Annelie Heuser
email: annelie.heuser@telecom-paristech.fr

Olivier Rioul
email: olivier.rioul@telecom-paristech.fr

Sylvain Guilley
email: sylvain.guilley@telecom-paristech.fr

Jean-Luc Danger
email: jean-luc.danger@telecom-paristech.fr

Information Theoretic Comparison of Side-channel Distinguishers Inter-class Distance, Confusion, and Success

corresponding Kolmogorov-Smirnov analysis (KSA) was studied in [START_REF] Whitnall | A Comprehensive Evaluation of Mutual Information Analysis Using a Fair Evaluation Framework[END_REF][START_REF] Whitnall | An Exploration of the Kolmogorov-Smirnov Test as a Competitor to Mutual Information Analysis[END_REF][START_REF] Zhao | Systematic Construction and Comprehensive Evaluation of Kolmogorov-Smirnov Test based Side-Channel Distinguishers[END_REF].

Although it has been highlighted in [START_REF] Whitnall | An Exploration of the Kolmogorov-Smirnov Test as a Competitor to Mutual Information Analysis[END_REF] that KSA may have disadvantages compared to MIA, a recent study [START_REF] Zhao | Systematic Construction and Comprehensive Evaluation of Kolmogorov-Smirnov Test based Side-Channel Distinguishers[END_REF] has identified variants of KSA that may perform better than MIA in some circumstances.

In [START_REF] Maghrebi | Comparison between Side Channel Analysis Distinguishers[END_REF], the authors suggested an alternative inter-class Kolmogorov-Smirnov analysis (IKSA) that compares the conditional distributions between themselves instead of comparing them with the global distribution of the leakage. This novel approach is shown to be of a different nature (non equivalent), and can outperform KSA in terms of success rate.

Similar ideas have also emerged in the literature: The single-bit DPA [START_REF] Paul | Differential Power Analysis[END_REF] can already be seen as a comparison of (means of) different classes without referring to the marginal distribution. Moreover, in [START_REF] Batina | Differential Cluster Analysis[END_REF] a cluster approach has been introduced that compares the inter-and intra-class variance of conditional classes. Also, in [START_REF] Veyrat-Charvillon | Generic Side-Channel Distinguishers: Improvements and Limitations[END_REF] a copula-based distinguisher has been introduced that compares each conditional distribution internally without referring directly to the global leakage distribution.

It is important to note that in general, a distinguisher's performance also depends on the choice of the leakage model. As pointed out in [START_REF] Whitnall | The myth of generic DPA...and the magic of learning[END_REF] a distinguisher would fail to distinguish if the model consists of a bijective function of the secret and plaintext. Therefore, in this paper, we restrict ourselves to leakage models for which the studied distinguishers are able to distinguish.

Because so many types of side-channel distinguishers have become available, their fair evaluation and comparison is an important topic. One cannot rely on one single experiment carried out on raw leakage measurements from one particular device to draw unequivocal conclusions about the relative efficiency of competing distinguishers (see e.g., the discussion in [START_REF] Standaert | Improving the Rules of the DPA Contest[END_REF]). Therefore, we seek to compare statistical procedures and methodologies in ideal scenarios with clearly defined and fixed leakage models, where in particular the signal-to-noise ratio can be varied as a parameter.

Now, there has been two distinct evaluation frameworks considered in the literature so far:

1. A theoretical framework proposed by Whitnall and Oswald [START_REF] Whitnall | A Comprehensive Evaluation of Mutual Information Analysis Using a Fair Evaluation Framework[END_REF] that uses the exact values of the distinguishers to evaluate the capability to recover the correct key hypothesis. One relevant metric is the so-called relative margin, that computes a normalized distance between the distinguisher's value for the correct key guess to that of its nearest rival. 2. An empirical framework proposed by Standaert et al. [START_REF] Standaert | A Unified Framework for the Analysis of Side-Channel Key Recovery Attacks[END_REF] in which the distinguishers are estimated on empirical data. The performance evaluation can be typically carried out using one of the following two metrics: the success rate, which estimates the probability of ranking the correct hypothesis first, and the guessing entropy, which estimates the average ranking of the correct hypothesis.

It should be emphasized that the theoretical framework is based on the exact computation of the distinguisher to evaluate its intrinsic distinguishing power-as if it was estimated on a infinite number of samples. In contrast, the empirical framework uses simulations or measurements to evaluate the ability of a distinguisher to succeed efficiently: it depends not only on the choice of the theoretical distinguisher, but also on the efficiency of its estimation. Roughly speaking, it can be said that the empirical framework encompasses the theoretical one plus the estimation algorithm.

For this reason, it appears to be more practical. On the other hand, the theoretical framework is more amenable to a mathematical analysis, since it only involves the distinguisher's values. So far, no link between the theoretical and empirical outcomes of a given distinguisher has been shown in the literature.

Our Contributions

Interclass Distinguisher

As a first contribution we introduce a new information-theoretic metric, referred to as inter-class information, that compares conditional probability density functions between themselves. Before applying it to side-channel analysis, we first carry out a detailed mathematical study on the metric itself. In particular, we show that inter-class information (II) shares similar properties with mutual information (MI). Interestingly, both can be computed from the same probability density estimates. But we also prove that the two metrics are not equivalent with a precise definition of the term. Next, we extend the inter-class information to the scenario of side-channel analysis and refer to the corresponding distinguisher as inter-class information analysis (IIA). We continue our mathematical investigation by proving soundness of IIA. Finally, we use the above-mentioned frameworks to investigate the efficiency of both MIA and IIA. From the theoretical framework we select the relative distinguishing margin as the relevant metric. From the empiral framework we select the success rate as the relevant metric. The results from both frameworks agree: IIA is shown to outperform MIA for the theoretical and empirical metric.

Success Metric

Second, we introduce a new metric, called success metric (SM), which evaluates estimated distinguishers while providing more feedback about the efficiency. Therefore, the SM is more suitable when comparing and evaluating distinguishers than the currently state-of-the-art. In fact, SM relies on the estimation parameters of the distinguisher affecting the theoretical success rate. To be precise, the key features of the success metric are:

• Monotony with the success rate (theoretically and empirically); • Quantification of the relationship between the distinguishing value of the correct key and its nearest rival; • Consideration of the noise probability distribution function (e.g., its variance), number of measurements, and estimation method

• Simplicity of the closed-form expressions for additive distinguisher (e.g., DPA, CPA) compared to the success rate; • Ability to derive a closed-form expression for MIA when estimated with histograms, which has not been shown for any other metric before.

Furthermore, we show further benefits of the closed-form expression of SM: We are able to connect the closed-form of the success metric for DPA/ CPA with properties of the sbox in case of a practical signal-to-noise ratio. Remarkably, unlike previous works [START_REF] Guilley | Differential Power Analysis Model and some Results[END_REF][START_REF] Prouff | DPA Attacks and S-Boxes[END_REF] we first not only derive bounds but achieve direct links, and second utilize a success rate/metric instead of only using properties of a distinguisher. However, our new metric, transparency metric, follows the same intuition as the transparency order introduced in [START_REF] Prouff | DPA Attacks and S-Boxes[END_REF], but is more reasonable and simple. Additionally, we are able to answer the question how the size of the keyspace is influencing the success metric and therefore the success rate.

Side-channel Analysis Model

Calligraphic letters (e.g., X ) denote finite sets, capital letters (e.g., X) denote random variables taking values in these sets, and the corresponding lowercase letters (e.g., x) denote their realizations. We write P{X = x} or p(x) for the probability that X = x and p(x|y) = P{X = x Y = y} for conditional probabilities.

Let k * denote the secret cryptographic key, k any possible key hypothesis from the keyspace K , and let T be the input or cipher text of the cryptographic algorithm. The mapping g : (T , K ) → I maps the input or cipher text and a key hypothesis k ∈ K to an internally processed variable in some space I that is assumed to relate to the leakage X. Usually, T , K , I are taken as F n 2 , where n is the number of bits (for AES n = 8).

Generally it is assumed that f is known to the attacker. A common consideration is g(T, k) = Sbox[T ⊕ k] where Sbox is a substitution box. The measured leakage X can then be written as

X = ψ(g(T, k * )) + N, (1)
where N denotes an independent additive noise.ψ is a device-specific deterministic function, which we assume to be known to the attacker in this contribution. For any key guess k ∈ K the attacker computes the sensitive variable

Y (k) = ψ(g(T, k)). (2)
Without loss of generality we may assume that Y is centered and normalized, i.e., E{Y } = 0 and Var{Y } = 1, and that the values in Y are regularly spaced with step ∆ y. For ease of notation, we let Y * = Y (k * ) and Y = Y (k).

A New Distinguisher Based on Intraclass Information

In this section, we introduce a new information-theoretic metric, referred to as inter-class information, that compares conditional probability density functions between themselves. Before applying it to side-channel analysis, we first carry out a detailed mathematical study on the metric itself. In particular, we show that inter-class information (II) shares similar properties with mutual information (MI). Interestingly, both can be computed from the same probability density estimates. But we also prove that the two metrics are not equivalent with a precise definition of the term. Next, we extend the inter-class information to the scenario of side-channel analysis and refer to the corresponding distinguisher as inter-class information analysis (IIA). We continue our mathematical investigation by proving soundness of IIA. Finally, we use the above-mentioned frameworks to investigate the efficiency of both MIA and IIA.

We review some information-theoretic tools to evaluate the dependence between two random variables X and Y, and refer to [START_REF] Cover | Elements of Information Theory[END_REF] for more details. We focus in this section on metrics and postpone the application to side-channel analysis to Sect. 4. However, since for this application one random variable (X) is continuous and the other (Y ) is discrete, we adopt this convention whenever it is possible.

Let p(x) be the probability density function of the continuous random variable X and p(y) = P{Y = y} be the probability mass function of the discrete random variable Y . The corresponding expectations are (3)

E(X) = ∞ -∞ x • p(x) dx and E(Y ) = ∑ y y • p(y), respectively. The variance is defined as σ 2 X = E{(X -E(X)) 2 },

Information Divergence

Definition 1 (Kullback-Leibler divergence [START_REF] Cover | Elements of Information Theory[END_REF]). Let q(x) be another probability distribution defined over the same space as p(x). The Kullback-Leibler divergence of q with respect to p is defined as

D KL [p q] = ∞ -∞ p(x) • log p(x) q(x) dx. ( 4
)
It is well known that D KL [p q] ≥ 0 and equals zero if and only if p(x) and q(x) coincide. The divergence is sometimes termed "distance" in the literature although it is not a distance in the mathematical sense of the word, because it is not symmetric:

D KL [p q] = D KL [q p]
and the triangle inequality is not satisfied in general. To achieve symmetry, Kullback and Hajek made the following definition: Definition 2 (Symmetric Kullback-Leibler divergence). The symmetric divergence between distributions p and q is defined as

δ KL (p q) = D KL [p q] + D KL [q p] 2 (5) = 1 2 ∞ -∞ (p(x) -q(x)) • log p(x) q(x) dx. ( 6
)
2.2 Conditional-to-Unconditional Metric To evaluate the dependence between X and Y , one possibility is to compute the distance between conditional probabilities p(x|y) and the unconditional probability p(x) = E(p(x|Y )) (see Fig. 1a). Using Kullback-Leibler divergence, we obtain

I(X;Y ) = E D KL [p(x|Y ) p(x)] (7) = ∑ y p(y)D KL [p(x|y) p(x)] (8) = ∑ y ∞ -∞ p(x, y) • log p(x|y) p(x) dx. ( 9
)
This is well-known as the mutual information between the two random variables X and Y . Mutual information can also be written as

I(X;Y ) = H(X) -H(X|Y ) (10)
where

H(X) = - ∞ -∞ p(x) • log p(x) dx (11)
is the (differential) entropy of X and

H(X|Y ) = ∑ y p(y) • H(X|Y = y) (12) = -∑ y ∞ -∞ p(x, y) • log p(x|y) dx ( 13
)
is the conditional entropy of X knowing Y . Note that unlike the (discrete) entropy [START_REF] Cover | Elements of Information Theory[END_REF], differential entropy can be negative and hence should not be interpreted as a measure of uncertainty 1 . For more details on the relationship between differential and discrete entropy and the absolute entropy we refer to [START_REF] Cover | Elements of Information Theory[END_REF].

Conditional-to-Conditional Metric

As suggested in [START_REF] Maghrebi | Comparison between Side Channel Analysis Distinguishers[END_REF], instead of referring to the average distribution p(x), a more direct strategy would be to consider all pairwise distances between conditional probabilities p(x|y) (see Fig. 1b). Therefore, we may replace the Kullback-Leibler divergence of p(x|y) with respect to the average distribution p(x) = E(p(x|Y )) by all Kullback-Leibler divergences between conditional probabilities p(X|Y = y) and p(X|Y = y ) for all pairs (y, y ). This yields to the following definition.

Definition 3 (Inter-class information). The inter-class information between random variables X and Y is defined as

II(X;Y ) = 1 2 E D KL [p(x|Y = y) p(x|Y = y )] (14)
where the summation over y = y has disappeared because divergence vanishes for identical distributions.

Proposition 1. The inter-class information can also be written in terms of the symmetric Kullback-Leibler divergence as

II(X;Y ) = E{δ KL (p(x|Y ) p(x))} (16) = 1 2 ∑ y ∞ -∞ (p(x, y) -p(x)p(y)) log p(x, y) p(x)p(y) dx. ( 17
)
Proof. We show equivalence between Eq. ( 15) and Eq. ( 16).

= E δ KL (p(x|Y ) p(x)) (21)
Equation ( 17) then follows easily from definition 1.

Interestingly, Eq. ( 16) is similar to Eq. ( 7) where the divergence (definition 1) is replaced by the symmetric divergence (definition 2). The latter is also sometimes referred to as inter-class divergence (see e.g., [START_REF] Saon | Minimum Bayes Error Feature Selection for Continuous Speech Recognition[END_REF]).

Moreover, similarly as for mutual information, it can be expressed in terms of entropies as shown in the following proposition.

Proposition 2. Let H (X | Y ) = -∑ y ∞ -∞ p(x)p(y) • log p(x|y) dx, (23)
be the conditional cross-entropy of X knowing Y . The inter-class information can be expressed as

II(X;Y ) = H (X | Y ) -H(X|Y ) 2 . ( 24
)
Proof. We show the equivalence between Eq. ( 17) and Eq. ( 24). Since

∑ y p(x, y) -p(x)p(y) = 0, (25)
we can remove p(x) inside the logarithm in [START_REF] Le | Mutual Information Analysis under the View of Higher-Order Statistics[END_REF]. Furthermore, since p(x,y) p(y) = p(x|y), we can write

1 2 ∑ x,y (p(x, y) -p(x)p(y)) log p(x, y) p(x)p(y) = 1 2 ∑ x,y (p(x, y) -p(x)p(y)) log p(x|y) (26) = H (X | Y ) -H(X|Y ) 2 (27)
It is important to notice that cross-entropy is, contrary to Eq. ( 13), averaged over the product distribution p(x)p(y) instead of the joint distribution p(x|y)p(y) = p(x, y).

Theoretical Analysis

Inter-class information has some important properties that are similar to well-known properties of mutual information. These are summarized in the following proposition. Proposition 3. For any two random variables X,Y :

(a) (Symmetry) II(X;Y ) = II(Y ; X) (b) (Independence) II(X;Y ) = 0 if and only if X, Y are independent (c) (Markov Chain Inequality) For any Markov chain X -Y -Z, the following hold: II(X;Y ) ≥ II(X; Z) and II(Y ; Z) ≥ II(X; Z) (d) (Relation to Mutual Information) 2II(X;Y ) = E{D KL [p(x|Y ) p(x)]} + E{D KL [p(x) p(x|Y )]} (28) = I(X;Y ) + E{D KL [p(x) p(x|Y )]} (29)
It follows in particular that II(X;Y ) ≥ 1 2 I(X;Y ).

Proof. The symmetry is obvious from Eq. ( 17). Independency is an obvious consequence of the following well-known property of (symmetric) divergence:

D KL [p q] ≥ 0 and D KL [p q] = 0 if and only if p = q [6]. Markov Chain In- equality: Recall that X → Y → Z forms a Markov chain if p(z|x, y) = p(z|y) for all x; in other words X and Z are independent given Y [6]. Since X → Y → Z is a Markov chain if and only if Z -Y -X is a Markov chain [6]
, it is sufficient to prove the first inequality II(X;Y ) ≥ II(X; Z). Furthermore we already have I(X;Y ) ≥ I(X; Z) from the corresponding property for mutual information. Since the latter is equivalent to the inequality H(X|Y ) ≤ H(X|Z), thanks to Proposition 2, it is sufficient to prove the inequality H (X|Y ) ≥ H (X|Z) for cross-entropies. Now since p(x|y) = p(x|y, z) by the Markov chain condition, it is easily checked that

H (X|Y ) = -∑ y,z p(x)p(y, z) log p(x|y, z) dx = H (X|Y, Z) (30)
which can be rewritten as

H (X|Y, Z) = ∑ z p(x)p(z) ∑ y p(y|z) log 1 p(x|y, z) dx. ( 31
)
By the strict concavity of the logarithm, we have the following inequality

H (X|Y, Z) ≥ ∑ z p(x)p(z) log 1 ∑ y p(y|z)p(x|y, z) dx (32) = ∑ z p(x)p(z) log 1 p(x|z) dx = H (X|Z) (33)
Finally, the relation to mutual information is obvious from the definition.

A Normal Example

In order to illustrate the difference between MI and II, we give the exact expression of I(X;Y ) and II(X;Y ) for two jointly normal random variables2 .

Proposition 4. Let the two random variables X,Y be identically distributed, zeromean and jointly normal, with covariance matrix σ 2 1 ρ ρ 1 , where |ρ| ≤ 1 is the correlation coefficient and σ 2 is the common variance of X and Y . One finds

I(X;Y ) = 1 2 log 1 1 -ρ 2 (34) II(X;Y ) = log e 2 ρ 2 1 -ρ 2 . ( 35
)
Proof. Since X follows the normal density N (0, σ 2 ), its differential entropy is easily computed as [START_REF] Cover | Elements of Information Theory[END_REF] H

(X) = -E{log p(X)} (36)
= log √ 2πσ 2 + (log e)E{X 2 /2σ 2 } (37) = 1 2 log(2πeσ 2 ). (38)
Now for every y, X given Y = y follows the density p(x|y) = p(x,y) p(y) which is easily seen to be the normal N (ρy, σ 2 (1ρ 2 )). It follows that

H(X|Y ) = 1 2 log(2πeσ 2 (1 -ρ 2 )). (39)
Subtracting Eq. ( 39) from Eq. ( 38) yields the announced expression for I(X;Y ) = H(X) -H(X|Y ).

To calculate inter-class information, we use Eq. ( 24). The conditional crossentropy can be similarly computed as

H (X|Y ) = - ∞ -∞ p(y) • E{log p(X|y)} dy (40) = 1 2 log(2πσ 2 (1 -ρ 2 )) + (log e) ∞ -∞ p(y)•E (X -ρy) 2 2σ 2 (1 -ρ 2 ) dy. (41)
Using [START_REF] Veyrat-Charvillon | Mutual Information Analysis: How, When and Why?[END_REF] and expanding E{(Xρy) 2 } = E(X 2 ) + ρ 2 y 2 -0 inside the integral, we obtain

H (X|Y ) = H(X|Y ) - log e 2 + (log e) σ 2 + ρ 2 E{Y 2 } 2σ 2 (1 -ρ 2 ) (42) = H(X|Y ) + (log e) σ 2 + ρ 2 σ 2 2σ 2 (1 -ρ 2 ) - 1 2 (43) = H(X|Y ) + (log e) ρ 2 1 -ρ 2 (44)
Subtracting H(X|Y ) and dividing by 2 yields the desired expression for

II(X;Y ) = 1 2 (H (X|Y ) -H(X|Y )).
The limit case ρ = 0 corresponds to independent random variables X, Y in this example, while ρ = 1 corresponds to total dependency. From proposition 4, both mutual and inter-class informations vanish when ρ = 0 in accordance with proposition 3 (b). However, when ρ → 1 -, II(X;Y ) is increasing to infinity much faster than I(X;Y ). This shows that II(X;Y ) is more sensitive in the dependency of the random variables. We found that this behavior is quite general for many probability distributions including the case of discrete random variables. This gives a first intuition, confirmed in the next section, why II may be more efficient than MI as a side-channel distinguisher.

Non-Equivalence of Mutual and Inter-Class Informations

Since I(X;Y ) and II(X;Y ) share similar properties (see proposition 3 (a)-(c)), and since we aim to compare these two informations as side-channel distinguishers to measure dependency between the measurements and the leakage model, it is important to assert generally whether I(X;Y ) and II(X;Y ) are equivalent or not. Although this does not reflect the ability to distinguish in the context of side-channel analysis, it would give a necessary condition whether II(X;Y ) could be applicable. For this we need a clear definition of equivalent metrics (see e.g., [START_REF] Rudin | Principles of mathematical analysis[END_REF]). Definition 4 (Equivalence). Two distances D(p, q) and D (p, q) are said to be equivalent if there exist finite constants α > 0 and β > 0 such that for any p, q,

D(p, q) ≤ α • D (p, q) and D (p, q) ≤ β • D(p, q). (45)
In particular, whenever one of two distances becomes small, so does the other and mathematically speaking, both "distances" define the same "topology" 3 .

Just to illustrate the usefulness of Definition 4 we provide the following example.

Example 1. Consider the linear correlation coefficient

ρ(X,Y ) = Cov(X,Y ) σ X σ Y ( 46
)
versus mutual information I(X;Y ). Although correlation implies dependence, it is possible that X and Y are linearly uncorrelated while still being dependenttake e.g., Y = X 2 where X ∼ N (0, 1). It follows that an inequality of the form

I(X;Y ) ≤ α • ρ(X,Y ) cannot hold.
Therefore, correlation and mutual information are not equivalent. The same conclusion goes unchanged if linear correlation is replaced by higher-order or nonlinear correlation-take e.g. X ∼ N (0, 1) and Y = ±X where the random sign is uniformly distributed and independent of X. This explains why correlation power analysis (CPA) and MIA are not equivalent.

Regarding IIA vs. MIA, proposition 3 (d) shows the inequality in one direction: I(X;Y ) ≤ 2 • II(X;Y ). However, we have the following.

Because the fraction λ -1 log λ is unbounded as λ → ∞, letting ρ → 1 -shows that no inequality of the form II(X;Y ) ≤ α • I(X;Y ) may hold for some finite constant α > 0.

The fact that mutual and inter-class informations are not equivalent and at the same time require the estimations of the same conditional probability distributions p(x|y) for their computation motivates for a formal comparison study in the context of side-channel analysis. This is investigated in the next section.

4 Side-Channel Analysis Scenario and Soundness

Side-Channel Scenario

There exists some necessary conditions on Y (k) for MIA-and hence IIA-to be able to distinguish. In particular, [START_REF] Prouff | Theoretical and Practical Aspects of Mutual Information Based Side Channel Analysis[END_REF][START_REF] Whitnall | The myth of generic DPA...and the magic of learning[END_REF] show that there should be at least one k ∈ K such that Y (k) is not an injective function of Z. Hence, if for all k, f ( • , k) is injective the attacker has to choose ϕ to be non-injective. In the following, we assume that these necessary conditions are satisfied. As in [START_REF] Prouff | Theoretical and Practical Aspects of Mutual Information Based Side Channel Analysis[END_REF][START_REF] Prouff | Theoretical and practical aspects of mutual informationbased side channel analysis[END_REF] we deduce the following scenario for wrong or correct key assumptions.

Wrong Key Assumption

The conditional distribution p(x|y) of the measured leakage X knowing the predicted leakage Y is given by

p(x|y) = ∑ y * p(y * |y) • p(x|y, y * ) (48) = ∑ y * p(y * |y) • p(x -y * |y) (49) = ∑ y * p(y * |y) • p N (x -y * ), (50)
where p N denotes the noise pdf and Eq. ( 48) follows from the law of total probability. The equivalence between Eq. (49) and Eq. (50) follows from the fact that N is independent of the leakage predictions Y . Thus, as proved in [START_REF] Prouff | Theoretical and Practical Aspects of Mutual Information Based Side Channel Analysis[END_REF], if the key guess is incorrect we have a nontrivial linear mixture of shifted noise densities, whose coefficients depend on the relationship between Y and Y * .

Correct Key Assumption

In contrast, if the key guess is correct, one obtains a Kronecker symbol p(y * |y) = δ y,y * so that the density mixture simplifies to

p(x|y) = p N (x -y * ), (51)
which is simply identically distributed as N + y * .

Soundness Proofs

Recall the following definition.

Definition 5 (Soundness).

A given distinguisher D is said to be sound if the value of the distinguisher for the correct key k * is strictly greater than for all other keys k = k * :

D(k * ) > D(k) (∀k = k * ) (52)
Under this condition, it is an easy consequence of the law of large numbers that the corresponding success rate tends to 1 as the number of measurements increases indefinitely. For mutual information used as a side-channel distinguisher [START_REF] Gierlichs | Mutual information analysis[END_REF]: D(k) = I(X;Y (k)), the soundness condition is expressed by the strict inequality I(X;Y * ) > I(X;Y ) for all k = k * . Proposition 6. Mutual information analysis is sound for arbitrary (not necessarily Gaussian) noise.

Proof. Moradi et al. [START_REF] Moradi | A Comparative Study of Mutual Information Analysis under a Gaussian Assumption[END_REF] proved that I(X;Y * ) ≥ I(X;Y ) which relies on the fact that Y → Y * → X forms a Markov chain [6, Thm 2.8.1]. Their paper [START_REF] Moradi | A Comparative Study of Mutual Information Analysis under a Gaussian Assumption[END_REF] was written (as the title states) "under a Gaussian [noise] assumption" but the argument goes unchanged for non-Gaussian noise; in fact, the Markov chain condition p(x|y, y * ) = p(x|y * ) relies only on the fact that N and Y are independent and not on the Gaussian nature of the noise.

To prove strict inequality, we use the fact that X given Y = y is a nontrivial linear mixture of densities p N (xy * ) of the same entropy as H(N). Since the entropy is strictly concave in the probability density function [6, Thm 2.7.3]4 we have the strict inequality

H(X | Y = y) > ∑ y * p(y * |y)H(N + y * ) = H(N) (53)
× p N (x -y * ) log 1 p N (x -y * ) dx (56) = H (X | Y * ). ( 57
)
Now as in the proof of Proposition 6, we still have H(X|Y ) > H(X|Y * ). Combining the two strict inequalities yields

II(X;Y ) = H (X | Y ) -H(X|Y ) 2 (58) < H (X | Y * ) -H(X|Y * ) 2 = II(X;Y * ), (59)
which is the required soundness statement for IIA.

Why Inter-class Information Analysis is more Discriminating than Mutual Information Analysis

In this section, we theoretically compare MIA and IIA under a Gaussian noise assumption using the scenario and the hypothesis of Sect. 4. We start by a theoretical investigation of I(X;Y * ) and II(X;Y * ), which is then extended with the help of some numerical calculation to I(X;Y ) and II(X;Y ).

Theoretical Comparison of I(X;Y * ) and II(X;Y * )

A key feature of IIA is that inter-class information is no less than mutual information for the correct key guess5 .

Proposition 8. Let X be as in Eq. (1) with Gaussian noise N ∼ N (0, σ 2 ). One has

II(X;Y * ) = log e 2 • σ 2 Y * σ 2 (60) and I(X;Y * ) ≤ II(X;Y * ) . (61)
Proof. To proof Eq. (60) we evaluate II(X;Y * ) using Eq. [START_REF] Prouff | Theoretical and practical aspects of mutual informationbased side channel analysis[END_REF]. Conditional crossentropy can be written as

H (X | Y ) = ∑ y p(y) p(x) log 1 p(x | y) dx. ( 62
)
Plugging the expressions p(x) = ∑ y p(y)p(x|y) and p(x|y) = ∑ y * p(y * |y)p N (xy * ) yields

H (X | Y ) = ∑ y,y p(y)p(y ) ∑ y * p(y * |y )• (63) p N (x -y * ) log 1 ∑ y * p(y * |y)p N (x -y * ) dx. ( 64
)
For k = k * this boils down to

H (X | Y * ) = ∑ y * ,y * p(y * )p(y * ) x p N (x -y * ) log 1 p N (x -y * ) dx ( * ) . (65)
Substituting ξ = xy * in ( * ) and assuming N ∼ N (0, σ 2 ) results in

p N (ξ ) log 1 p N (ξ + y * -y * ) dξ = 1 2 log(2πσ 2 ) + log(e) 2σ 2 E (N + y * -y * ) 2 (66) = 1 2 log(2πσ 2 ) + log(e) 2σ 2 σ 2 + (y * -y * ) 2 (67) = H(N) + log(e) 2σ 2 (y * -y * ) 2 . ( 68
)
So, by letting Y * denote a random variable independent and identically distributed as Y * ,

H (X | Y * ) = H(N) + log(e) 2σ 2 ∑ y * ,y * p(y * )p(y * )(y * -y * ) 2 (69) = H(N) + log(e) 2σ 2 E((Y * -Y * ) 2 ) (70)
where

E((Y * -Y * ) 2 ) = 2E((Y * -E(Y * )) 2 ) (71) = 2σ 2 Y * . (72)
Combining using Eq. ( 24) and that fact that H(X|Y * ) = H(N) for k = k * gives the announced formula:

II(X;Y * ) = H (X | Y * ) -H(N) 2 = log e 2 • σ 2 Y * σ 2 . ( 73
)
To prove Eq. ( 61) we use the fact that the differential entropy is maximum for normal densities [START_REF] Cover | Elements of Information Theory[END_REF]:

H(X) ≤ 1 2 log(2πeσ 2 X ) (74)
Since X given Y * is normal, we obtain

I(X;Y * ) = H(X) -H(X|Y * ) (75) ≤ 1 2 log(2πeσ 2 X ) - 1 2 log(2πeσ 2 X|Y * ) (76) = 1 2 log σ 2 X σ 2 X|Y * (77) = 1 2 log σ 2 Y * + σ 2 σ 2 (78) ≤ log e 2 σ 2 Y * σ 2 = II(X;Y * ) (79)
where we have used the well-known inequality log x ≤ (log e)(x -1).

Distinguishability of I(X;Y ) and II(X;Y )

We now investigate the ability to distinguish between the correct key k * and the incorrect keys k = k * for MIA and for IIA. For this purpose, we use the theoretical metric given by the relative distinguishing margin introduced in the SCA evaluation framework in [START_REF] Whitnall | A Comprehensive Evaluation of Mutual Information Analysis Using a Fair Evaluation Framework[END_REF] and defined by

RelMarg(D) = D(k * ) -max k =k * D(k) Var D(K) . ( 80
)
where K is the random variable uniformy distributed in the keyspace K . The theoretical evaluation for both MIA and IIA involves the determination of the Gaussian density mixture of the leakage X given each possible input Z, with mean value y * and variance σ 2 . That of the conditional densities of p(x|y) follow similarly Fig. 2: Relative distinguishing margin for MIA (black) and IIA (red) for various SNRs for all possible values of y. Given the expressions for p(x) and p(x|y), we are able to compute the required entropies given in Eq. ( 11), Eq. ( 13) and Eq. ( 23) with the help of numerical integration with arbitrary precision. To compute Eq. (80) we have chosen the following practical side-channel scenario:

Y (k) = HW (SBox -1 P [Z ⊕ k * ]) (81) X = Y (k * ) + N, (82)
where SBox -1 P is the inverse substitution box operation in PRESENT (F 4 2 → F 4 2 ), HW is the Hamming weight, and N ∼ N (0, σ 2 ).

Figure 2 displays the relative distinguishing margin for various signal-to-noise ratios (SNR), defined as

SNR = Var(Y * ) Var(N) = 2 σ 2 . ( 83
)
It is clearly observed that RelMarg(IIA) lies essentially above RelMarg(MIA) for high SNR while at smaller SNR the two curves tend to the same asymptote.

Simulation Results

In order to compare the practical and theoretical evaluations, we consider the same leakage scenario as before (Eq. ( 81) and Eq. ( 82)). Again N ∼ N (0, σ 2 ) with σ = {1, 4} in our simulations. Although the assumption of additive white Gaussian noise may not be always realistic, it is common in numerous works in the community.

The maximum distinguisher's value gives the key prediction k * , viz.,

k * = arg max k I(X;Y ) or k * = arg max k II(X;Y ). (84)
To compare the performance of MIA and IIA empirically we used the first-order success rate (SR), which we computed over a set of 230 independent experiments for σ = 1 and 120 experiments for σ = 4, where the secret key is chosen randomly for each experiment. In order to guarantee a fair comparison, we choose the same data set for both MIA and IIA.

We used the kernel density estimation to estimate the required probability densities. The parameters were chosen as recommended in previous publications (see e.g., [START_REF] Veyrat-Charvillon | Mutual Information Analysis: How, When and Why?[END_REF][START_REF] Prouff | Theoretical and practical aspects of mutual informationbased side channel analysis[END_REF][START_REF] Batina | Mutual Information Analysis: a Comprehensive Study[END_REF]). To be specific, the bandwidth was chosen according to normal scale rule [START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF] and we used the normal kernel.

Moreover as suggested in [START_REF] Maghrebi | Comparison between Side Channel Analysis Distinguishers[END_REF], we highlight the standard deviation of the SR by computing error bars. More precisely, since SR follows a binomial distribution for multiple retries R with variance SR(1-SR) R , we obtain confidence intervals

SR - SR(1 -SR) R , SR + SR(1 -SR) R
that are drawn as error bars to provide a fair comparison.

Figures 3a shows the success rate with error bars for σ = 1. One can see that IIA reaches the threshold of the SR of 0.9 before MIA. The success rate for σ = 4 is displayed in Figures 3b, which again highlights the same classification for MIA and IIA. Interestingly, one can see that the difference between MIA and IIA is smaller for low SNR than for high SNR. Thus, the empirical results confirm our theoretical results and mathematical study made in the previous sections.

7 Comparing Side-Channel Distinguishers

Existing Evaluation Metrics

Comparing Empirical Distinguishers

The success rate (SR) is a classical evaluation metric when comparing empirical side-channel distinguishers D m (K). In most publications, SR is derived empirically as defined in Def. 6 (e.g. in [START_REF] Mangard | One for All -All for One: Unifying Standard DPA Attacks[END_REF][START_REF] Maghrebi | Comparison between Side Channel Analysis Distinguishers[END_REF][START_REF] Doget | Univariate side channel attacks and leakage modeling[END_REF]). Moreover, in [START_REF] Standaert | A Unified Framework for the Analysis of Side-Channel Key Recovery Attacks[END_REF] the authors tackled the essential question how to compare two implementations? or how to compare two side-channel adversaries? by presenting an empirical framework including the empirical success rate.

SR( D m ) = 1 r r ∑ i=1 1 k * = ki . (85)
Even if the empirical success rate directly describes the practical outcome of a distinguisher, the given feedback is very limited. In particular, it only outputs the average probability of success without revealing influencing factors or quantifying how close the outcome of the correct key to its rivals is.

Apart from comparing the empirical SR, contributions tackled the questions on determining the theoretical success rate of distinguishers: Definition 7 (Theoretical success rate). The theoretical success rate is defined as

SR( D m ) = P D m (X;Y (k * )) > D m (X;Y (k)) (∀k = k * ) (86) = P ∆ m (k * , k) > 0 (∀k = k * ) . (87)
In [START_REF] Rivain | On the Exact Success Rate of Side Channel Analysis in the Gaussian Model[END_REF] Rivain determined the theoretical 6 SR for CPA and Bayesian attacks. Recently in [START_REF] Fei | A Statistical Model for DPA with Novel Algorithmic Confusion Analysis[END_REF], Fei et al. provided a closed-form expression for the theoretical success rate of DPA. Interestingly, their approach consists in estimating the theoretical success rate depending on the relationship between the correct and incorrect key hypothesis (named as confusion), the number of measurements and the SNR. Following this approach, Thillard et al. [START_REF] Thillard | Success through Confidence: Evaluating the Effectiveness of a Side-Channel Attack[END_REF] extended the idea of confusion coefficients to the general case and reformulated the theoretical success rate of [START_REF] Rivain | On the Exact Success Rate of Side Channel Analysis in the Gaussian Model[END_REF]. Thus, it is possible to determine the success rate without the need of measurements or simulations. Even more, the influencing factors of the success rate as the number of measurements, SNR and the confusion due to the leakage model are determinable. Unfortunately, the computation of the closed-form is not straightforward as mentioned in [START_REF] Rivain | On the Exact Success Rate of Side Channel Analysis in the Gaussian Model[END_REF] and it again gives no quantification of the goodness of the distinguisher. Further, up to now only closed-forms for DPA and CPA exists.

Comparing Theoretical Distinguishers

A different approach to classify the efficiency of side-channel distinguishers has been presented in [START_REF] Whitnall | A Fair Evaluation Framework for Comparing Side-Channel Distinguishers[END_REF]. The authors aim at characterizing the behavior of theoretic distinguishers D(K) instead of D m (K). Thus, the distinguisher is provided with full information about the leakage distribution without the need of estimation. The framework overall consists in six metrics, however, the most common metric is the relative distinguishing margin (RDM) that has been used as a reference in [START_REF] Whitnall | A Comprehensive Evaluation of Mutual Information Analysis Using a Fair Evaluation Framework[END_REF][START_REF] Whitnall | An Exploration of the Kolmogorov-Smirnov Test as a Competitor to Mutual Information Analysis[END_REF] 7 : Definition 8 (Relative distinguishing margin [START_REF] Whitnall | A Fair Evaluation Framework for Comparing Side-Channel Distinguishers[END_REF]). Let D(k * ) be the theoretical distinguishing value of the correct key and D(k) the theoretical distinguishing value of any incorrect key hypotheses, then the relative distinguishing margin RDM is defined as

RDM(D) = D(k * ) -max k =k * D(k) Var(D(K)) = min k =k * D(k * ) -D(k) Var(D(K)) . ( 88
)
The RDM gives a quantified feedback about the margin between the correct key D(k * ) and its nearest rival, unfortunately, no link between the outcome of an empirical and a theoretical distinguisher has been shown so far. Apart from this, the denominator in Eq. ( 88) is highly dependent on the number of key hypothesis used. For example, Var(D(K)) with K = F 8 2 (8-bit key hypothesis) will be smaller than for K = F 4 2 (4-bit key hypothesis) and so RDM will be smaller for smaller key spaces than vice versa, which does not seem intuitive and we prove in Subsect. 8.2 the contrary. Thus, it is not possible to make reasonable comparisons between different cryptographic algorithms or implementations.

A Novel Approach to Compare Distinguishers

As pointed out above, both state-of-the art approaches, the SR and the RDM, have significant drawbacks, which shows the need of a new metric. Our aim is to develop a novel metric that on the one hand coincides with the empirical outcome of distinguishers, like the SR, but on the other hand gives more quantified feedback as the RDM. Our new metric, called success metric, captures the relevant parameters of the theoretical success rate. We provide all necessary approximations from the theoretical success rate to the success metric. In particular, we first define the failure rate as the contrary to the success rate to apply the union bound. Following, we give two different approximations identifying the same relevant influencing factors with different convergence rate and, finally, we utilize a first order approximation to achieve the success metric in Def. 11.

Theoretical Foundation

Complementary to the theoretical success rate (see Def. 7) we define: Definition 9 (Failure rate). The failure rate is defined as

FR( D m ) = 1 -SR( D m ) = P ∃k = k * / ∆ m (k) ≤ 0 . ( 89
)
We first use the union bound (Boole's inequality) to achieve an upper bound of the failure rate:

P ∃k = k * / ∆ m (k) ≤ 0 ≤ ∑ k =k * P ∆ m (k) ≤ 0 . ( 90
)
Next, we give two different approximations that both indicate the same properties but with different convergence rates and pre consumptions. Definition 10. Let X ∼ N (0, 1). The Q-function is defined as

Q(x) = 1 2π ∞ x e -t 2 /2 dt (91) = P(X > x). ( 92
)
Under the assumption of

∆ m (k * , k) ∼ N (∆ (k * , k), EV(k * , k)) we use the Q- function to approximate P ∆ m (k * , k) ≤ 0 , i.e., P ∆ m (k * , k) ≤ 0 (93) = P ∆ m (k * , k) -E( ∆ m (k * , k)) EV(k * , k) ≤ - (∆ (k * , k) + EB(k * , k)) EV(k * , k) (94) = Q ∆ (k * , k) + EB(k * , k) EV(k * , k) , (95) since Q
(x) = 1 -Q(-x). Accordingly, if EB(k * , k) is small with respect to ∆ (k * , k),
we have

P( ∆ m (k * , k) ≤ 0) -→ 0 (96)
exponentially as

∆ (k * , k) + EB(k * , k) EV(k * , k) -→ ∞ (97)
increases for large m. We recall the Chebyshev bound [START_REF] Tchebichef | Des valeurs moyennes[END_REF]: Let ρ > 0, then

P(X > E(X) + ρ) ≤ P(|X -E(X)| > ρ) ≤ Var(X) ρ 2 . (98)
Accordingly, we achieve

FR = P( ∆ m (k * , k) ≤ 0) (99) = P ∆ m (k * , k) ≤ E{ ∆ m (k * , k)} -∆ (k * , k) -EB(k * , k) -ρ (100)
≤ EV(k * , k) (EB(k * , k) + ∆ (k * , k)) 2 . (101)
As ρ -→ 0 the term

EV(k * ,k) (EB(k * ,k)+∆ (k * ,k)) 2 -→ 0 exponentially.
Note that, a similar usage of the Chernov bound [START_REF] Chernoff | A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations[END_REF] allows to prove exponentially convergence. Further, since we achieved exponentially convergence of P( ∆ m (k * , k) ≤ 0) against 0, we use the following first order approximation

∑ k * =k P( ∆ m (k * , k) ≤ 0) ≈ max k =k * P( ∆ m (k * , k) ≤ 0). (102)
Concluding, using the relationship between success and failure rate, we define the success metric as Definition 11 (Success Metric (SM)).

SM(D, D

m ) =min k =k * ∆ (k * , k) + EB(k * , k) EV(k * , k) (103)
=min k =k * E{ ∆ m (k * , k)} Var( ∆ m (k * , k)) . (104)
Interestingly, the success metric includes the minimum distance between the correct key and its nearest-rival as the RDM, however, it is, of course, based on the estimated distinguisher and thus includes the variance of the estimated difference ∆ m (k * , k) in the denominator.

Remark 1. From Appendix 7.2.1 one can see that SR can be approximated from SM. More precisely, As the success rate, the success metric can be derived empirically from simulations/ measurements or theoretically from closed-form expressions. In the next subsection we develop closed-form expressions for additive distinguisher (e.g., DPA, CPA). Even more, in Subsect. 7.4 we derive a closed-form expression of the information theoretic distinguisher MIA for the success metric, which has not be done for any metric so far and cannot be straightforwardly extended to the success rate.

SR . = 1 -exp - 1 2 SM 2 , (105)

Closed-Form Expression for Additive Distinguishers

Definition 13 (Additive distinguisher). We call an estimated distinguisher D m (k) additive if it is unbiased (i.e., EB(k * , k) = 0) and takes the form

D m (k) = 1 m m ∑ i=1 D(X i ,Y i (k)), (107)
where D(X i ,Y i (k)) is a deterministic function of the i.i.d. sequence (X i ,Y i (k)) and, therefore

E{ D m (k)} = D(k). (108)
Remark 2. This definition implicitly assumes that the distribution of Y (k) is identical for all k ∈ K . In other words, knowing the distribution of Y (k) does not give any evidence about the secret (see [START_REF] Heuser | How a symmetry metric assists side-channel evaluation -a novel model verification method for power analysis[END_REF][START_REF] Prouff | Statistical Analysis of Second Order Differential Power Analysis[END_REF] for similar assumptions). Thus, Var{Y (k)} is constant for all k ∈ K . Furthermore, without loss of generality we assume that the sensitive variable Y is normalized such that E{Y (k)} = 0 and Var{Y (k

)} = E{Y (k) 2 } = 1.
Proposition 9. Considering Remark 2 one can simplify both D mDPA [START_REF] Paul | Differential Power Analysis[END_REF] and

D mCPA [3] to 1 m m ∑ i=1 X i Y i (k). (109)
Proof. A proof for D mCPA is given in the following. As formalized in [START_REF] Doget | Univariate side channel attacks and leakage modeling[END_REF] D mDPA and D mCPA can be directly translated into each other. Recall the definition of CPA:

D mCPA (k) = 1 m ∑ m i=1 (X i -X)(Y i (k) -Y (k)) 1 m ∑ n i=1 (X i -X) 2 1 m ∑ m i=1 (Y i (k) -Y (k)) 2 , (110)
where

X = 1 m m ∑ i=1 X i Y (k) = 1 m m ∑ i=1 Y i (k). (111)
Due to Remark 2, (for large m) we have

Y (k) = 0 and 1 m ∑ m i=1 (Y i (k) -Y (k)) 2 = 1.
Straightforward computation yields Proposition 9 for D mCPA (k). For more details on CPA (and side-channel distinguisher) we refer to [START_REF] Standaert | Introduction to Side-Channel Attacks Secure Integrated Circuits and Systems[END_REF][START_REF] Standaert | An Overview of Power Analysis Attacks Against Field Programmable Gate Arrays[END_REF].

To formulate a closed-form expression for the success metric for any additive distinguisher, we extend the idea of confusion similar to [START_REF] Thillard | Success through Confidence: Evaluating the Effectiveness of a Side-Channel Attack[END_REF], which we call general 2-way confusion coefficients.

Definition 14 (General 2-way confusion coefficients). For

k = k * we define κ(k * , k) = E Y (k * ) -Y (k) 2 2 , (112)
κ (k * , k) = E Y (k * ) 2 Y (k * ) -Y (k) 2 2 . ( 113
)
Remark 3. The confusion coefficient introduced in [START_REF] Thillard | Success through Confidence: Evaluating the Effectiveness of a Side-Channel Attack[END_REF] is defined as κ

• (k * , k) = E{Y (k * )Y (k)}
and we obtain the following relationship

κ • (k * , k) = 1 -2κ(k * , k). (114)
Note that, our definition is consistent and a natural extension of the work in [START_REF] Fei | A Statistical Model for DPA with Novel Algorithmic Confusion Analysis[END_REF]. We now precise our side-channel model from Eq. (1) and Eq. ( 2)) in case of additive distinguishers. As these distinguishers are most usually used when the leakage X is linearly depend on Y * , we assume X = αY * + N8 .

Proposition 10 (SM for CPA). Let ε = 2α. The success metric for any additive distinguisher takes the closed-form expression

SM(D, D m ) = min k =k * εκ(k * , k) ε 2 (κ (k * , k) -κ 2 (k * , k)) + 4σ 2 κ(k * , k) √ m. (115)
Proof. We first give the following proposition.

Proposition 11. The first two moments of ∆ m (k * , k) are given by

E{ ∆ m (k * , k)} = 2ακ(k * , k), (116)
Var( ∆ m (k * , k)) = 4[α 2 (κ (k * , k) -κ 2 (k * , k)) + σ 2 κ(k * , k)]. ( 117
) Proof. Recall ∆ m (k * , k) = (αY (k * ) + N)(Y (k * ) -Y (k)).
Since E{Y (k * ) 2 } = 1 (see Remark 2), we obtain

E{Y (k * )(Y (k * ) -Y (k))} = 1 -E{Y (k * )Y (k)} (118) = 2E{ Y (k * ) -Y (k) 2 } (119) = 2κ(k * , k). ( 120
) Because N is independent of Y (k), E{N • (Y (k * ) -Y (k))} = E{N} • E{Y (k * ) -Y (k)} = 0. ( 121
)
Therefore we obtain

E{ ∆ m (k * , k)} = 2ακ(k * , k). (122)
For the variance we obtain

E{ ∆ m (k * , k) 2 } = E{(XY * -XY ) 2 } (123) = 2E{N 2 (Y * -Y ) 2 } + α 2 E{Y * 2 (Y * 2 -Y ) 2 } (124) = 4σ 2 κ(k * , k) + α 2 4κ (k * , k), (125)
since all cross terms with N vanish. Hence, we have

Var( ∆ m (k * , k)) = E{ ∆ m (k * , k) 2 } -E{ ∆ m (k * , k)} 2 (126) = 4[α 2 (κ (k * , k) -κ 2 (k * , k)) + σ 2 κ(k * , k)]. (127)
Plugging Proposition 11 into the success metric given in Eq. ( 103) and considering the normalizing factor of the variance √ m (see Eq. ( 107)) directly derives Proposition 10.

For DPA with one-bit variables Y (k) we can further simplify the success metric such that it can be expressed directly through the SNR, number of measurements and 2-way confusion coefficient κ(k * , k): Proposition 12 (SM for 1-bit DPA). Let ε = 2α, Y a one-bit variable (e.g., Y ∈ {±1}) and D m (k) an additive distinguisher, then

SM(D, D m ) = √ m max k =k * 1-κ(k * ,k) κ(k * ,k) + 1 κ(k * ,k) SNR , (128)
with SNR = Var(signal) Var(noise) = ε 2 σ 2 , since ε = 2α is the difference between X when Y = 1 and Y = -1.

Proof. When Y (k) ∀k ∈ K is a one-bit variable, we achieve the following simplification:

κ(k * , k) = E{ Y (k * ) -Y (k) 2 2 } = E{Y (k * ) 2 Y (k * ) -Y (k) 2 2 } = κ (k * , k). ( 129
)
From this, Proposition 12 follows directly.

Remark 4. Estimating the success rate from confusion coefficients includes a computation of a multivariate normal cumulative distribution function [START_REF] Rao | Linear Statistical Inference and its Applications[END_REF] for which (contrary as stated in [START_REF] Fei | A Statistical Model for DPA with Novel Algorithmic Confusion Analysis[END_REF]) no closed-form expression exists. Moreover, we discovered that the calculated covariance matrices 9 that directly depend on the confusion coefficients are not of full rank. This effect was similarly discovered for CPA by Rivain in [START_REF] Rivain | On the Exact Success Rate of Side Channel Analysis in the Gaussian Model[END_REF], where the author propose to use Monte-Carlo simulation to overcome this problem.

According to Remark 4, we stress that the computation of the success metric as a closed-form expression is more convenient than using the closed-form expression for the success rate for DPA and CPA, since only 2-way confusion coefficients (κ(k * , k), κ (k * , k)) without multivariate distributions are involved.

Additionally, with the help of κ(k * , k) we can give a closed-form expression for RDM (see Eq. ( 88)) for any additive distinguisher: Proposition 13. For additive distinguisher the RDM(D) can be simplified as

RDM(D) = min k =k * κ(k * , k) Var(κ(k * , K)) . ( 130
)
Proof Sketch: As the RDM takes as a input the theoretical value of a distinguisher D, κ(k * , k) directly describes the difference between D(k * ) and D(k) for any k ∈ K. Thus, Prop. 13 directly follows.

The comparison of the closed-form expressions of RDM in Eq. ( 130) and SM in Eq. ( 115) again highlights the different aspects of both metrics.

Closed-form Expression for Mutual Information Analysis

Definition 15. The Mutual Information Analysis distinguisher (MIA) [START_REF] Gierlichs | Mutual information analysis[END_REF] between a continuous variable X and a discrete variable Y is defined by

I(X;Y ) = H(X) -H(X|Y ), (131)
where

H(X) = -∞ -∞ f (x)•log f (x) dx is the (differential) entropy of X and H(X|Y ) = ∑ y p(y) • H(X|Y = y) = -∑ y p(y) ∞ -∞ f (x|y) • log f (x|y) dx is the conditional entropy of X knowing Y .
In practice, I(X;Y ) has to be estimated, while unlike for CPA or DPA the estimation of MIA is a nontrivial problem. For a detailed evaluation of estimation methods of mutual information distinguishers we refer to [START_REF] Veyrat-Charvillon | Mutual Information Analysis: How, When and Why?[END_REF]. In the following, we consider the estimation with histograms in order to formulate a closed-form expression. To estimate MIA with histograms (H-MIA), one has to partition the leakage X into h distinct bins b i of width ∆ x with i = 1, . . . , h. Note again that, Y is already discrete.

For simplification, we consider in the following only the negative conditional entropy -Ĥ(X|Y ) as a distinguisher, since Ĥ(X) does not depend on a key hypothesis. Additionally, we reasonably assume that the distribution of Y is know to the attacker and thus we use p(y) instead of p(y). So, H-MIA simplifies to

Note that, since we estimate the differential entropy the additional term log ∆ x arises, which is eliminated in Eq. ( 132). For more information on differential entropy and mutual information we refer to [START_REF] Cover | Elements of Information Theory[END_REF]. First, we develop a closed-form expression for EB{ ∆ m (k * , k)}: Since Y is discrete the bias only arise due to the discretization of X and the limited number of measurements m. Thus, we utilize the approximations given for the bias of Ĥ(X) in [START_REF] Moddemeijer | On estimation of entropy and mutual information of continuous distributions[END_REF] (3.14) to calculate E{ D m (k)} and E{ ∆ m (k * , k)} for H-MIA. To be specific, let h define the number of bins and ∆ x their width, then

E{ D m (k)} = -E{ Ĥ(X|Y )} = -∑ y p(y)E{ Ĥ(X|Y = y)}, (134)
≈ -∑ y p(y) H(X|Y = y) + ∆ x 2 24 J(X|Y = y) - h -1 2m , ( 135
) E{ ∆ m (k * , k)} ≈ ∑ y p(y) H(X|Y = y) + ∆ x 2 24 J(X|Y = y) -∑ y * p(y * ) H(X|Y * = y * ) + ∆ x 2 24 J(X|Y * = y * ) , (136)
with J(X|Y ) = ∑ y p(y)J(X|Y = y) and J(X|Y = y) being the Fisher information

∞ -∞ [ d dx p(x|y)] 2
p(x|y) dx [START_REF] Fisher | Statistical Methods for Research Workers[END_REF]. Next, to calculate Var{ D m (k)} we use the law of total variance [START_REF] Kardaun | Classical Methods of Statistics[END_REF] (Eq. (137) ⇔ Eq. ( 138)) and the approximations for the variance given in [START_REF] Moddemeijer | On estimation of entropy and mutual information of continuous distributions[END_REF] (4.9) for Eq. (138) ⇒ Eq. (139) and Eq. (140) ⇒ Eq. ( 141):

Var{ D m (k)} = Var{ Ĥ(X|Y )}} = Var{E{ Ĥ(X|Y = y)}} (137) = Var{ Ĥ(X)} -E{Var{ Ĥ(X|Y = y)}} (138) ≈ Var{H(X)} - 1 m ∑ y p(y) Var{-log f (x|y)} (139) Var{ ∆ m (k * , k)} = Var{E{ Ĥ(X|Y = y}} -Var{E{ Ĥ(X|Y * = y * }} (

SM(D, D

m ) min k * =k ∆ (k * , k) + ∆ x 2 24 J(X|Y ) -J(X|Y * ) √ m ∑ y p(y) Var{-log f (x|y)} + ∑ y * p(y * ) Var{-log f (x|y * )} , ( 143
) with ∆ (k * , k) = H(X|Y ) -H(X|Y * ), J(X|Y ) = ∑ y p(y)J(X|Y = y) while J(X|Y = y) is the Fisher information ∞ -∞ [ d dx f (x|y)] 2 f (x|y) dx [9].
Interestingly, the SM of MIA involves the number of traces as the √ m in the nominator like DPA and CPA, which seems reasonable. Remark 6. Remarkably, the variance is approximately independent of the size of ∆ x. Only in extreme cases like ∆ x = 1 and ∆ x → ∞ is affecting the variance. Also see [START_REF] Moddemeijer | On estimation of entropy and mutual information of continuous distributions[END_REF] for more information. Interestingly, all linear terms have disappeared in the expression of the SM.

8 Features of SM Expressions

Linking the Success to Properties of the Sbox

All previous studies about the relationship between the sbox properties and sidechannel analysis considered the direct link between a metric on a distinguisher itself and the sbox. In [START_REF] Guilley | Differential Power Analysis Model and some Results[END_REF], Guilley et al. use as a metric the maximal value of the distinguisher divided by its standard deviation (SNR). The authors demonstrate that for DPA the SNR is lower bounded by quantities that are expected to be large for sboxes resisting against linear differential cryptanalyses. Prouff introduces in [START_REF] Prouff | DPA Attacks and S-Boxes[END_REF], an alternative metric for CPA, called the transparency order, that is defined as the difference between the maximal value of CPA and the average of all rivals. Besides, the power model is not the Hamming weight, but the Hamming distance; however, strangely enough, the sensitive variable is not the Hamming distance, but instead the average of the initial state exclusive-ored with all possible final states. This leakage model is, to our best knowledge, rather unusual in practice. In both previous works the relationship is only stated as an expected outcome but not proven. The results have been further investigated by Carlet in [START_REF] Carlet | On Highly Nonlinear S-Boxes and Their Inability to Thwart DPA Attacks[END_REF].

In the following, we not only bound but directly link the success metric and the sbox in case of low SNR (practical conditions). As DPA is a special case of CPA, we further concentrate on the closed-form expression of CPA and simplify Eq.

From Eq. (112), κ(k * , k * ) = 0 and κ(k * , k) ≥ 0, thus the argument of the square root in Eq. ( 145) is always positive. Besides, by the Cauchy-Schwarz theorem, we also have that κ(k * , k) ≤ 1. Now, the objective to minimizing min k =k * κ(k * , k) (i.e., making side-channel attacks as hard as possible) is tantamount to maximizing max k =k * E(Y (k * )Y (k)). In the following, we assume that Y * and Y explicitly depend on an sbox (or inverse sbox) and a Hamming weight (w H ) leakage model 10

As ∀a ∈ {0, 1}, (-1) a = 1 -2a, the goal to make CPA difficult is to minimize the following quantity, that we call the transparency metric

min k =k * n ∑ i, j=0 ∑ t∈F n 2 S i (t ⊕ k * ) ⊕ S j (t ⊕ k). ( 147
)
Remark 7. Note that, for single-bit attacks (n = 1), the criteria of Eq. ( 147) simplifies to the one-sided criteria discovered in [START_REF] Heuser | A Theoretical Study of Kolmogorov-Smirnov Distinguishers: Side-Channel Analysis vs. Differential Cryptanalysis[END_REF].

So, minimizing the objective on the sbox in Eq. ( 147) is equivalent to minimizing min k =k * κ(k * , k), which can be understood intuitively on the illustration of Fig. 3. The key corresponding to the nearest rival, i.e., argmin k =k * κ(k * , k), shall have a confusion coefficient as high as possible.

To further illustrate the transparency metric and show the relationship to the transparency order [START_REF] Prouff | DPA Attacks and S-Boxes[END_REF], we use the same three sboxes as in [START_REF] Heuser | A Theoretical Study of Kolmogorov-Smirnov Distinguishers: Side-Channel Analysis vs. Differential Cryptanalysis[END_REF]: Let ⊕ and be respectively the inner addition and multiplication of the Galois field F 2 8 of 256 elements, then the sboxes are given by CPA, whereas S 254 has the highest min k =k * κ(k * , k) being the most vulnerable. Tab. 1 displays the transparency metric and order. The transparency metric is different from the transparency order, nonetheless, it remains consistent with it, meaning that the order of S 1 , S 101 and S 254 is the same for both metrics and consistent with the rating through κ(k * , k). Hardware devices are known to leak approximately in Hamming distance. This makes leakage models complicated, because they involve two consecutive states of the cipher. Let us consider the example of an AES-128 computed one round per clock period. The plaintext is P, the cipher C, and the first (resp. last) round key K 1 (resp. K 11 ). On the one hand, the uncentered and non-normalized leakage model at the first round for the byte at position 0 is:

Y 1 (T, K 1 ) = w H (T 0 ⊕ 02 • S(T 0 ⊕ K 1 0 ) ⊕ 01 • S(T 5 ⊕ K 1 5 ) (148)
⊕ 01 • S(T 10 ⊕ K 1 10 ) ⊕ 03 • S(T 15 ⊕ K 1 15 )) , (149)
where 01, 02 and 03 are the MixColumns constants, and S is the SubBytes operation. Clearly, a guess for this model requires an hypothesis on 4 bytes of the key K 1 .

On the other hand, the uncentered and non-normalized leakage model at the last round for the byte at position 0 is:

Y 10 (C, K 10 ) = w H C 0 ⊕ S -1 (C 0 ⊕ K 10 0 ) , (150)
where S -1 is the InvSubBytes operation. So, a guess for the model requires simply one hypothesis on a key byte (namely K 10 0 ). This is due to the absence of MixColumns at the last round.

The transparency order (resp. metric) of InvSubBytes is 7.85 (resp. 7964), meaning that it is very close to that of SubBytes. So, the confusion coefficient associated to Y 1 and to Y 10 have similar distributions, meaning that the data complexity (the number of traces m) of the attack is similar at either end of the AES. Specifically, the minimal nonzero confusion coefficient for Y 1 is 0.468750, whereas it is 0.404297 for Y 10 . The most crucial difference is the computational complexity, owing to the largest key space to explore at the first round.

  and similarly for Y . Let p(x|y) = p(x|Y = y) be the conditional probability distribution of X knowing that Y = y and p(x, y) be the joint probability distribution of X and Y . Notice that the marginal distribution p(x) becomes the average over Y of the conditional distribution p(x|y): p(x) = ∑ y p(x, y) = ∑ y p(y)p(x|y) = E(p(x|Y )).

( a )

 a Conditional vs Unconditional. (b) Conditional vs Conditional.

Fig. 1 :

 1 Fig. 1: Illustration of comparing probability distributions (the "distance" is depicted with an arrow).

  )p(y )D KL [p(x|y) p(x|y )]

Proposition 5 .

 5 Mutual information I(X;Y ) and inter-class information II(X;Y ) are not equivalent. Proof. It is sufficient to give the following counterexample. Consider X,Y as in § 3.1. Letting λ = 1 1-ρ 2 we have 2I(X;Y ) = log λ and 2II(X;Y ) = (λ -1) log e.

  for all y. Taking expectations over Y yields H(X|Y ) > H(N) = H(X|Y * ), that is, I(X;Y * ) > I(X;Y ). For inter-class information used as a side-channel distinguisher: D(k) = II(X;Y (k)), soundness is similarly expressed by the strict inequality II(X;Y * ) > II(X;Y ) for all k = k * . Proposition 7. IIA is sound for arbitrary noise. Proof. Let k = k * . By strict concavity of the logarithm (or by strict convexity of function x → log(1/x)): H (X | Y ) = ∑ y,y p(y)p(y ) ∑ y * p(y * |y ) × p N (xy * ) log 1 ∑ y * p(y * |y)p N (xy * ) )p(y ) ∑ y * ,y * p(y * |y )p(y * |y) × p N (xy * ) log 1 p N (xy * ) dx (55) = ∑ y * ,y * p(y * )p(y * )

  (a) Success rate for MIA (red) and IIA (black) with error bars using σ = 1. (b) Success rate for MIA (red) and IIA (black) with error bars using σ = 4.

Definition 6 (

 6 Empirical success rate). Let k = arg max k D m (K) denote the key guess maximizing the experimental distinguisher D m (K) for one experiment and let k = [ k1 , . . . , kr ] define a vector of key guesses of r independent experiments. Then the empirical success rate is defined as

Definition 12 .

 12 so SM is the first order exponent of SR regarding the following definition of equivalence [6, page 63, Eqn. (3.76)]: The notation a m. implies that a m and b m are equal to the first order in the exponent.

Definition 16 .

 16 Let p(x) = #b i m with x falling into bin b i and let p(x|y) be the estimated probability knowing Y = y, then Îm (X;Y ) = -∑ x p(x) log p(x) + ∑ y p(y) ∑ x p(x|y) log p(x|y).

H

  -MIA(X,Y ) = ∑ y p(y) ∑ x p(x|y) log p(x|y) + log ∆ x.

  140) -2 Cov(E{ Ĥ(X|Y = y}}, E{ Ĥ(X|Y * = y * }}) ≈ 1 m ∑ y p(y) Var{-log f (x|y)} + 1 m ∑ y p(y * ) Var{-log f (x|y * )} (141) -2 Cov(E{ Ĥ(X|Y = y}}, E{ Ĥ(X|Y * = y * }}) Var{-log f (x|y)} + ∑ y p(y * ) Var{-log f (x|y * )} (142) Using the closed-form expressions for EB{ ∆ m (k * , k)} and EV{ ∆ m (k * , k)} we formulate the following proposition. Proposition 14 (SM for H-MIA).

Remark 5 .

 5 If N is normal distributed with variance σ 2 we can further simplify H(X|Y * = y * ) = 1 2 log(2πeσ 2 ) since p(x|y * ) = p N (xy * ). Moreover, J(X|Y * = y) = 1 σ 2 and Var{-log f (x|y * )} = 1 2m .

  (115) when σ α. More precisely, SM(D, D m ) ≈ mink =k * 4α 2 κ 2 (k * , k)m σ 2 4κ(k * , k) * , k).

1 √nFig. 3 : 2 (- 1 )

 1321 Fig. 3: Illustration of the confusion coefficients for CPA

  1. A "bad" Sbox[•], termed S 1 , of equation y → a y ⊕ b, 2. An "average" Sbox[•], termed S 101 , of equation y → a y 101 ⊕ b, 3. A "good" Sbox[•], termed S 254 and used in AES, of equation y → a y 254 ⊕ b.

Fig. 4 Fig. 4 :

 44 Fig. 4 displays the confusion coefficient for S 1 , S 101 and S 254 . One can see, that the minimal min k =k * κ(k * , k) is achieved by S 1 , which is the hardest to attack with

Table 1 :

 1 Comparison of side-channel metrics for sboxes How does the Size of the Key Space Influence the SM/SR?

		Transparency Order [22] Transparency Metric (Eqn. (147))
	S 1	5.84	7424
	S 101	7.50	7936
	S 254	7.86	8000
	8.2

Another reason is that differential entropy is not "coordinate-free" -it depends on the underlying coordinate system.

Note that, unlike in our previous definitions, the random variable Y is also continuous in this example. Thus sums have to be replaced by integrals.

Note that this equivalence of metrics is not the same as the equivalence between distinguishers stated in[START_REF] Doget | Univariate side channel attacks and leakage modeling[END_REF].

A well-known information-theoretic property commonly referred to as "mixing increases entropy".

Interestingly, it is not true that II(X;Y ) ≥ I(X;Y ) for general random variables X and Y . For example, we can find a counterexample when X,Y are binary variables with small p(x|y) for all x, y = 0.

In[START_REF] Rivain | On the Exact Success Rate of Side Channel Analysis in the Gaussian Model[END_REF] the term exact instead of theoretical is used.

Note that, in some publications, the relative distinguishing margin is also called nearest-rival distinguishing score.

Note that, a similar model was also implicitly used in[START_REF] Fei | A Statistical Model for DPA with Novel Algorithmic Confusion Analysis[END_REF][START_REF] Thillard | Success through Confidence: Evaluating the Effectiveness of a Side-Channel Attack[END_REF].

Namely [κ(k * , i, j)] (i, j)∈K \{0} and [κ(k * , i) × κ(k * , j)] (i, j)∈K \{0} .

Acknowledgement

Annelie Heuser is partly founded by the Google Doctoral European Fellowship in the field of privacy.


Florence Epron
email: florence.epron.cognet@univ-poitiers.fr

Florence Gauthard

Carole Pinéda

Jacques Barbier

Catalytic Reduction of Nitrate in

Keywords: nitrate, nitrite, Pt-Cu catalysts, catalytic reduction

published or not. The documents may come

Introduction

Intensive agricultural activities and specially over-fertilization induce an increase of nitrate concentration in groundwater. Nitrates consumed with drinking water can be converted into nitrites in human body and may cause methaemoglobinaemia in new-borns, as well as other sides effects like hypertension. European Community has set the maximum permitted level of nitrate in drinking water to 50 mg.l -1 . To comply with the legislation, a specific treatment of drinking water is necessary. Nowadays, two types of processes are developed :

(i) physico-chemical processes using ion exchange, electrodialysis or reversed osmosis and (ii) biological processes, consisting in nitrate reduction by microorganisms into gaseous nitrogen (1,2).

The electrolytic reduction of nitrate anions has been under investigation for a long time [START_REF] Plieth | Encyclopedia of electrochemistry of the Elements[END_REF][START_REF] Kinza | [END_REF](5)(6)(7)(8)(9) and constitutes an important part of the electrochemistry of nitrogen compounds.

Chemical reduction of nitrate in water has also been studied (10,11) in presence of ferrous ions and a copper catalyst, or in presence of metallic iron as reducing agent. More recently, catalytic denitratation has been developed [START_REF] Tacke | Dechema Biotechnology Conferences[END_REF][START_REF] Vorlop | [END_REF](14)(15)(16)(17)(18)(19)(20)(21)(22)(23)[START_REF] Prüsse | Preparation of catalysts[END_REF][START_REF] Hählein | Preparation of catalysts[END_REF][START_REF] Daum | [END_REF] to obtain the selective reduction of nitrate into nitrogen, with the undesired formation of ammonia.

It was demonstrated [START_REF] Tacke | Dechema Biotechnology Conferences[END_REF][START_REF] Vorlop | [END_REF](14)(15)(16)) that nitrites are reduced by hydrogen either to nitrogen or to ammonia on various hydrogenation catalysts like Pd/C, Pd/Al2O3, Pt/C, Ru/Al2O3, Ir/Al2O3 or Rh/Al2O3. Among these five catalysts, those based on palladium show the highest activity and selectivity. On alumina support, the activity ranking is 5% Pd>5% Rh>2% Ir> 2% Ru and the selectivity sequence is 5% Pd>2% Ru>2% Ir = 5% Rh (15). From these results, palladium based catalysts were used in all subsequent studies reported in the literature. Nevertheless, the activity and selectivity of alumina supported platinum based catalysts was no longer studied.

To reduce nitrates, it is necessary to activate the precious metal based catalysts by addition of a second metal as copper or silver. In the early studies, the best activity and selectivity obtained with palladium based bimetallic catalysts were found with palladiumcopper catalysts [START_REF] Tacke | Dechema Biotechnology Conferences[END_REF][START_REF] Vorlop | [END_REF](14)(15)(16). The catalytic liquid-phase hydrogenation of aqueous nitrate solution on these bimetallic Pd-Cu catalysts was the subject of kinetical studies (17,19). More recently, other promoting metals like Sn and In (22,23,[START_REF] Hählein | Preparation of catalysts[END_REF], Zn [START_REF] Hählein | Preparation of catalysts[END_REF][START_REF] Daum | [END_REF] were investigated. In terms of activity, palladium-tin and palladium-indium catalysts seem to be more suited for nitrate reduction than their palladium-copper equivalents minimizing at the same time the formation of ammonia.

With the aim of gaining a deeper understanding of the mechanism of nitrate reduction on bimetallic platinum based catalysts, we have studied in this work the behavior of the monometallic platinum and copper catalysts. Palladium-based catalysts are not active for the liquid phase nitrate reduction (14) and this was verified in the case of monometallic platinum catalysts. Then, we tested the activity of platinum catalysts for the liquid-phase nitrite hydrogenation, whereas the activity of monometallic and bimetallic copper based catalysts were evaluated for both nitrate and nitrite reduction. To determine the role of the copperplatinum interaction in the reaction, two different preparation techniques were chosen to favor the deposition of copper on a parent platinum catalyst, either on the platinum particles or on the support. The influence of Pt/Cu atomic ratio was also studied.

Experimental

Catalysts

Support

The support was an -alumina (Procatalyse) with a specific surface area (BET method) of 215 m 2 /g, a total pore volume of 0.55 cm 3 /g and an isoelectric point at pH7. The support was crushed and sieved in order to retain particles with sizes between 0.04 and 0.08 mm.

Catalysts

Monometallic platinum and copper catalysts were prepared by impregnation of the support with Pt(NH3)4(OH)2 or Cu(NO3)2, 3 H2O in an aqueous solution brought to pH11 by addition of ammonia. This pH value was chosen to ensure the cationic exchange between the precursor salt and the alumina surface (27,28), leading therefore to a well-dispersed catalyst after activation. The mixture was stirred for 12 hours and then ammonia and water were evaporated by heating on a sandbath. After drying overnight at 120°C, catalysts were calcined in flowing air for 4 h, at 450°C for platinum and 300°C for copper catalysts (to avoid copper aluminate formation). Then the oxidized platinum catalyst was reduced in flowing pure hydrogen at 500°C for 8 h. For copper catalyst, this reduction step was carried out at 320°C, in situ, i.e. in the same reactor than the one used for the nitrate or nitrite reduction test. This allowed to avoid any contact with air, thereby maintaining copper species into their metallic form.

To prepare Pt-Cu bimetallic catalysts, two techniques were chosen to deposit copper onto the monometallic 'parent' platinum catalyst either on platinum particles or on the support :

-Catalytic reduction : this technique, widely used in our laboratory (29)(30)(31)(32), was developed in order to favor the deposition of the second metal on the prereduced parent metal. A platinum monometallic catalyst was modified by the surface reaction between chemisorbed hydrogen and the second metal, according to the overall reaction :

Cu 2+ + 2 Pt-H Pt2Cu + 2 H + [1]
A given amount of the prereduced parent platinum catalyst was introduced in an aqueous solution previously degassed by bubbling nitrogen and continuously stirred by a counter-current hydrogen flow (1 h). The solution of the copper salt (Cu(NO3)2, 3 H2O), containing a well-defined amount of copper, was then added and maintained in contact with the catalyst under hydrogen flow for 2 hours. We verified, by atomic absorption spectroscopy (Perkin Elmer 3300), that copper was totally absent in solution after the deposition reaction.

Bimetallic catalysts were then filtered, washed with water and dried at 120°C overnight. In the following, this type of catalysts is referred to as 'CR catalysts'.

-Successive impregnation by cationic exchange : this technique was used to favor the copper deposition on the support and therefore to minimize interactions between platinum and copper. The experimental procedure was the same as the one described above for the monometallic copper catalyst. This type of catalyst was called 'SI catalysts'.

Both types of bimetallic catalysts were activated by direct reduction at 380°C under flowing pure hydrogen for 2h.

The monometallic and bimetallic catalysts contained 1 wt.% of metal (platinum and/or copper on alumina).

Characterization methods

Determination of platinum dispersion

Platinum accessibility was determined by O2/H2 titration (33,34). After reduction in H2 at 500°C, the catalyst was flushed for 1 h 30 with helium at the same temperature to remove physically adsorbed hydrogen, and then cooled to room temperature. Pulses of oxygen were then introduced until full saturation of the catalyst. The chimisorbed oxygen was titrated by hydrogen. The amount of hydrogen consumed in this way corresponds to the total hydrogen (reversibly and unreversibly bound) HT1. The catalyst was then degassed for 20 min at room temperature, and a second hydrogen chemisorption was carried out corresponding to the reversibly bound and/or physisorbed hydrogen HT2. The amount of adsorbed hydrogen on the metal surface (HT) is simply the difference between HT1 and HT2.

The gases used in this study were of ultra high purity grade.

This method was effective to determine the platinum accessibility in the monometallic catalyst, but unsuitable for the bimetallic catalysts.

Electron microscopy

Electron microscopy was mostly used to characterize bimetallic catalysts.

Transmission electron microscopy (TEM) in a bright field mode was carried out in a Philips CM120 electron microscope operating at 120 kV with a theoretical resolution of 0.35 nm. The sample was embedded in a polymeric resin (spurr) and cut into a section as small as 40 nm with an ultramicrotome equipped with a diamond knife. Cuts were then deposited on an Al grid previously covered with a thin layer of carbon. EDX analysis of elements was carried out in the STEM mode of the microscope using a Si (Li) Super UTW detector.

X-ray diffraction

XRD analysis were carried out in a Siemens D500 powder diffractometer using CuK1 radiation, but only peaks due to support phases were detected, which can be explained on the one hand by the small amount of metal loading, and, on the other hand, by the high dispersion of metallic phases.

Fourier transform infrared spectroscopy

Carbon monoxide adsorption on bimetallic catalysts was followed by Fourier-transform infrared (FTIR) experiment using a Nicolet Magna-750 spectrometer, on a sample of about 20 mg compressed into a disk and placed into a glass cell. After reduction in situ in flowing pure hydrogen at 400°C for 2 h, the pellet was outgassed for 1 h at the same temperature and then cooled. CO was admitted in the cell by injecting pulses at room temperature until a pressure of 10 Torr (1 Torr = 133.3 N.m -2 ) corresponding to the apparent saturation of the platinum surface. Then, the samples were evacuated at room temperature for 1 h.

Nitrate and nitrite reduction reactions

The reactions were carried out in a 250-mL batch reactor. In a typical experiment, the catalyst was introduced in the reactor and degassed under N2 upflowing. 90 mL of ultrapure water with predissolved hydrogen was then added and the temperature was lowered to 10°C.

Then, 10 mL of a concentrated aqueous solution of nitrate (10 -2 mol/L) were introduced in the reactor to start the reaction. Ultrapure water (18.2 MΩ) was obtained by a Millipore purifying system. Operating conditions are reported in Table 1. The catalyst dispersion in the aqueous medium was operated by the hydrogen bubbling flow through a porous glass located at the bottom of the reactor. It was checked that the hydrogen flow and the resulting stirring were sufficient to ensure that the reaction was not rate-limited by reactant diffusion. By varying experimental parameters, like reaction temperature, catalyst amount, metal or catalyst loading, hydrogen flow rate …, it was demonstrated that the kinetics of the overall process is not influenced by diffusion limitations. For example, Figure 1 demonstrates that the activity (in mol/min) for nitrate reduction is proportional to the catalyst loading up to 2 g/100 mL of catalyst. This confirms that the nitrate disappearance rate was not influenced by diffusion phenomena with the catalyst loading of 0.8 g/100 mL, applied in the present study.

At appropriate time intervals, an aliquot of the reaction mixture was sampled and filtered. Nitrate and nitrite intermediate concentrations were determined by HPLC analysis, on an Interchrom Inertsil ODS2 column coupled with UV-VIS spectrophotometer (Thermo Separation Products, Spectra SERIES UV100). The mobile phase was a mixture of acetonitrile, diamminohydrogenophosphate, N-Octylamine (High-purity grades) and ultrapure water. The pH was kept between 6 and 6.5 using 86% H3PO4 (35). The use of Noctylamine in the eluent allows the separation of inorganic anions. Operating conditions consisted of a flow rate of 1.2 mL/min and a detection wavelength of 210 nm.

The pH of the initial and final solutions were determined by means of a digital pH meter ( Consort).

The amount of ammonium ions was measured at the end of the reaction using a selective electrode (Mettler Toledo). In this study, the selectivity towards ammonium ions was always lower than 30%, except for nitrate reduction on monometallic copper catalyst.

However, considering both the pH value at the end of the reaction (near 9) and the further possibility of cationic adsorption on alumina at this pH value, the amount of total ammonia (basic and acidic forms) was probably much higher. Still, by comparing the amount of the ammonium ions produced, it was possible to assess the performance of various catalysts in this work, since the pH value of the solution at the end of the reaction was nearly the same.

To identify gaseous products during nitrate reduction, a separate experiment was conducted in a batch reactor in the presence of CR bimetallic catalyst, with an atomic ratio Cu/Pt of 1. Before addition of nitrate, the solution and the reactor volume were saturated with hydrogen and the gas was recirculated from the top to the bottom of the reactor to create mixing conditions equivalent to a semi-batch set-up with hydrogen bubbling. We checked that the nitrate disappearance rate was unchanged compared to typical experimental conditions.

Gases were analyzed by gas chromatography equipped with a catharometer, after separation on a molecular sieve 5A column coupled with a Porapak Q column. Hydrogen accounting for a majority of the gas phase, we chose this gas as a carrier to prevent its detection. Dinitrogen was the only gas detected by this method, although J. Daum and K-D. Vorlop [START_REF] Daum | [END_REF] identified dinitrogen oxide as an intermediate product of nitrate reduction on Pd-Cu bimetallic catalysts.

The absence of dinitrogen oxide in gas phase in the present case could be explained either by the change of bimetallic catalyst or by the high solubility of N2O in water, which is 40 to 45 times higher than those of N2 and H2, or by amount of formed dinitrogen oxide below the detection limits of GC analysis.

Results and discussion

1 wt.% Cu/Al2O3 catalysts

At first 1% Cu/Al2O3 catalysts were prepared as described in the experimental part but were reduced ex-situ and stocked several days under air before the reaction test in the presence of nitrate or nitrite and hydrogen. In the same conditions, Pintar and Kajiuchi (18) observed that the catalyst is totally inactive for nitrate reduction. If we consider the redox potential of the principal species involved in this reaction [START_REF] Pourbaix | Atlas d'équilibres électrochimiques à 25[END_REF][START_REF] Maloy | Standard potentials in aqueous solution[END_REF], nitrate or nitrite should be reduced by metallic copper. This inactivity can be explained by a surface oxidation of copper in contact with air, leading to CuO, Cu2O or CuCO3.Cu(OH)2 basic copper carbonate species [START_REF] Lee | Concise Inorganic Chemistry[END_REF] giving a green color to this catalyst. To avoid this phenomenon, the catalyst was reduced in situ, i.e. in the reactor, at 320°C for 150 min. This temperature is sufficient to reduce copper, as it is proved by the TPR profile presented Figure 2d, where the peaks at 260 and 320°C could be respectively assigned to superficial Cu2O and CuO reduction to metallic copper, respectively [START_REF] Granger | [END_REF]. After copper reduction nitrate or nitrite solution was added. As expected, nitrate and nitrite were reduced, principally to ammonia, under both hydrogen and nitrogen atmospheres, confirming a redox mechanism between metallic copper and nitrate or nitrite. The evolution of nitrate concentration in presence of 1 wt.% Cu/Al2O3 is plotted as a function of time on Figure 3. Although the amount of copper was sufficient, nitrate were not totally converted, copper being rapidly passivated by CuO or Cu2O formation in water, occurring even in presence of hydrogen [START_REF] Pourbaix | Atlas d'équilibres électrochimiques à 25[END_REF][START_REF] Granger | [END_REF]. The same result was obtained during nitrite reduction. At the end of the reaction, the concentration of copper ions in aqueous solution was below the detection limit of atomic absorption spectroscopy. According to the pH value at the beginning of the reaction (pH  5.5) the thermodynamically favored reactions lead to NO (g), N2O(g), N2(g), NO2 -(aq) and NH4 + (aq). For example, the nitrate and nitrite reduction leading to nitrogen, could be written according to the following reactions : Consequently, catalysts were aged by hydrogen bubbling in water before nitrite introduction to avoid the platinum accessibility modification during the reaction.

The nitrite evolution during the reaction is presented on Figure 4.

The initial reaction rate is approximately equal to 6.10 -6 mol/min.gcat, which corresponds to a turnover frequency of 7.5.10 -3 s -1 , calculated from the observed specific reaction rate and the fraction of exposed platinum atoms (26%). The selectivity towards NH4 + is 2%. The apparent activation energy of this reaction, calculated from the Arrhenius plot (Figure 5) between 10 and 26°C, is equal to 45 ± 2.5 kJ.mol -1 , which is higher than the value found for 5 wt.% Pd/-Al2O3 by Tacke and Vorlop (14). Temperature programmed reduction (TPR) profiles of fresh CR and SI are presented on Figure 2 and compared to those of corresponding monometallic catalysts. They were established after preparation and exposition to air at ambient temperature. In the case of CR catalyst, one can observe an important hydrogen uptake between 50 and 250 °C, with two maxima at about 115°C and 205°C followed by a lower hydrogen uptake, between 270 and 400°C. For SI catalyst, two hydrogen uptakes can be identified with maxima at 100°C and 345°C approximately. In this case, the TPR profile is essentially the overlap of the individual contributions of monometallic platinum and copper catalysts. The profile of CR catalyst is significantly different and the first domain, with maxima at 115°C and 205°C could be attributed to the reduction of mixed Pt-Cu oxidized species where both metals are mutually interacting [START_REF] Noroha | [END_REF]. From the TPR profiles, one can deduce that a temperature of 380°C is sufficient to reduce all species present in the SI and CR catalysts.

The average composition of bimetallic particles in CR and SI catalysts is presented on Figure 6, for a global Pt Cu atomic ratio equal to 1, corresponding to 50 at.% of copper. TEM pictures of these catalysts are shown on Figure 7.

 CR catalyst

The particle size of CR catalyst was determined from TEM pictures (Figure 7a) by means of histogram as illustrated on Figure 8. The particle diameter, calculated as dp = nidi 3 /nidi 2 , where dp is the average diameter of the particles and ni the number of particles having diameter in the range (di  2.5 Å), is equal to 25 ± 2.5 Å.

According to the preparation method, catalytic reduction favors the deposition of copper onto platinum particles. 80% of individual particles analyzed by EDX have the average composition of the whole catalyst, which proves the strong interaction between copper and platinum. Given the platinum dispersion in the parent catalyst (26%) and the particle composition ( Pt Cu = 1), this indicates that the amount of deposited copper is approximately four times higher than the amount of accessible platinum surface Pts. Assuming that each adsorbed copper atom may occupy one adsorption site of hydrogen on platinum (42), and recalling the similar size of platinum and copper atoms, this corresponds to bulk copper deposition on platinum under hydrogen atmosphere. This type of deposition on platinum can occur via the electrons generating by the hydrogen ionization on the noble metal, according to the following scheme :

This mechanism implies that platinum surface is not totally covered by copper, even for such an amount of second metal added, and that some sites of platinum remain accessible to hydrogen. Using electrochemical methods, Furuya and Motoo (43) studied copper deposition on a platinum electrode in acidic medium. They demonstrated, that, close to the reversible

Pt 2e- H2 2 H + Cu Cu 2+
hydrogen potential, the following reaction take place before all platinum sites are occupied by copper adatoms :

Cu 2+ aq + 2 e -+ Cu-Pts Cu-Cu-Pts [START_REF] Plieth | Encyclopedia of electrochemistry of the Elements[END_REF] For larger amounts of copper, we can assume that copper deposition on copper leads to tridimensional structure sufficiently organized to be stable.

Another possibility involves the diffusion of copper atoms into the platinum particles during copper layer formation, since copper and platinum form series of stable intermetallic compound [START_REF] Hansen | Constitution of binary alloys[END_REF].

To verify the accessibility of platinum in the bimetallic CR catalyst, we studied the adsorption of CO by FTIR spectroscopy. The spectra of CO adsorbed onto the bimetallic CR catalyst with a global Pt Cu atomic ratio equal to 1 were recorded after adsorption of CO at a pressure of 10 torr following by an evacuation for 1 h at room temperature (Figure 9). The band at about 2055 cm -1 can be attributed to CO bonded to surface platinum atoms in the linear form [START_REF] Primet | [END_REF], that proves that platinum surface remains accessible to gases in the bimetallic catalyst.

 SI catalyst

Contrary to CR catalyst, the particle composition of SI catalyst is very heterogeneous, the analyzed sample (Figure 6) being composed of copper or platinum monometallic particles and of either platinum-rich or copper-rich bimetallic particles. TEM analysis on this type of catalyst confirmed that samples are constituted of small particles (average size 30 Å) of pure platinum or copper, favored by the ion exchange preparation, but larger particles were found, with a bimetallic composition. Figure 7b shows an example of this type of bimetallic particle where copper agglomerates surround platinum. This could be explained by the broad distribution of platinum and copper particle size, preferred interactions taking place between the largest particles of each metal.

b-Effect of the preparation method on the nitrate and nitrite reduction reaction 2.

Higher activities being obtained with CR catalyst, these results demonstrate that the preparation method of the bimetallic catalyst is of major importance when using the same impregnation sequence (platinum deposited first). If the catalyst activity for the nitrate reduction can be directly related to the interactions between the noble metal and copper, the difference of activity between SI and CR catalysts may be explained only by the number of bimetallic active sites, which are optimum on the CR catalyst. Once nitrates converted into nitrites, these intermediates could be reduced either on the bimetallic particles or on platinum monometallic particles. Nitrite evolution vs. nitrate conversion is presented on Figure 5b.

Whereas the curves are nearly similar, the maximum amount of nitrite present in solution is higher for SI catalyst (0.53.10 -3 mol/L) than for CR catalyst (0.43.10 -3 mol/L). This maximum is reached at about 80 percent nitrates conversion for both catalysts. Considering the catalyst activity for nitrite reduction (Table 2), we can deduce that the higher amount of nitrite in solution during nitrate reduction on SI catalyst may be partly explained by the lower activity of this catalyst for nitrite reduction. Alternatively, another explanation could be the favored nitrite desorption on SI catalyst. If we now compare the activity of SI bimetallic and monometallic catalysts at similar platinum content (0.75 wt.%) (Table 2), we observe that they are quite similar. SI catalyst being constituted of mostly small monometallic platinum or copper particles, together with large bimetallic particles (see § 2-4-3), we can conclude that the nitrite reduction occur principally on monometallic platinum sites on SI catalyst. The CR catalyst activity being much higher, this indicates that bimetallic sites are more active for nitrite reduction.

These results are quite different from those obtained for palladium based catalysts by J. Batista et al. (46) who found that the relative location of palladium and copper in the catalyst has no influence on the observed activity for the nitrate reduction. This different behavior could be attributed to the precious metal change or by the different preparation procedure.

In the present study, it was shown that monometallic platinum catalyst is inactive for nitrate reduction, while metallic state copper is able to reduce nitrate according to a redox reaction, although it is rapidly deactivated. Therefore, we can deduce that the first step in the nitrate reduction is probably a redox reaction between Cu° and NO3 -, leading to nitrite intermediate or directly to N2 or NH4 + , and to an oxidized form of copper. Subsequently, the role of the precious metal is to activate hydrogen, allowing the reduction of copper according to the following scheme :

NO 3 - Cu 2+ Cu NO 2 - Pt Pt-H H 2
With regard to the nitrite intermediates, they can be reduced either on copper, according to the same redox process than the one described above, or on the precious metal by catalytic reduction.

Van de Moesdijk [START_REF] Van Der Moesdijk | Catalysis in organic reactions in Chemical Industries[END_REF] proposed the same type of redox mechanism between the promoter and nitrate for the reduction of nitrate in water in the presence of Pd/C and/or Pt/C catalysts and

(N2 or NH4 + )
GeO2 in solution as promoter. It was suggested that, in the presence of the noble metal and hydrogen, GeO2 rapidly disappears from the solution and an intermetallic compound, Pd2Ge, was identified by X-ray diffraction. The metallic state of germanium was very surprising since it is a non-precious-metal, which is thermodynamically unstable in the presence of water. However it was verified later by XPS that germanium, present in sub-monolayer concentration on palladium, is completely reduced to the metallic state under hydrogen atmosphere [START_REF] Pijpers | [END_REF], validating this type of mechanism.

Concerning the catalyst stability, the amount of metal ions present in solution after nitrate reduction in the presence of CR and SI catalysts was below the detection limit of the atomic absorption spectroscopy.

c-Effect of Pt Cu atomic ratio

We varied the copper and the platinum amount in the catalyst to obtain an Pt Cu atomic ratio between 0.5 and 9, with a constant metal loading equal to 1 wt.-%. The influence of copper loading on the initial activity of CR and SI catalysts is presented on Figure 6. Irrespective of the preparation method, the best initial activity is obtained for 1 Pt Cu  atomic ratio corresponding to 50 percent of copper in the metallic phase. This result is different from the one obtained by Hörold et al. (15) demonstrates that the performance of different bimetallic catalysts should be compared at the optimal composition rather than at a given composition.

For both type of platinum-based catalysts (CR and SI), when 1 Pt Cu  , the activity decreases as copper content increases, to reach the activity of monometallic copper catalyst when 90 percent of copper is present in the metal phase. This can be explained by the low accessibility of platinum sites when copper amount is too important. The selectivity towards ammonium is increasing with the increase of copper content. Then, although copper is essential in the bimetallic catalyst to reduce nitrate to nitrite, it has a negative influence on selectivity.

Conclusion

Copper can reduce nitrates and nitrites according to a redox process but deactivates rapidly, copper being passivated by CuO or Cu2O formation in water. Monometallic platinum catalysts are inactive for nitrate reduction.

Nitrates are totally and rapidly reduced on Pt-Cu bimetallic catalyst when both metals are in close contact. The catalytic reduction preparation method, consisting in Cu 2+ reduction by hydrogen adsorbed on platinum, is particularly adapted to favor this interaction.

In the bimetallic catalyst, the role of copper is to reduce nitrate to nitrite according to a redox process. In this step, the interaction between copper and platinum is of major importance, in order to maintain copper in the metallic state by way of hydrogen adsorbed on platinum.

Although copper is essential in the bimetallic catalyst to reduce nitrate to nitrite, it has a negative influence on the selectivity towards nitrogen.

2. 4 . 3 .

 43 1 wt.% Pt-Cu/Al2O3 bimetallic catalysts a-Characterization of CR and SI catalysts ( atomic ratio Pt Cu of 1)

  Nitrate and nitrite evolution during the reaction in the presence of CR or SI catalysts ( Pt Cu = 1) is presented on Figure 10 a and b. Nitrite evolution is expressed as a function of nitrate conversion. The initial activities of CR and SI catalysts are reported in Table

Figure 1 :

 1 Figure 1 : Activity of CR catalyst ( with Cu/Pt =1) for nitrate reduction (in mol/min) vs.

Figure 2 :Figure 3

 23 Figure 2 : TPR profile obtained after catalyst preparation and exposition to ambient air.

Figure 4 :

 4 Figure 4 : Nitrite concentration vs. time during nitrite reduction in the presence of 1 wt.%

Figure 5 :

 5 Figure 5 : Arrhenius plot used to obtain the activation energy relative to the nitrite reduction

Figure 6 :Figure 7 :

 67 Figure 6 : Number of particles vs. copper percentage in metallic particles on SI ( ) and CR

Figure 8 :Figure 9 :

 89 Figure 8 : Metal particle size distribution for CR catalyst (number of analyzed particles 665)

Figure 10 :

 10 Figure 10 : Nitrate concentration vs. time (a) and nitrite concentration vs. nitrate conversion X(NO3 -) (b) in the presence of 1 wt.% Pt-Cu/ Al2O3 ( ) CR catalyst ( ) SI catalyst.

Figure 11 :

 11 Figure 11 : Initial activity () and selectivity towards ammonium ions formation ( )

Figure 7 Figure 11

 711 Figure 7 a and b

  on palladium based catalysts who presented the better Results are presented on Table3and compared with those obtained with Pt-Cu/Al2O3 catalysts containing the same molar amount of noble metal (about 1.5.10 -4 mole per 100 g of alumina). As we obtained the same optimal composition as Hörold et al. for palladium-based catalysts, we can conclude that this value depends mainly on the nature of metals present in the bimetallic phase. The comparison with the results obtained on platinum-based catalysts

	activity for	Pd Cu 	42 . 0	atomic ratio. This different behavior can be explained either by the
	noble metal change or by the different experimental conditions (preparation method and/or
	reaction test conditions). To verify one of these hypothesis, we prepared and tested different
	palladium-copper catalysts according to the experimental conditions described in the present
	study.

Table 2 :

 2 Activity of CR and SI bimetallic and platinum monometallic catalysts for nitrate and absolute error  0.5.10 -6 mol/min.gcat (by taking into account the error on sample weighing during preparation procedure, and reaction test, on solution volumes, on HPLC analysis…..)

	nitrite reduction
				Cu	Activity* for	Selectivity
	Catalyst	Cu	Pt	Pt	nitrate reduction,	(% of NH4 + )	n
		( wt.%)	(wt.%)	(atomic ratio )	(mol/min.gcat)	for nitrate reduction,
	CR Pt-Cu/Al2O3	0.25	0.75	1	11.10 -6	5
	SI Pt-Cu/Al2O3	0.25	0.75	1	4.10 -6	7
	Pt/Al2O3	0	0.75	0	0	-
	Pt/Al2O3	0	1	0	0	-

*

Table 3 :

 3 Activity for nitrate reduction of Pd-Cu/Al2O3 and Pt-Cu/Al2O prepared by catalytic reduction (CR) with the same molar amount of noble metal (about 1.5.10 -4 mole per 100 g of

	alumina)
	Cu	Pd or Pt	Cu	Activity* for	Selectivity
	(wt.%)	(wt.%)	Pd	or Pt	nitrate reduction,	(% of NH4 + )
			(atomic ratio)	(mol/min.gcat)	for nitrate reduction,
	 Pd based catalysts
	3.9	1.7		4	2.10 -5	-
	1	1.7		1	4.10 -5	4
	0.7	1.7	0.71	5.10 -5	2
	0.5	1.7	0.5	8.10 -5	1
	0.4	1.7	0.4	7.10 -5	1
	 Pt based catalysts
	1	3		1	9.10 -5	4
	0.5	3	0.5	3.10 -5	1

*absolute error  0.5.10 -6 mol/min.gcat (by taking into account the error on sample weighing during preparation procedure, and reaction test, on solution volumes, on HPLC analysis…..)

ACKNOWLEDGMENTS

The authors thank Dr. Sandrine Labruquère for her work on Electron microscopy and Dr Olivier Georjon for his helpful comments. F.G. and C.P. thank La Région Poitou-Charentes for research fellowships.