * * *	0.017 * * *	0.0002	-0.0001	0.002 * * *	0.011 * * *	0.0002
		(0.003)	(0.002)	(0.0003)	(0.0002)	(0.001)	(0.002)	(0.002)
	Observations	188,484	188,484	188,484	188,484	188,484	188,484	188,484
	R 2	0.664	0.706	0.573	0.371	0.658	0.505	0.635
	Département dummies	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	year dummies	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	controls	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Number of pollution days	Yes	Yes	Yes	Yes	Yes	Yes	Yes

Note: * p<0.1; * * p<0.05; * * * p<0.01

Table 6 :

 6 Robustness checks : economic indicators in dense urban area (number of pollution days) : Table reports OLS estimates of equation (1) in the dense urban sub-sample. The dependent variable is the number of days in the year for which the health threshold concentration (WHO) of the considered pollutant is exceeded. The dependent variable ALL is the number of days in the year when at least one of the pollutants exceeds its respective threshold. Robust standard errors are clustered at the local departement level. * p<0.1; * * p<0.05; * * * p<0.01

			DENSE URBAN
		NO2	PM25	PM10	O3	ALL
		(1)	(2)	(3)	(4)	(5)
	poverty rate	1.375 * * *	0.231 *	0.049	-0.020	1.227 * * *
		(0.453)	(0.135)	(0.039)	(0.016)	(0.357)
	Observations	5,129	5,129	5,129	5,129	5,129
	R 2	0.799	0.813	0.821	0.648	0.797
	first decile	-0.006 * * * -0.001 * * *	-0.0001	0.00004	-0.005 * * *
		(0.002)	(0.0003)	(0.0001)	(0.00005)	(0.002)
	Observations	5,267	5,267	5,267	5,267	5,267
	R 2	0.798	0.814	0.821	0.649	0.797
	ninth decile	-0.0003	-0.0001	-0.00002 -0.00000	-0.0003
		(0.0002)	(0.0001)	(0.00002) (0.00002)	(0.0002)
	Observations	5,267	5,267	5,267	5,267	5,267
	R 2	0.794	0.814	0.821	0.649	0.793
	share of social benefits	-1.117	0.172	0.077	0.007	-0.566
		(0.897)	(0.331)	(0.105)	(0.049)	(0.620)
	Observations	5,267	5,267	5,267	5,267	5,267
	R 2	0.794	0.813	0.821	0.649	0.792
	share of income from assets	-0.751 * *	-0.368 * *	0.036	0.022	-0.679 * * *
		(0.348)	(0.163)	(0.107)	(0.021)	(0.191)
	Observations	5,267	5,267	5,267	5,267	5,267
	R 2	0.795	0.814	0.821	0.649	0.793
	Département dummies	Yes	Yes	Yes	Yes	Yes
	year dummies	Yes	Yes	Yes	Yes	Yes
	controls	Yes	Yes	Yes	Yes	Yes

Note

Table 7 :

 7 Robustness checks : economic indicators in urban intermediate density area (number of pollution days) : Table reports OLS estimates of equation (1) in the urban intermediate density sub-sample.The dependent variable is the number of days in the year for which the health threshold concentration (WHO) of the considered pollutant is exceeded. The dependent variable ALL is the number of days in the year when at least one of the pollutants exceeds its respective threshold. Robust standard errors are clustered at the local departement level. * p<0.1; * * p<0.05;

		URBAN INTERMEDIATE DENSITY
		NO2	PM25	PM10	O3	ALL
		(1)	(2)	(3)	(4)	(5)
	poverty rate	0.055	-0.103	0.016	-0.061 * *	-0.072
		(0.212)	(0.145)	(0.036)	(0.029)	(0.174)
	Observations	13,175	13,175	13,175	13,175	13,175
	R 2	0.711	0.813	0.715	0.599	0.808
	first decile	-0.0001	0.0001	-0.0001	0.0001	0.00003
		(0.001)	(0.0004)	(0.0001)	(0.0001)	(0.001)
	Observations	15,490	15,490	15,490	15,490	15,490
	R 2	0.712	0.812	0.715	0.600	0.803
	ninth decile	0.0001	-0.0002 * * -0.0001 * * *	0.00003	0.00001
		(0.0003)	(0.0001)	(0.00002)	(0.00003) (0.0002)
	Observations	15,490	15,490	15,490	15,490	15,490
	R 2	0.712	0.813	0.716	0.600	0.803
	share of social benefits	-0.124	0.777 *	0.160	-0.189 * *	0.115
		(0.596)	(0.400)	(0.119)	(0.090)	(0.512)
	Observations	15,490	15,490	15,490	15,490	15,490
	R 2	0.712	0.813	0.716	0.600	0.803
	share of income from assets	0.151	-0.109	0.002	-0.023 * *	0.035
		(0.096)	(0.072)	(0.031)	(0.012)	(0.077)
	Observations	15,490	15,490	15,490	15,490	15,490
	R 2	0.712	0.812	0.715	0.600	0.804
	Département dummies	Yes	Yes	Yes	Yes	Yes
	year dummies	Yes	Yes	Yes	Yes	Yes
	controls	Yes	Yes	Yes	Yes	Yes

Note* * * p<0.01

Table 8 :

 8 Robustness checks : economic indicators in urban intermediate density area (share of cocktail days)Table reports OLS estimates in each of the sub-samples of dense urban and urban intermediate density municipalities. The dependent variable is the the share of cocktail days (days with at least two pollutants above the health thresholds) among pollution days (at least one pollutant above its health threshold). Robust standard errors are clustered at the local departement level. * p<0.1; * * p<0.05; * * * p<0.01

			Share of cocktail days
		DENSE URBAN URBAN INTERMEDIATE DENSITY
		(1)	(2)
	poverty rate	-0.0004	-0.0001
		(0.001)	(0.001)
	Observations	5,129	13,175
	R 2	0.741	0.754
	first decile	0.00000	0.00000
		(0.00000)	(0.00000)
	Observations	5,267	15,490
	R 2	0.741	0.761
	ninth decile	0.00000	0.00000
		(0.00000)	(0.00000)
	Observations	5,267	15,490
	R 2	0.741	0.761
	share of social benefits	0.001	0.0003
		(0.001)	(0.002)
	Observations	5,267	15,490
	R 2	0.741	0.761
	share of income from assets	-0.0002	0.0001
		(0.001)	(0.001)
	Observations	5,267	15,490
	R 2	0.741	0.761
	Département dummies	Yes	Yes
	year dummies	Yes	Yes
	controls	Yes	Yes
	Number of pollution days	Yes	Yes

Note:

Table 10 :

 10 Robustness checks : spatial regression models (SEM)

	NO2	PM2.5	PM10	O3	Share of cocktail
	(1)	(2)	(3)	(4)	(5)

Table A1 :

 A1 Income and number of pollution days at national level * * -46.864 * * * -13.339 * * * -0.657 * * * -35.897 * * * Note: Robust standard errors are clustered at the local departement level. All regressions are weighted by the population of the municipalities. * p<0.1; * * p<0.05; * * * p<0.01 D Density, income, economic activity and pollution

		NO2	PM25	PM10	O3	ALL
		(1)	(2)	(3)	(4)	(5)
	Income	-0.001 * *	-0.0004 * *	-0.00001	0.0001	-0.001 * * *
		(0.0005)	(0.0001)	(0.00004)	(0.00005)	(0.0004)
	2012	REF	REF	REF	REF	REF
	2013	-4.345 * *	-8.380 * *	-2.440 * *	1.814 * * *	-4.961 *
		(1.903)	(3.318)	(0.979)	(0.338)	(2.874)
	2014	-17.101 (2.273)	(3.542)	(1.313)	(0.242)	(3.419)
	2015	-17.788 * * * -38.229 * * * -13.259 * * *	2.794 * * *	-31.055 * * *
		(1.912)	(2.030)	(1.421)	(0.570)	(1.545)
	2016	-17.754 * * * -51.161 * * * -13.790 * * *	-0.495 *	-43.960 * * *
		(1.883)	(3.332)	(1.207)	(0.294)	(2.783)
	2017	-19.347 * * * -58.698 * * * -13.793 * * *	0.189	-51.386 * * *
		(1.882)	(2.617)	(1.469)	(0.291)	(2.306)
	2018	-34.000 * * * -65.145 * * * -18.219 * * *	7.341 * * *	-50.663 * * *
		(2.217)	(4.810)	(1.410)	(0.656)	(4.144)
	sparsely populated autonomous rural	REF	REF	REF	REF	REF
	very sparsely populated autonomous rural	0.221	-0.398	-0.018	-0.022	-0.480
		(1.443)	(0.509)	(0.145)	(0.121)	(1.170)
	rural under weak influence of a pole	-1.012	0.590	0.174 *	-0.025	-0.319
		(1.035)	(0.468)	(0.097)	(0.103)	(0.871)
	rural under strong influence of a pole	1.687	1.900 * * *	0.529 * * *	-0.239 * *	1.471
		(1.436)	(0.608)	(0.138)	(0.121)	(1.136)
	dense urban	39.723 * * *	10.535 * * *	2.564 * * *	-1.100 * * *	26.686 * * *
		(3.929)	(1.238)	(0.360)	(0.350)	(3.056)
	urban intermediate density	7.459 * * *	3.807 * * *	0.871 * * *	-0.249 *	5.162 * * *
		(1.469)	(0.576)	(0.156)	(0.142)	(1.030)
	Departement dummies	Yes	Yes	Yes	Yes	Yes
	Controls	Yes	Yes	Yes	Yes	Yes
	Observations	217,317	217,317	217,317	217,317	217,317
	R 2	0.907	0.825	0.741	0.588	0.900

*

See[START_REF] Aguilar-Gomez | This is Air: The "Non-Health" Effects of Air Pollution[END_REF] for a literature review. See[START_REF] Anderson | As the Wind Blows: The Effects of Long-Term Exposure to Air Pollution on Mortality[END_REF] and[START_REF] Deryugina | The Mortality and Medical Costs of Air Pollution: Evidence from Changes in Wind Direction[END_REF] for a causal analysis of the effect of air pollution on mortality,[START_REF] Carneiro | The Effects of Air Pollution on Students' Cognitive Performance: Evidence from Brazilian University Entrance Tests[END_REF] and[START_REF] Ebenstein | The Long-Run Economic Consequences of High-Stakes Examinations: Evidence from Transitory Variation in Pollution[END_REF] for a recent analysis of the consequences of pollution on educational outcomes, Graff Zivin and Neidell (2012) and[START_REF] Chang | The Effect of Pollution on Worker Productivity: Evidence from Call Center Workers in China[END_REF] for an analysis of the effects of pollution on productivity.

Beyond the addition of risks induced by exposure to multiple pollutants, the mixture of pollutants can also generate synergistic effects between pollutants and further increase the health risk. This cocktail effect on health remains very poorly documented.

i.e. days on which at least two pollutants are above the health thresholds.

Measured through the standard of living: household disposable income divided by the number of consumption units.

See Banzhaf, Ma, and Timmins (2019b) for a review of the literature on measuring these inequalities and analyzing the mechanisms of their formation.

These studies used geographically fine-grained data at the ZIP code tabulation area (ZCTA) and census block level.

Banzhaf, Ma, and Timmins (2019b) emphasize the importance of using the least aggregated data possible to avoid the ecological fallacy problem.

carbon monoxide (CO), nitrogen dioxide (NO2), fine particulate matter (PM2.5), and sulfur dioxide (SO2).

The estimates account for spatial autocorrelation with a model similar to[START_REF] Champalaune | Inequality in Exposure to Air Pollution in France: Evidence from a City-Level Public Policy[END_REF].

Over the period studied, several municipalities merged with each other. We have therefore retained in the database the municipalities after the merger and excluded the data of the formerly autonomous municipalities. In the pre-merger period, for the new communes resulting from the mergers, we aggregate

See www.airparif.asso.fr, fine particles, origin and sources of pollution, Fine particles -Airparif.

Splm package on R.

Calculated on the sub-sample.

The results are similar in the different types of rural areas (as in the first part), so we consider the rural municipalities as a whole.

The variable is standardized at the national level, which implies that the standard deviation of the variable in the subsamples is not equal to one.

CEE-M Working Papers / Contact : laurent.garnier@inrae.fr  RePEc https://ideas.repec.org/s/hal/wpceem.html  HAL https://halshs.archives-ouvertes.fr/CEE-M-WP/

Note: Each coefficient corresponds to the p-value of the associated test performed on OLS error terms. To perform these tests we use the spatial weighting matrix of the k nearest neighbors, using k=5. Regressions at the national level are weighted by the population of the municipalities.

Appendix

A Composition of each pollutant Figure A.1 shows for each pollutant the average share of each mixture in the total number of days above the threshold for that pollutant. According to these estimates, on 29% of the days on which the PM2.5 threshold is exceeded, the threshold for at least one other pollutant is also exceeded (Figure A.1). Thus, studies assessing the effect of PM2.5 concentration on health actually capture the effect of pollutant cocktails and not the pure effect of PM2.5.

Similarly, studies assessing exposure inequalities do not differentiate between exposure to pure PM2.5 particles and cocktails. In the case of PM10 particles, in 100% of the cases, the days above the threshold for PM10 particles are mixed with other pollutants (the PM10 PM2.5 and NO2 mixture represents 57% of PM10 pollution days on average, the PM10 PM2.5 mixture represents 43% of days on average). The PM2.5 NO2 mixture represents 67% of the nitrogen dioxide (NO2) pollution days on average, while the pollution days for this mixture represent only 19% of the PM2.5 pollution days (Figure A.1). Ozone (O3) is pure in 64% of cases on average. Figure A.2 shows the composition of pollution days in three types of areas. Overall, the composition of pollution days changes between urban and rural areas. In the case of fine particles and ozone, the share of cocktail days is higher in urban than in rural areas.

E Spatial autocorrelation

In order to test the robustness of our results, we estimate a spatial error model to explicitly take into account the spatial autocorrelation problem. Following Bouayad Agha, Le Gallo, and Vedrine (2018) the fixed effects SEM with two dimensional data can be written as :

With y ct , the number of pollution days in the year of municipality i in year t, with c = 1,2,...,N and t = 1,2,...,T, λ c an time fixed effect and ε ct the error term. ρ is the spatial error autocorrelation parameter and N j=1 w cj ε jt represents the spatially displaced error term. w cj is a spatial weighting matrix of dimension (N,N) that describes the spatial relationship between the municipalities, with c and j two different municipalities. The spatial relationships described by the spatial weighting matrix W can take several forms. We retain the k nearest neighbors matrix by retaining k=5. For each municipality, the 5 closest municipalities are identified geographically. The 5 closest municipalities have a value (weight) of 0.2 with the commune considered in the weighting matrix W.

Following [START_REF] Agha | Chapter 7. Spatial econometrics on panel data[END_REF], we extend this model to data in three dimensions:

With y cdt , the number of pollution days in the year of municipality c of département d in year t, with c = 1,2,...,N, d = 1,2,...,M and t = 1,2,...,T. ρ is the spatial parameter to be estimated and M g=1 Ng h=1 w cd,hg ε hgt the spatially displaced error term. w cd,hg are the elements of the spatial weighting matrix of dimension (N,N), with cd representing municipality c in département d and hg a municipality h in département g.

CEE-M Working Papers


www.ijeas.org  Abstract-In this paper, we present the model of programmable resistors based-on memristor component. We also show the interest to use programmable resistor in the case of analog electronic circuits. In the first of paper, we present the memristor. In fact before 1971, the passive electrical components is limited to three: resistor, capacitor and inductor. In 1971, Chua predicts the existence of a fourth element that links the charge to the magnetic flux. After a discussion on the modeling of memristor and critics of the scientific community about the existence of the memristor and its modeling, in the second part, we will present Matlab simulation results of HP lab model. We will also show the interest of using memristor in analog electronic circuits through a programmable resistor model.

Index Terms-Programmable resistors, Memristor, modeling, programmable resistor applications.

I. INTRODUCTION

The memristor, or memory resistor, is a two-terminal passive electronic component. It was introduced in 1971, by Professor Chua, in an article of the IEEE journal Transaction on Circuit Theory [START_REF] Chua | Memristor -Missing Circuit Element[END_REF]. Chua noted that there are four different mathematical relations connecting of the three passive electrical components: resistor, capacitor and inductor. The physical variables allowing the relation between these three components are: current, voltage, electrical load and magnetic flux. These relations are deduced from Faraday's law. Chua compared the above model to Aristole's model of matter.

Thus, depending on the above model he saw a striking resemblance and predicted the existence of the fourth electrical component called memristor, which link the electrical charge to magnetic flux.

II. MEMRISTOR MODELING

A. Discussion on the modeling of memristor Two pairs of equations define the memristor: electrical load control and the magnetic flux control [START_REF] Chua | Memristor -Missing Circuit Element[END_REF].

For electrical load control:

(1)

For magnetic flux control:

(2)

Kebbati, Laboratoire de Physique et Chimie de l'Environnement et de l'Espace LPC2E/CNRS, Université d'Orléans, Orléans, France According to the mathematical relations, the electrical resistance of the memristor is not constant but depends on the current that had previously circulated through the device. In 1976, L. Chua introduced dynamic equations to define a memristive system with the property of "pinched hysteresis loop" between current and voltage [2] [3]. According to Biolek, this footprint of the memristif is "self crossing" which corresponds to a cross with coordinates (0 : 0) [START_REF] Bioleck | Pinched Hysteresis Loops of ideal Memristors, Memcapacitors and Meminductors must be Self Crossing[END_REF].

In 2008, an HP lab manufactures the first memristor component with layers of TiO 2 between two layers of Pt [START_REF] Strukov | The missing memristor found[END_REF]. HP also introduces a mathematical model of component operation. The scientific community in several points has criticized this model:

-The use of the term "Fourth Component" is questioned [START_REF] Meuffels | Fundamental Issues and Problems in the Realization of Memristors[END_REF][7][8] [START_REF] Vongehr | The Missing Memristor has Not been Found[END_REF]. Some experimental evidence contradicts the generalization of Chua, since a non-passive nanobattery effect is observable in the resistance switching memory [START_REF] Valov | Nanobatteries in redox-based resistive switches require extension of memristor theory[END_REF].

-Williams argued that MRAM, phase change memory and RRAM were memristor technologies [START_REF] Mellor | HP and Hynix to produce the memristor goods by 2013[END_REF]. Some researchers have argued that biological structures such as blood [START_REF] Courtland | Memristors...Made of Blood?[END_REF] and skin [START_REF] Johnsen | Memristive model of electro-osmosis in skin[END_REF] [14] fit the definition. Others have argued that the memory device developed by HP Labs and other forms of RRAMs were not memristors but rather part of a larger class of variable resistor systems [START_REF] Clarke | Memristor brouhaha bubbles under[END_REF]. Thus, the broader memristor was an unjustifiable scientific position for patents [START_REF] Marks | Online spat over who joins memristor club[END_REF].

-In 2011 and in 2012, Meuffels indicates the existence of an erroneous hypothesis concerning the ionic conduction [11][17]. This error is the result of shortcomings in the electrochemical modeling presented in the HP Lab article.

-Meuffels and Soni also revealed an important inconsistency: If a current-controlled memristor with the property of non-volatility exists, its behavior would violate the Landauer principle: minimal amount of energy required to change the "information" states of a system [START_REF] Meuffels | Fundamental Issues and Problems in the Realization of Memristors[END_REF] [START_REF] Chua | Resistance Switching Memories are Memristors[END_REF]. Indeed, the storage of nonvolatile information requires the existence of energy barriers. In the opposite case, the system would arbitrarily oscillate from one memory state to another just under the influence of thermal fluctuations. Di Ventra and Pershin finally adopted this criticism in 2013 [START_REF] Ventra | On the physical properties of memristive, memcapacitive and meminductive systems[END_REF].

A. Memristor modeling

As discussed previously, there is no scientific consensus regarding the definition of the memristor as well as its modeling. Thus, in this article as the first approach, we have chosen to use the HP Labs model to define the operation of a memristor. Our objective being twofold: this model will enable us to simulate the variation of the memristor value as a function of the voltage as well as the "pinched hysteresis loop" characteristic between current and voltage. It will also allow us to show the interest of the memristor in analog electronics through its use in a number of typical circuits.

The HP model's is based on the "modulation" of the size of Programmable Résistors : Model and Applications Y. Kebbati www.ijeas.org the doping zone w of the memristor between the value R on and R off as a function of the electrical loads (current) that runs through the component.

(

) 3
With M (q), the instantaneous value of the memristor, w the size of the doped zone, D the total size of the doped zone, μ v the mobility of the electric charge. As discussed before, the footprint of the memristor appear under sinusoidal input. Thus, the "pinched hysteresis loop" between current and voltage is shown in Fig. 2. For this simulation, we choose the electrical load control.

III. PROGRAMMABLE RESISTORS

In this section, we will present two examples of programmable resistor applications in analog electronic circuits.

First, we introduce a model of programmable resistor that is built using a memristor as shown in Fig. 4. This programmable resistor consists of a memristor, two switches (S1 and S2) for programming the desired resistance value on the memristor, a programming voltage generator. Both capacitors (C1, C2) allow DC decoupling of the memristor. The programmable resistor will be connected to the analog circuit between the points A and B. To illustrate the use of the programmable resistor, we will use it in an amplifier circuit and a second-order bandpass filter. www.ijeas.org

The chosen amplifier is an instrumentation amplifier widely used in analog electronics. Fig. 5 shows the circuit and the evolution of the gain as a function of the value of the programmable resistor. The results obtained for both applications (amplifier and filter) show a real interest in using the memristor-based programmable resistors. Indeed, it is easy to obtain different functional characteristics of analog circuits with a programming of the value of the memristor.

IV. CONCLUSION

In this article, we discuss the modeling of memristors and define the programmable resistors model and their interest in analog electronic circuits.

In the first part of the article, we presented the principle of memristors developed by Professor Chua. We also presented the model developed by HP lab. A discussion showed the different points of view and the mistrust of the scientific community on the existence of the memristor as well as on its modeling.

In the second part, we modeled the memristor using Matlab software. We have shown the current-voltage characteristic, which is the fingerprint of a memristor device, in the case of a sinusoidal polarization. After, we developed the circuit of a programmable resistor based-on memristor component.

The interest to use the programmable resistors in analog electronic circuits is show through two examples. Indeed, it is easy to obtain different functional characteristics of analog electronic circuits with a programming of the value of the memristor.

Fig. 1

 1 Fig. 1 The transient response of the memristor value under ramp input source v(t)=tV, for different initial value: 500, 1000 and 1500 Ω Fig.1 illustrates the change of the memristor value versus time under rampe input voltage source, for three initial values 500, 1000, 1500 Ω and R on , R off are fixed to 100, 16000 Ω. Also as see on fig1, the low value of memristor is limited to R on .As discussed before, the footprint of the memristor appear under sinusoidal input. Thus, the "pinched hysteresis loop" between current and voltage is shown in Fig.2. For this simulation, we choose the electrical load control.

Fig. 2 Fig. 3

 23 Fig.2 Pinched hysteresis loop appear between current and voltage for sinusoidal input with frequency f=0.1 Hz

Fig. 4

 4 Fig. 4 Programmable resistor

Fig. 5 Fig. 6

 56 Fig. 5 Schematic of the instrumentation amplifier and evolution the gain versus value of the programmable resistorFig.6shows the evolution of the high and low cut pulsations in the case of a second-order band pass filter.

Fig. 7 Fig. 7

 77 Fig. 7 shows the Bode diagram of the filter when the value of the programmable resistor is set to 1000 Ω


Jean-Loup Hatchikian-Houdot

Constant Time Secure Embedded Systems Through Hardware/Software Cooperation

Keywords: Side-channel attacks, Constant-time secure, ISA, Compilation

Side-channel attacks exploit power consumption, execution time, or any other physical effect caused by an implementation. Our work focuses on timing attacks (side-channel attacks exploiting only execution time). There are already several countermeasures to prevent or limit timing attacks, either on the hardware part or on the software part. However, these security mechanisms are working often separately from each other. The software part only knows the functional behavior of the hardware, but with little knowledge of the micro-architecture details, and the hardware does not know the security expected by the software. The goal of our Ph.D. is to establish a contract between the hardware and the software. This contract would allow cooperation between them to have better control regarding timing security and achieve full resistance against timing attacks at a minimal cost.

I. INTRODUCTION

Timing attacks are especially threatening because they can be done remotely [START_REF] Brumley | Remote timing attacks are practical[END_REF]. A malicious program running on the target system, even with low privileges, can learn critical secrets of a victim program running on the same system. The attacker does not need physical access to the target system, contrary to most attacks exploiting other side-channels like EM-radiations or power consumption, which require physical proximity with the targeted device.

Our Ph.D. started in October 2021 in the context of the SCRATCHS project funded by the Labex CominLabs. SCRATCHS is a collaboration between researchers in the fields of formal methods (Celtique, Inria Rennes), security (Cidre, CentraleSupélec Rennes), and hardware design (Lab-STICC). The project's goal is to co-design a RISC-V processor and a compiler to ensure by construction that a securitysensitive code is immune to timing side-channel attacks while running at maximal speed. This work only considers monocore in-order processors typically used in Internet-of-Things objects.

In the rest of this article, we first present a simple timing vulnerability and define the fundamental concepts. Then we describe some existing protections against these vulnerabilities and their limitations. Finally, we explain how we expect to achieve high-level security regarding timing attacks while limiting the performance costs of the protections.

II. AN EXAMPLE OF TIMING VULNERABILITY

The function of Figure 1 computes modular exponentiation b e mod m. Some implementations of the RSA cryptographic algorithm have been relying on this implementation of the modular exponentiation (like GnuPG [START_REF] Yarom | FLUSH+RELOAD: A high resolution, low noise, l3 cache Side-Channel attack[END_REF]), with the exponent e being the secret key.

However, the exponent is used in the end condition of a while loop and in a branching condition. This means that the execution time will vary depending on the value of the exponent. An attacker able to precisely measure the execution time or to test which operations are executed could retrieve the value of the exponent, thus breaking these RSA implementations. This kind of attack can be done if the attacker runs one of his programs on the same device as this implementation of RSA [START_REF] Schindler | A timing attack against RSA with the chinese remainder theorem[END_REF], [START_REF] Kocher | Cryptanalysis of diffie-hellman, rsa, dss, and other systems using timing attacks (extended abstract)[END_REF]. In this case, the value of the exponent is leaking, because it impacts a behavior observable by attackers.

III. LEAKAGE AND NON-INTERFERENCE PROPERTY

We call leakage any data observable by attackers from sidechannels. In the attacker model we consider, all branching choices and memory access indexes are observable through timing side-channel.

The non-interference property states that the leakage of a program must only depend on its public inputs, but not on the secret ones. This implies that attackers cannot deduct any secret from the leakage. Here, sec 1 and sec 2 are the confidential data. We also assume that the value of pub 1 , pub 2 , leak 1 , leak 2 , and P rog can be observed by anyone, including attackers.

The vulnerable implementation of Figure 1 does not respect this property because two executions with the same public input might have a different timing leakage, depending on the secret value of the exponent.

IV. EXISTING COUNTERMEASURES

There are currently two main approaches to prevent timing attacks: those implemented in hardware [START_REF] Purnal | Systematic analysis of randomization-based protected cache architectures[END_REF], [START_REF] Page | Partitioned cache architecture as a side-channel defence mechanism[END_REF], and those implemented in software [START_REF] Almeida | Verifying Constant-Time implementations[END_REF].

A. Timing Attacks Countermeasures in Hardware

Timing attacks often rely on resource sharing, especially stateful resources like caches. In this case, the potential victim first alters the content of a cache when using it. Then, this content can impact the execution time of other programs, potentially controlled by the attacker, making it a vector for timing attacks [START_REF] Su | Survey of CPU cache-based side-channel attacks: Systematic analysis, security models, and countermeasures[END_REF].

Different hardware protections against timing attacks have been proposed, such as cache randomization [START_REF] Purnal | Systematic analysis of randomization-based protected cache architectures[END_REF] or cache partitioning [START_REF] Page | Partitioned cache architecture as a side-channel defence mechanism[END_REF]. For example, attackers cannot exploit such vulnerabilities if the hardware strongly isolates each task and does not allow them to share any resource.

Unfortunately, such hardware protections have a significant impact on performance. For example, complete partitioning would be too costly since resource sharing is a standard and efficient optimization. A better solution would be to allow sharing for every public data but to make a small requisition of resources when a task is handling a secret (e.g., data used in cryptographic operations).

B. Timing Attacks Countermeasures in Software

On the software side, the state-of-the-art defense against timing attacks is to write programs that are Constant Time Secure (CTS) [START_REF] Almeida | Verifying Constant-Time implementations[END_REF].

Constant time programming achieves the non-interference property. A CTS program never uses a secret value in a branching condition or as a memory index. In practice, when the program would need to branch on a condition depending on a secret, it will instead execute both sides of the branching and then ignore results from the unwanted branch. An attacker observing timing leakage from such a program should not be able to deduce any secrets, as the leakage does not depend on any secret.

Modern compilers like GCC or Clang do not always preserve the CTS of a program during its compilation. Indeed, they implement aggressive optimizations, which could revert the code to a non-CTS state. However, a specialized compiler like Jasmin [START_REF] Almeida | Jasmin: Highassurance and high-speed cryptography[END_REF] or a modified version of CompCert [START_REF] Barthe | Formal verification of a constant-time preserving C compiler[END_REF] preserves the constant time security of a program during its compilation.

Non-interference property is sufficient to prove there is no leakage. However, depending on how leakages are defined, this property can be very restrictive. In the attacker model we use, all branching choices and memory access indexes are in the leakage, i.e., we consider attackers can infer which branch the program takes at every branching point and the value of memory indexes for every memory access.

Unfortunately, this attacker model is weaker than some real attackers. Thus, respecting the non-interference property for our leakage definition is not enough to prevent all timing attacks. The compiled program is executed with microarchitecture instructions defined by the ISA (Instruction Set Architecture). But this ISA only defines the functional behavior of the instructions, not their temporal behaviors. (e.g., some instructions like division take a variable amount of time to execute, which depends on their operands). The microarchitecture used to run software can introduce new timing leakage that is not considered by our leakage model. Thus, real attackers can exploit this vulnerability to deduct secrets, despite constant time security on the software part [START_REF] Aciic ¸mez | Micro-architectural cryptanalysis[END_REF].

C. Hardware / Software Cooperation

Both existing approaches prevent timing attacks, but they both have limitations. Several previous works argue that a contract between hardware and software is required. This contract would restrict the behavior of the hardware such that our current attacker model is sufficient to include every possible timing attack. Thus, software and hardware both respecting this contract could not leak a value unless it is a branching condition or a memory access index).

Heiser et al. [START_REF] Ge | No security without time protection: We need a new hardware-software contract[END_REF], [START_REF] Heiser | For safety's sake: We need a new hardware-software contract![END_REF] demonstrate that current processors are still exposed to timing attacks because of this lack of cooperation, hence the need for this contract to truly achieve constant time security. But they do not propose an implementation of this contract.

Yu et al. [START_REF] Yu | Data oblivious ISA extensions for side channel-resistant and high performance computing[END_REF] propose a Data Oblivious ISA (OISA) which would not only design the intended functional behavior of instructions, but also the security they provide regarding sidechannel leakage. This OISA would be sufficient for a software programmer to make secure programs, without the need to care about how this OISA is implemented in the underlying hardware. They provide and implementation of this OISA for an out-of-order RISC-V processor, but this adds restrictions only on the hardware part.

V. OUR CONTRIBUTION Our Ph.D. will focus on the software part of the solution. We propose to define a secure ISA specification based on the RISC-V instruction set. This contract will define several requirements both on the hardware and software parts.

A. Hardware Requirement: Instruction Operands Safety

The OISA will define which instruction operands are safe regarding timing leakages [START_REF] Yu | Data oblivious ISA extensions for side channel-resistant and high performance computing[END_REF]. A compiler will then be able to use the right instruction depending on the confidentiality of the arguments. For example, consider a division. Depending on the criticality level of the operands, a compiler would emit either a classical division instruction (which leaks the value of its operands) or a leak-free (but potentially slower) division for sensitive input.

The execution time of an operation with safe operands (i.e., operands that do not interfere with the timing behavior of the operation [START_REF] Yu | Data oblivious ISA extensions for side channel-resistant and high performance computing[END_REF]) will probably be the worst-case execution time of its unsafe version. So the compiler will use the unsafe operation whenever this operation is done on public variables.

B. Hardware Requirement: Security Mechanisms for Resources Management

A resource is any component used during the computation. However, our primary concerns are storage components like caches, prefetchers, or buffers. Such stateful components of a computer are common targets of timing attacks. A critical task must have mechanisms to protect its secret data from leaking through timing side-channels. This mechanism could either be via partitioning as previously discussed or via a reset at every context switch.

The compiler will have to choose the appropriate mechanism for each case. The partitioning would have a memory space overhead, and a reset at every context switch would increase execution time.

Other resources will also need protection. Stateful resources are the primary target of known attacks, but stateless resources like transfer or computation resources could also leak data :

The availability of a stateless resource can reflect how and when it is used by other processes [START_REF] Tan | Invisible probe: Timing attacks with pcie congestion side-channel[END_REF]. This can leak data about these other processes if their resources usage depends on variables.

We will investigate potential vulnerabilities in these components during this thesis, and security solutions when needed.

These data protection mechanisms must have a constant execution time (their execution time must not depend on the protected data). Otherwise, this would defeat their purpose.

C. Software Requirements

The contract will also dictate prerequisites that the software needs to satisfy to get a security guarantee. A simple example could be on a memory load. Currently, memory loads expose the memory index to side-channel attacks. As a result, a CTS program must never perform a memory load using a secret value as an index. This memory index can leak because a memory load is not performed in constant time. For example, if the data fetched in memory is already present in a cache, the memory load will be faster. A naive solution could be a constant time load with no optimizations, but this would severely reduce the program's efficiency. Our contract could establish a protocol for safe memory loads: the software must signal what load it intends to do in advance and if it wants it to be constant time.

Of course, this software will have to be written in a CTS way and compiled with a specialized compiler. One of the main objectives of our Ph.D. will be to upgrade CompCert so it can compile code complying with this contract. We will also have to prove that the produced version of CompCert works as intended. We are considering using Coq proof assistant for this last part.

VI. CONCLUSION

Timing attacks, which can be done remotely, are a significant threat to confidentiality. There are already countermeasures for the hardware and software parts, but the lack of cooperation between hardware and software results in costly and imperfect protection.

Our project is to establish a contract between both sides to achieve non-interference property while minimizing the impact on the performances. A program should achieve noninterference, thus be immune to timing attacks if it respects the following requirements. It must be written in a Constant Time Secure way, compiled by a compiler complying with our contract, and executed on a hardware device complying with the same contract.

Our objective is to modify CompCert [START_REF] Barthe | Formal verification of a constant-time preserving C compiler[END_REF] to implement this contract. The first step would be to specify the safety of operands for each instruction of the ISA and to use this specification in CompCert.

Fig. 1 .

 1 Fig. 1. Vulnerable modular exponentiation

Fig. 2 .

 2 Fig. 2. Two executions of the same program with different inputs