Andrés Cuberos

Esterina Masiello

Véronique Maume-Deschamps

COPULAS CHECKER-TYPE APPROXIMATIONS: APPLICATION TO QUANTILES ESTIMATION OF AGGREGATED VARIABLES

Keywords: risk aggregation, empirical copulas, checkerboard copula, checkmin copula, quantile estimation. 1

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Introduction

Consider a random vector X = (X 1 , . . . , X d ) and a measurable function Ψ : R d → R + , called the aggregation function. In the context of quantitative risk management X is known as a risk vector and generally represents the profit-losses of a portfolio at a given future date. Ψ(X), the aggregated risk, represents its total future position. The main examples of aggregation functions are: the sum, max, weighted sums or a slightly more complex function that may include stop-loss reinsurance type function on each of the marginal distributions. In this paper, we will be essentially concerned with Ψ = , which is the most commonly studied aggregation function. We are interested here in the estimation of p-quantiles, 0 < p < 1, of Ψ(X): Q p (Ψ(X)) = inf{x ∈ R, F Ψ (x) ≥ p}, where we denote by F Ψ (x) = P (Ψ(X) ≤ x) the distribution function of the aggregated random variable. In financial or insurance contexts, the Q p 's are called Value at Risk and denoted VaR p . We will assume that the distributions F 1 , . . . , F d of the marginal variables X 1 , . . . , X d are known and that some information on the dependence between them is given. Usually this information is available via some observations of the joint distribution and also via expert opinion.

In practice, neither the marginals nor the dependence of the risk vector X are known. However, in many cases, the information available on the marginal distributions is much more important than the one on the dependence structure. For example, when some observations of the vector X are available, inferences one can do on the marginal distributions give better results than inferences one can do on the multivariate distribution. Also, samples available for marginal laws may be much larger than those available for the joint distribution. Moreover, on each marginal risk, some extra information, as for example expert opinion or prior information, may be available. These situations arise e.g. for environmental data or in insurance contexts. So, even if the assumption of the knowledge of marginal distributions may seem not realistic, by simplification we shall assume here that the marginal distributions F i are known. However it may be well the case that these distributions are actually estimations based on data or extra marginal information.

When the marginals are known but the dependence is unknown, the rearrangement algorithm (introduced in special cases in [START_REF] Rüschendorf | Solution of a statistical optimization problem by rearrangement methods[END_REF] and [START_REF] Rüschendorf | Random variables with maximum sums[END_REF]) allows to obtain bounds on the distribution of Ψ(X) ( [START_REF] Puccetti | Computation of sharp bounds on the distribution of a function of dependent risks[END_REF]) for d ≥ 30. By improving the re-arrangement algorithm, bounds on the VaR are obtained in ( [START_REF] Embrechts | Model uncertainty and var aggregation[END_REF]) in high dimensional (d ≥ 1000) inhomogeneous portfolio. These bounds are usually too wide to be useful in practice. Cases in which some kind of dependence information is available lead to narrower bounds ( [START_REF] Bernard | Value-at-Risk bounds with variance constraints[END_REF][START_REF] Bernard | A new approach to assessing model risk in high dimensions[END_REF]) for the risk measure at hand. Bounds are also derived in ( [START_REF] Cossette | A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks[END_REF]) for dependence structures described by different copula models. A general mathematical framework which interpolates between marginal knowledge and full knowledge of the distribution function of X is considered in ( [START_REF] Embrechts | Risk aggregation[END_REF]).

In this paper, we propose to use the check-min-erboard copula (as introduced in [START_REF] Li | Strong approximation of copulas[END_REF] in dimension 2 and developed in [START_REF] Mikusinski | Some approximations of n-copulas[END_REF] in higher dimension) to merge the information given by a small sample of the distribution of X with the known marginal distributions. Related kinds of copulas approximation appeared already in [START_REF] William F Darsow | Copulas and markov processes[END_REF] and have been studied in [START_REF] Genest | A primer on copulas for count data[END_REF][START_REF] Neslehová | On rank correlation measures for non-continuous random variables[END_REF][START_REF] Genest | On the empirical multilinear copula process for count data[END_REF] for discrete variables. Durante et al. [START_REF] Fabrizio Durante | Convergence results for patchwork copulas[END_REF] presented patchwork copulas which give a general framework for studying piecewise approximations of copulas. We consider generalized versions of the checkerboard copula: the partition considered needs not to coïncide with the one given by the sample. Moreover, we introduce the checkerboard copula with information on the tail and with information on a sub-vector, to take into account some additional informations which may improve the quantile estimation (see Section 3). We introduce empirical versions of the above approximations that are proper copulas and allow to obtain efficient estimations of aggregated quantiles.

In Section 2, we recall basic definitions on copulas and we present the checkmin-erboard copulas and their empirical version. We also prove some convergence results. In Section 3, we show how additional information (on the tail or on the law of a sub-vector) may be pushed into the check-minerboard copula. A simulation study is presented in Section 4. Conclusions are provided in Section 5.

Copulas approximations

Let F be the distribution function of X = (X 1 , . . . , X d ), where X 1 , . . . , X d are assumed to be random variables living on an atomless probability space. By Sklar's Theorem, there exists a copula distribution C on [0, 1] d such that

F(x 1 , . . . , x d ) = C(F 1 (x 1 ), . . . , F d (x d )),
where the F i 's are the distribution functions of the X i 's. When the marginal random variables X i are absolutely continuous this copula C is unique. We will assume that the marginals of X are absolutely continuous. Remark that if G is a distribution function on [0, 1] d , it is a copula if and only if the marginal distributions are uniform on [0, 1], that is,

G(x) = x k for any x ∈ [0, 1] d such that x i = 1, i = k.
The aggregation function Ψ : R d → R + is considered to be the sum:

Ψ(X) = d i=1 X i .
As above, F Ψ (x) = P (Ψ(X) ≤ x) will denote the distribution function of the aggregated random variable. The check-min-erboard copula as introduced in [START_REF] Li | Strong approximation of copulas[END_REF] in dimension 2 and developed in [START_REF] Mikusinski | Some approximations of n-copulas[END_REF] in higher dimension, is a flexible tool to approximate copulas. These copula's approximations are particular cases of patchwork copulas presented in [START_REF] Fabrizio Durante | Convergence results for patchwork copulas[END_REF]. In this section, we present the check-min-erboard copulas and their empirical version.

2.1. Check-min-erboard Copulas. As above, let F denote the cumulative distribution function (c.d.f.) of X, C its copula function and F i the c.d.f of X i , i = 1, . . . , d. Let µ C be the probability measure associated to C, i.e such that:

µ C d i=1 [0, u i ] = C(u 1 , . . . , u d ) for any u = (u 1 , . . . , u d ) ∈ [0, 1] d . By a µ-decomposition of a set A ⊂ R d we mean a finite family of measurable sets {A i ⊂ A} such that (1) µ(A i ∩ A j ) = 0 whenever i = j (2) i µ(A i ) = µ(A). Definition 1. A measure µ * is a checkerboard approximation for a copula C if there exists a λ-decomposition A = {(a i , b i )} of I d , the d-dimensional unit cube, made out of d-intervals such that for all i, (1) µ * is uniform on (a i , b i ); (2) µ * (A) = µ C (A) for any A ∈ A.
I i,m = d j=1 i j -1 m , i j m , i = (i 1 , . . . , i d ), i j ∈ {1, . . . , m}.
µ * m is the checkerboard approximation associated to the regular decomposition I m .

We shall denote by C * m the checkerboard copula associated to the measure µ * m . The definition of the checkerboard copula may then be rewritten as: ) and then sampling uniformly in I i,m . This leads us to consider also the checkmin copula: instead of sampling uniformly in I i,m , one may sample in I i,m with respect to the comonotonic copula. The checkmin copula is then given by:

C * m (x) = i m d µ(I i,m )λ([0, x] ∩ I i,m ) where [0, x] = d i=1 [0, x i ], for x = (x 1 , . . . , x d ) ∈ [0, 1] d and
C † m (x) = i mµ(I i,m ) min x j - i j -1 m , 1 m .
In what follows, C o m denotes either C * m or C † m and is called check-min-erboard copula.

In [START_REF] Mikusinski | Some approximations of n-copulas[END_REF], it is proved that C o m is a copula that approximates C. The following proposition gives a more precise bound on the approximation of C by C o m by a factor 2, than the one presented in dimension 2 in [START_REF] Li | Strong approximation of copulas[END_REF], page 613, or than the result obtained in [START_REF] Fabrizio Durante | Convergence results for patchwork copulas[END_REF].

Proposition 2.1. Let C o m be either the checkerboard copula or the checkmin copula defined above. We have:

sup x∈[0,1] d |C o m (x) -C(x)| ≤ d 2m .
Proof. This is clear that for any

x ∈ [0, 1] d with x = i m , i ∈ {1, . . . , m} d , C o m (x) = C(x).
We present the computations for C * m , the same computations give the result for C † m . For a ∈ {1, . . . , m} and k ∈ {1, . . . , d}, we denote by B k+ a and B k- a the (half)-strips:

B k+ a = x ∈ [0, 1] d , a m - 1 2m < x k ≤ a m and
B k- a = x ∈ [0, 1] d , a m < x k ≤ i k m - 1 a . If x ∈ I i,m with i = (i 1 , . . . , i d ) then, |C * m (x) -C(x)| ≤ d k=1 |µ * m (B k- i k ) -λ(B k- i k )|1 B k- i k (x) + d k=1 |µ * m (B k+ i k ) -λ(B k+ i k )|1 B k+ i k (x) ≤ d k=1 min(µ * m (B k- i k ), λ(B k- i k ))1 B k- i k (x) + d k=1 min(µ * m (B k+ i k ), λ(B k+ i k ))1 B k+ i k (x) = d 2m since µ *
m and λ are both associated to a copula,

µ * m (B k- i k ) = λ(B k- i k ) = µ * m (B k+ i k ) = λ(B k+ i k ) = 1 2m .
The announced result follows.

In what follows, we will define an empirical version of the check-minerboard copula defined above, by using the empirical copula.

2.2.

Empirical check-min-erboard copulas. The empirical copula, introduced by Deheuvels ([8]), may be used to estimate non parametrically the copula. Definition 2. Let X 1 , . . . , X n be n independent copies of X. Each of them writes X j = (X j 1 , . . . , X j d ). Let R 1 i , . . . , R n i , i = 1, . . . , d be their marginals ranks, i.e.,

R j i = n k=1 1{X (j) i ≥ X (k) i }, i = 1, . . . , d, j = 1, . . . , n where
X (1) i < • • • < X (n) i
are the order statistics associated to the ith coordinate sample X 1 i , . . . X n i . The empirical copula C n of X 1 , . . . X n is defined as

C n (u 1 , . . . , u d ) = 1 n n k=1 1 1 n R k 1 ≤ u 1 , . . . , 1 n R k d ≤ u d .
It is well known (see [START_REF] Fermanian | Weak convergence of empirical copula processes[END_REF] e.g.) that the empirical copula may be used to estimate C. Nevertheless, it is not a proper copula as its marginal laws are discrete. We shall use the empirical copula C n and the empirical probability measure µ associated to C n , to define an empirical version of the checkerboard copulas introduced above. Note that in [START_REF] Deheuvels | La fonction de dépendance empirique et ses propriétés[END_REF] an interpolation of the empirical copula in order to get an estimation of C which is a proper copula is also mentioned. As we shall see below, the main advantage of our approach is that it is very easy to simulate a sample from the empirical checkerboard copula. Definition 3. Let X (1) , . . . , X (n) be n independent copies of X. The empirical checkerboard copula (ECBC) C * m is defined by

C * m (x) = i m d µ(I i,m )λ([0, x] ∩ I i,m ).
The empirical checkmin copula (ECMC) C † m is defined by

C † m (x) = i µ(I i,m )m min x j - i j -1 m , 1 m .
In what follows, C o m denotes either the ECBC or the ECMC and is called empirical check-min-erboard copula.

Remark. If U 1 , . . . , U n is an i.i.d sample distributed as C, it is known (see e.g. [START_REF] Fermanian | Weak convergence of empirical copula processes[END_REF]) that the resulting empirical distribution function and C n defined above coincide. This remark justifies the use of the Donsker property below.

In the next proposition we show that C o m , defined above, is a copula whenever the integer m from the length size of the partition I m divides n, the sample size.

Proposition 2.2. The empirical chek-min-erboard copula C o m defined on the regular partition I m and based on an i.i.d sample of size n is a copula, if and only if m divides n.

Proof. We give the details of the proof for C * m . The proof is the same for C † m . Suppose that m ≤ n and that m divides n. By definition the empirical checkerboard copula is a distribution function, we should simply check that the marginals are uniform. Without losing generality we show only that the projection on the first coordinate of the measure induced by C * m is uniform, or equivalently that C * m (x) = x 1 for any x ∈ [0, 1] d with x j = 1 for j = 1. For ∈ {1, . . . , m}, consider the strip B 1 :

B 1 = x ∈ [0, 1] d , -1 m < x 1 ≤ m = -1 m , m × [0, 1] d-1 .
The empirical copula is concentrated on n points of [0, 1] d whose coordinates are of the form j n , j = 1, . . . , n. Moreover, there is exactly one mass on each strip B 1 j , j = 1, . . . , n. So that if k = n/m, then the number of masses of C n on each strip

B 1 , = 1, . . . , m is exactly k, which means that µ(B 1 ) = k n = 1 m . Let x = (x 1 , 1, . . . , 1), x 1 ∈ [0, 1], x ∈ B 1 with -1 m < x 1 ≤ m . C * m (x) = i∈{1,...,m} d m d µ(I i,m )λ([0, x] ∩ I i,m ) = j< µ(B 1 j ) + I i,m ⊂B 1 m d µ(I i,m ) (x 1 --1 m ) m d-1 = j< µ(B 1 j ) + m x 1 - -1 m µ(B 1 ) = -1 m + x 1 - -1 m = x 1 ,
which shows that the first marginal is uniform. On the other direction, it is easy to see that if m does not divide n then the number of masses of C n on each strip B 1 , = 1, . . . , m is not the same and thus the uniform distribution for the margin is lost.

We have proved that both the ECBC and ECMC are proper copulas provided that m divides n. The following result shows that they go to C at rate

1 √ n + 1 m . Proposition 2.3. Let m divide n, we have: sup t∈[0,1] | C o m (t) -C(t)| ≤ O P 1 √ n + d 2 m .
Proof. Once more, we give the details for C * m , the proof is the same for

C † m . | C * m (t) -C(t)| ≤ | C * m (t) -C * m (t)| + |C * m (t) -C(t)|. From Proposition 2.1, we know that sup t∈[0,1] d |C * m (t) -C(t)| ≤ d 2 m . Furthermore, | C * m (t) -C * m (t)| ≤ i∈{1,...,m} d m d λ([0, t] ∩ I i,m )| µ(I i,m ) -µ(I i,m )|. Let I = {I i,m , i ∈ {1, . . . ,
| µ(I i,m ) -µ(I i,m )| = O P 1 √ n .
Hence the result.

Since C o m (t) converges to C(t), it is natural to estimate P(Ψ(X) ≤ t) = F Ψ (t) by P(Ψ(T -(U o m )) ≤ t) = F o m (t) where U o m ; C o m and T -(u 1 , . . . , u d ) = (F 1 (u 1 ), . . . , F d (u d )).
The following result is close to that of Mainik [START_REF] Mainik | Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions[END_REF], where P(Ψ(X) ≤ t) is estimated by using the empirical copula and the empirical distribution functions of the marginal laws. For t ∈ R, let

A t = {x ∈ R d , d i=1 x i ≤ t}.
We shall assume the following regularity condition on C. This condition is satisfied for copulas with bounded density, as well as Clayton copulas in dimension 2 and Gaussian copula in dimension 2, with ρ > 0 (see [START_REF] Mainik | Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions[END_REF] for a more complete discussion on this condition). Simple symmetry considerations show that it will also be satisfied for survival Clayton copulas.

Assumption 2.1. For t ∈ R, B t = {u ∈ [0, 1] d , T -(u) ∈ A t } and ∂(B t ) its boundary, the regularity of F j , j = 1, . . . , d, implies that ∂(B t ) is a d -1 hyper-surface in [0, 1] d . For δ > 0, U δ (B t ) is the δ neighborhood of ∂(B t ).
The regularity condition on C is :

(2.1) µ(U δ (B t )) = O(δ).
Proposition 2.4. Under Assumption 2.1

sup t∈R |F Ψ (t) -F o m (t)| = O P 1 √ n + O 1 m .
Proof. We present the proof for the checkerboard copula. We have:

|F Ψ (t) -F * m (t)| = i∈{1,...,m} d µ(I i,m )m d λ(I i,m ∩ B t ) -µ(I i,m ∩ B t ) ≤ i∈{1,...,m} d | µ(I i,m ) -µ(I i,m )|m d λ(I i,m ∩ B t ) (1)
+ i∈{1,...,m} d µ(I i,m )m d |λ(I i,m ∩ B t ) -µ(I i,m ∩ B t )| (2)
.

The term (1) is (4) Estimate the distribution function F o Ψ of Ψ(X o ) empirically using the above sample. We will denote F o Ψ the empirical distribution function from the sample above.

O P ( 1 √ n ) as in Proposition 2.

Checkerboard copulas with additional information

We define two kinds of checkerboard copula with additional information and then present their empirical version. First of all, we consider the case where the distribution of a sub-vector

X J = (X i ) i∈J , J ⊂ {1, . . . , d}, is known, |J| = k < d. Denote C J the copula of X J . Let µ J be the probability measure on [0, 1] k associated to C J . For i = (i 1 , . . . , i d ), let x = (x 1 , . . . , x d ) ∈ [0, 1] d , x J = (x j ) j∈J , x -J = (x j ) j ∈J and
I J i,m = x ∈ [0, 1] d / x j ∈ i j -1 m , i j m , j ∈ J , I -J i,m = x ∈ [0, 1] d / x j ∈ i j -1 m , i j m , j ∈ J .
The checkerboard copula with information on X J is defined below.

Definition 4. Consider the probability measure on [0, 1] d defined by

µ J m ([0, x]) = i⊂{1,...,d} m d-k µ J (I J i,m ) µ(I i,m )λ([0, x -J ] ∩ I -J i,m )µ J ([0, x J ] ∩ I J i,m ).
Let C J m , the checkerboard copula with additional information on X J , be defined by

C J m (x) = µ J m ([0, x]). Proposition 3.1. C J m is a copula, it approximates C: sup x∈[0,1] d |C J m (x) -C(x)| ≤ d 2m .
If X J and X -J are independent then,

sup x∈[0,1] d |C J m (x) -C(x)| ≤ d -k 2m .
Proof. The definition of C J m ensures that it is a cumulative distribution function on [0, 1] d . The fact that C J m is a copula then follows from an easy computation to get that C J m (x) = x k whenever x = (x j ) j=1,...,d , with x j = 1 for j = k. The rest of the proof is done as in that of Proposition 2.1.

We may also add information on the tail and so define the following particular checkerboard copula.

J i,m = p • I d m,i for d-tuple i = (i 1 , . . . , i d ) in {0, 1 m , . . . , m-1 m } d . Define C Ep m
as the checkerboard copula associated to the λ-decomposition of the unit d-cube J m ∪ E p , that is

C Ep m (x) = µ C (E c p )µ * m ([0, x]/t)1 E c p (x) + E∈Ep µ C (E) λ(E) λ([0, x] ∩ E).
This is the checkerboard copula with extra information on the tail.

Empirical versions.

• The ECBC with information on a sub-vector X J is defined by

C J m (x) = i⊂{1,...,d} m d-k µ J (I J i,m ) µ(I i,m )λ([0, x -J ] ∩ I -J i,m )µ J ([0, x J ] ∩ I J i,m
).

• The ECBC with information on the tail is defined by:

C E m (x) = µ C (E c p ) C * m (x/t)1 E c p (x) + E∈Ep µ C (E) λ(E) λ([0, x] ∩ E).
In Figure 3 we present a simulation of the ECBC with information on the tail for different values of m. Remark. Checkmin copulas with additional information (on a subvector or on the tail) may be defined in the same way as checkerboard with additional information.

Numerical Application

In this section we use the estimator of the distribution function F Ψ , as defined in Section 2.3, in order to estimate the quantiles Q p (S) for S = X 1 + • • • + X d at different confidence levels 0 < p < 1. We will consider the Pareto-Clayton model, defined in Section 4.1, because, in that case, the exact value of Q p (S) can be calculated, so that we may compare our simulation results to the exact one. A simulation study will also be presented for another model with gaussian copulas and lognormal marginal laws. In that case, the exact value of Q p (S) is not known so that we shall use huge sample (of size 10 7 ) to get a fine approximation of Q p (S) and then compare it with our estimation using the empirical check-min-erboard copulas based on relatively small samples. In Section 4.2, we shall see that our method is performant in high dimension (d = 25, 50, 100) with relatively small sample size. In Section 4.3, we provide some examples to show the impact of the information added in the checkerboard copula.

4.1.

The Pareto -Clayton model. We consider X = (X 1 , . . . , X d ) such that:

P(X 1 > x 1 , . . . , X d > x d | Λ = λ) = d i=1 e -λx i ,
that is, conditionally to the value of Λ the marginals of X are independent and exponentially distributed. If Λ is Gamma distributed, then the X i 's are Pareto distributed with dependence given by a survival Clayton copula. If Λ is Levy distributed, then the X i 's are Weibull distributed with a Gumbel survival copula. These models have been studied by [START_REF] Oakes | Bivariate survival models induced by frailties[END_REF] and [START_REF] Yeh | The frailty and the Archimedean structure of the general multivariate Pareto distributions[END_REF] [START_REF] Oakes | Bivariate survival models induced by frailties[END_REF][START_REF] Yeh | The frailty and the Archimedean structure of the general multivariate Pareto distributions[END_REF]. In the context of multivariate risk theory, they have been used e.g. in [START_REF] Véronique Maume-Deschamps | Impact of dependence on some multivariate risk indicators[END_REF] and [START_REF] Dacorogna | Explicit diversification benefit for dependent risks[END_REF]. In what follows, we consider that Λ ; Γ(α, β), so that the X i 's are Pareto (α, β) distributed and the dependence structure is described by a survival Clayton copula with parameter 1/α. In [START_REF] Satya | Compound gamma, beta and F distributions[END_REF], it is shown that, in this case, S follows the so-called Beta prime distribution:

F S (x) = F β x 1 + x
where F β is the distribution function of the Beta(dβ, α) distribution. The inverse of F S (or quantile function of S) can also be expressed in terms of the inverse of the Beta distribution

F S (p) = F β (p) 1 -F β (p) .
From these results (see also [START_REF] Cuberos | High level quantile approximations of sums of risks[END_REF]), we may compute Q p (S).

4.2.

Simulation study in dimension 25, 50 and 100. In this section we will consider several Pareto-Clayton models and several Gaussian-Lognormal models. Nevertheless, at the beginning we will presentat results in dimension 25, for one Pareto-Clayton model and then for one Gaussian-Lognormal model. One open question is the choice of the partition size m. Following Coeurjolly [START_REF] Coeurjolly | Median-based estimation of the intensity of a spatial point process[END_REF], we propose to keep the median of the estimations obtained for all the m divisor of n for each quantile. The study done in Section 4.2.1 indicates that this choice performs well, it is why in the other subsections, we just present the median result (and not the results for all m). We also perform simulations in dimension 50 and 100 (Section 4. 4 and5 give the results in terms of Relative Mean Square Error (RMSE) over a run of s = 100 different simulations of the initial sample. Let q k be the estimations of q on the kth simulation run, then

RMSE = 1 s s k=1 ( q k -q) 2 q .
The size of the checkerboard samples is N = 10000. The RMSE is computed for each value of m for the checkerboard (ECBC) and the chekmin (ECMC) approximations. Alternatively, the median choice of m is done for each of the 100 simulations and the RMSE is then computed for the corresponding estimations. The check-min-erboard estimations are compared with parametric ones: we adjust to the samples three copulas (survival Clayton, Gaussian, Clayton), simulate an aggregated sample of size 10000 with the estimated copula and the known marginals to estimate the aggregated quantile. Finally, we put in the table the result obtained with the empirical copula and the known margins. In Tables 4 and5, we remark that the checkmin approximation performs significantly better than the cherckerboard one. This may be explained by the fact that the survival Calyton copula is closer to a comonotonic copula than to the independent copula for the upper tails. Tables 6 and7 give the results for a Gaussian -lognormal model with correlation ρ = 0.1 for all pairs, the parameters of the lognormal margins are different for each of the 25 coordinates. It is expected in that case that the checkerboard copula will give better results than the checkmin. As already mentioned, in the case of a Gaussian -lognormal model, the exact value of the aggregated quantile is unknown. We estimate it with huge samples (of size 10 7 ) and use this estimate (called near exact value) to compute the RMSE's and compare the estimations.

Tables 6 and7 show that, as expected, in this case, the checkerboard copula gives better results than the checkmin copula. This may be explained by the fact that for a correlation coefficient ρ = 0.1 for all pairs, the model is closer to the independence than to the comonotony. We now give results in higher dimension (d = 50, d = 100).

4.2.2. Simulations in higher dimensions. We now perform simulations in dimension 50 and 100 for samples of size n = 400. We consider the Pareto-Clayton model with parameters 2 and 1, and a Gaussian-lognormal model with correlations equal to 0.25, 0.5, 0.75 (one third of the coordinates of X for each correlation value). The boxplots corresponding to 100 iterations of the estimation algorithm are presented in Appendix C. In the case of the Pareto-Clayton model, the horizontal line is the real value.

Both Figures 4 and5 show that the median ECMC estimation is performant for quantile levels 0.995 and 0.999. It is close in mean to the real VaR and much less dispersed than the empirical estimation. Of course the correct parametric model (here the survival Clayton copula) performs better but our approach will be interesting in cases where the parametric models are not well suited to data.

In the case of Gaussian-lognormal model, we consider the estimation with the gaussian copula as a reference. As above, Figures 6 and7 show that the checkmin estimator is performant also for this model.

4.3.

Adding information (on the tail or on a sub-vector). We consider a Pareto-Clayton model in dimension 2, with β = 1 and α = 2. The multivariate sample is of size n = 30 and for each presented method we performed N = 1000 estimations of the p-quantile at different confidence levels. Table 1 presents the mean and the root mean square error of the N = 1000 estimations. The estimations were calculated using the ECBC with and without information on the tail on different λ-regular decompositions I m for m = 6, 15, 30. The information on the tail is introduced on E p , for p = 0.95, 0.99, by giving to each E in E p the measure µ C (E) where C is the survival Clayton copula with parameter 1/2. For comparison a direct estimation from the empirical distribution of S is given. 1. The mean and the RMSE in % of the exact value for 1000 estimations of the quantile for a Pareto-Clayton sum in dimension 2.

All the methods we proposed perform much better than the empirical estimation based on the multivariate sample alone. The estimations based on the ECBC with λ-decomposition I m , perform better when m = 6 and m = 15 than when m = 30. ECBC with m = 6 performs slightly better than ECBC with m = 15 for the estimation of the quantiles with confidence levels lower than 99.5% and slightly worst on the higher levels. When the information on the tail is introduced on E p with p = 0.95 the estimation on the quantile with confidence level > 0, 95 is significantly improved. When it is introduced on E p with p = 0.99 the estimations improve on the higher confidence levels 99.5% and 99.9%.

Remark 1. It is known (see e.g. Proposition 2 in [START_REF] Embrechts | Model uncertainty and var aggregation[END_REF]) that the supremum of the aggregated VaR of level α over distributions X = (X 1 , X 2 ) with X 1 and X 2 having common distribution function F is:

2F -1 1 + α 2 .
For the above example (Pareto margins of parameters 1 and 2), we get the following bounds: These bounds have to be compared with the real values above. This comparison shows that the use of the bounds is not sharp for quantile estimations, even if they are very interesting to bound the uncertainty risk.

In order to assess the gain that the knowledge of the information on a sub-vector may give to the estimation, we performed here the following simulation study. Let X = (X 1 , X 2 , X 3 ) be the model where X 1 = X 2 = Y /2, and X 3 ∼ Y where Y is Pareto distributed with α = 2. We assume that (Y , X 3 ) is a Pareto-Clayton model. That is, X 1 and X 2 are comonotonic (or fully dependent) and the dependence between X 1 and X 3 is given by a survival Clayton of parameter 1/2. Clearly the distribution of the sum S = X 1 + X 2 + X 3 is equal to the distribution of the sum of the Pareto-Clayton model in dimension 2, with parameters α = 2 and β = 1 and thus the exact value of the quantiles can be easily computed. We compare the results on the quantiles estimation using the ECBC method without and with information on the sub-vector (X 1 , X 2 ) and λ-decompositions I m for m = 6, 15, 30. As before the multivariate sample is of size n = 30 and for each method we performed N = 1000 estimations of the quantile at different confidence levels. The results are presented in Table 2.

It can be noticed that the RSME of the quantile estimation is lower when the information on (X 1 , X 2 ) is introduced in the ECBC of dimension 3 and the gap is more important on higher confidence levels.

Remark. Simulating with respect to the empirical checkerboard copula with information on a sub-vector may be a difficult task because one has to simulate with respect to a given copula conditionally to belonging to a given set I i,m . In the case of a comonotonic sub-vector, this becomes trivial because we only need to simulate one coordinate uniformly.

Simulation results with the same kind of model in dimension 6 are presented in Table 3. We assumed X = (X 1 , . . . , X 6 ) with X 1 = X 2 = Y /2 and X 3 , X 4 , X 5 and X 6 distributed as Y , a Pareto r.v. with parameter α = 2. The copula of X is assumed to be a survival Clayton of parameter 1/2. As above, the size of the multivariate sample is n = 30 and for each method we performed N = 1000 estimations of the quantile at different confidence levels.

Again, by introducing the information on the sub-vector (X 1 , X 2 ) we get a smaller RMSE than in the case where no information is added. On the other hand, we also remark that by increasing the dimension (from d = 3 to 3. The mean and the RMSE in % of the exact value for 1000 estimations of the quantiles in dimension 6. d = 6) we get higher RMSE, for the same sample size, which is an expected behavior.

Conclusion

In this paper, we have constructed empirical check-min-erboard copulas with and without additional information on the joint law. We have used them to get efficient estimations of the quantiles of the sum when using a (relatively) small sample of the joint law and the knowledge of the marginal laws. Remark that a sample of size 200 in dimension 25 is small (and so does a sample of size 500 in dimension 100) and we get nevertheless rather good estimations. Our procedure provides a flexible tool for estimation of quantiles of sums from a small sample. This situation arises in many applications. In order to perform well, one has to figure out from the sample whether it is closer to an independent or comonotonic copula. Moreover, if one has partial information on the vector (copula of a sub-vector, information on the tail), it may be plugged in to improve the estimation. It is remarquable that the method remains quite efficient in high dimension. Of course, the interest of this non-parametric method would be for data for which a parametric estimation is not well suited. We are aware that many theoretical and practical questions have to be studied further, among which the points below.

• The optimal choice of m with respect to the sample size n. The choice of the median value could allow avoid this problem. Simulations indicate that this may be a good choice when the data is not too close to independence. In that case, the checkmin copula should be preferred to the checkerboard one and the median estimation performs well. For data closer to independence, the checkerboard copula should be chosen and then it seems that the higher m the better the estimation. These claims would require theoretical support; • The quantification of the impact of plugging additional information in the empirical checkerboard copula; • Developing efficient algorithms to simulate with respect to the empirical checkerboard copula with information on a sub-vector, for other copulas than the comonotonic one; 7. RMSE in % of the exact value for the Gaussian lognormal model with correlation ρ = 0.1 for all pairs, in dimension 25, for a sample size n = 200.

For

  m ∈ N, let us consider the regular λ-decomposition of the unite cube [0, 1] d denoted as I m and consisting of m d d-cubes with side length 1/m :

3 . 3 )Figure 1 .

 331 Figure 1. Simulation of empirical checkerboard copulas (N = 10000) based on the same sample of size n = 30, with m = 5 (left) and m = 15 (right). The red points are the sample rank points.

Figure 2 .

 2 Figure 2. Simulation of empirical checkmin copulas (N = 10000) based on the same sample of size n = 30, with m = 6 (left) and m = 10 (right). The red points are the sample rank points.

Definition 5 .

 5 For p ∈]0, 1[, let E p = d i=1 [0, p] d c and E p the λ-decomposition of E p consisting of the hyper rectangles [a 1 , b 1 ] × • • • × [a d , b d ] where [a i , b i ] = [0, p] or [a i , b i ] = [p, 1] for all i = 1, . . . , d with at least one of [a i , b i ] = [p, 1]. We assume that µ C (A) is known for each A ∈ E p . Consider the λdecomposition of the d-cube [0, p] d given by J m consisting of the elements

Figure 3 .

 3 Figure 3. Simulation of empirical checkerboard copulas with information on the tail, with n = 30, p = 0.95 and m = 5 (left), m = 15 (right). The red points are the sample rank points

2 . 2 ). 4 . 2 . 1 .

 22421 . Tables and boxplots showing the simulation results are postponed to Appendix B and Appendix CSimulations in dimension 25. We first perform simulations for a Pareto-Clayton model of parameters β = 3 and α = 1. We vary the sample size: n = 80, n = 200. Recall that for the Pareto-Clayton model, the real value of the quantiles is known. Tables

Figure 5 .

 5 Figure 5. Pareto-Clayton model with parameters β = 2 and α = 1, in dimension 100.

Figure 6 .

 6 Figure 6. Gaussian-lognormal model with correlations 0.25, 0.5, 0.75, in dimension 50.

Figure 7 .

 7 Figure 7. Gaussian-lognormal model with correlations 0.25, 0.5, 0.75, in dimension 100.

Table 2 .

 2 The mean and the RMSE in % of the exact value for 1000 estimations of the quantiles in dimension 3, with or without using the knowledge of the comonotonic dependence between X 1 and X 2 .

		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Exact	2.5	4.1	6.4	16.0	23.2	53.4
	ECBC (m=6)
	No information	2.7	4.6	6.6	14.0	19.1	40.7
		(13%)	(13%)	(7%)	(13%)	(18%)	(24%)
	Information on	2.6	4.4	6.6	14.8	20.8	45.7
	(X 1 , X 2 )	(9%)	(8%)	(6%)	(8%)	(11%)	(15%)
	ECBC (m=10)
	No information	2.5	4.6	7.0	14.5	19.8	41.3
		(12%)	(13%)	(12%)	(11%)	(15%)	(23%)
	Information on	2.5	4.3	6.7	15.2	21.2	46.1
	(X 1 , X 2 )	(11%)	(9%)	(9%)	(8%)	(10%)	(15%)
	ECBC (m=30)
	No information	2.5	4.2	6.8	15.9	21.4	43.3
		(14%)	(16%)	(19%)	(14%)	(14%)	(21%)
	Information on	2.5	4.2	6.6	15.8	21.9	47.1
	(X 1 , X 2 )	(13%)	(16%)	(17%)	(13%)	(13%)	(14%)
		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Exact	6.1	9.8	14.9	36.4	52.4	120.1
	ECBC (m=6)
	No information	7.2	11.0	15.0	28.2	37.3	74.3
		(19%)	(14%)	(7%)	(23%)	(29%)	(38%)
	Information on	7.1	10.9	15.0	28.9	38.4	77.5
	(X 1 , X 2 )	(17%)	(13%)	(7%)	(21%)	(27%)	(36%)
	ECBC (m=10)
	No information	6.8	11.3	16.1	30.0	39.2	76.3
		(16%)	(18%)	(13%)	(19%)	(26%)	(37%)
	Information on	6.7	11.1	16.0	30.6	40.3	79.7
	(X 1 , X 2 )	(15%)	(16%)	(12%)	(17%)	(24%)	(34%)
	ECBC (m=30)
	No information	6.3	10.4	16.3	33.7	43.4	81.3
		(15%)	(19%)	(20%)	(18%)	(22%)	(33%)
	Information on	6.3	10.4	16.2	34.0	44.1	84.2
	(X 1 , X 2 )	(15%)	(18%)	(19%)	(18%)	(20%)	(31%)
	Table

•

  Developing efficient algorithms to simulate with respect to the empirical checkerboard copula with information on the tail in dimension larger than 2.

	Appendix B. Tables
		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Exact value	16.43	23.08	31.28	59.10	76.41	135.89
	ECBC, m = 2	2%	16%	29%	51%	58%	68%
	ECBC, m = 4	10%	5%	17%	43%	51%	64%
	ECBC, m = 5	11%	4%	14%	40%	48%	63%
	ECBC, m = 8	12%	6%	8%	33%	43%	59%
	ECBC, m = 10	12%	8%	7%	30%	40%	58%
	ECBC, m = 20	10%	9%	9%	21%	31%	52%
	ECBC, m = 40	8%	9%	11%	18%	26%	48%
	ECBC, m = 80	8%	9%	12%	23%	25%	44%
	ECBC, median estimation	9%	5%	8%	31%	41%	59%
	ECMC, m = 2	2%	7%	12%	18%	20%	24%
	ECMC, m = 4	1%	2%	3%	5%	5%	13%
	ECMC, m = 5	2%	3%	4%	6%	7%	13%
	ECMC, m = 8	4%	3%	5%	10%	12%	16%
	ECMC, m = 10	5%	3%	6%	11%	13%	19%
	ECMC, m = 20	7%	5%	6%	14%	17%	23%
	ECMC, m = 40	7%	6%	7%	15%	19%	27%
	ECMC, m = 80	8%	7%	10%	16%	21%	32%
	ECMC, median estimation	3%	3%	4%	9%	11%	15%
	Gaussian copula	6%	3%	10%	27%	34%	48%
	Survival Clayton copula	1%	2%	3%	5%	6%	12%
	Clayton copula	5%	10%	23%	46%	54%	66%
	Empirical copula	8%	9%	12%	23%	31%	56%

Table 4 .

 4 RMSE in % of the exact value for the Pareto-Clayton model of parameters β = 3 and α = 1, in dimension 25, for a sample size n = 80.

		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Exact value	16.43	23.08	31.28	59.10	76.41	135.89
	ECBC, m = 2	1%	16%	29%	51%	58%	68%
	ECBC, m = 4	10%	4%	17%	42%	50%	64%
	ECBC, m = 5	11%	3%	13%	39%	48%	63%
	ECBC, m = 8	11%	6%	6%	31%	41%	59%
	ECBC, m = 10	11%	7%	5%	27%	38%	56%
	ECBC, m = 20	8%	8%	7%	17%	28%	50%
	ECBC, m = 25	7%	8%	8%	15%	24%	47%
	ECBC, m = 40	5%	6%	9%	12%	19%	43%
	ECBC, m = 50	5%	6%	9%	12%	18%	40%
	ECBC, m = 100	5%	6%	8%	15%	18%	35%
	ECBC, m = 200	5%	6%	8%	16%	20%	32%
	ECBC, median estimation	6%	5%		18%	28%	50%
	ECMC, m = 2	2%	7%	11%	18%	20%	25%
	ECMC, m = 4	1%	2%	2%	4%	6%	11%
	ECMC, m = 5	1%	2%	3%	5%	6%	12%
	ECMC, m = 8	3%	2%	5%	9%	10%	14%
	ECMC, m = 10	4%	2%	5%	9%	11%	15%
	ECMC, m = 20	5%	4%	4%	11%	14%	19%
	ECMC, m = 25	5%	4%	4%	11%	15%	20%
	ECMC, m = 40	4%	5%	6%	10%	14%	22%
	ECMC, m = 50	4%	5%	6%	10%	14%	23%
	ECMC, m = 100	5%	5%	7%	11%	14%	23%
	ECMC, m = 200	5%	6%	8%	13%	15%	28%
	ECMC, median estimation	3%	3%	4%	8%	11%	14%
	Gaussian copula	6%	2%	10%	27%	34%	48%
	Survival Clayton copula	1%	1%	2%	4%	5%	10%
	Clayton coupla	5%	10%	23%	46%	53%	65%
	Empirical coupla	5%	6%	9%	17%	23%	42%

Table 5 .

 5 RMSE in % of the exact value for the Pareto-Clayton model with parameters α = 3 and β = 1, in dimension 25, for a sample size n = 200.

		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Near exact value	93.94	111.65	129.81	176.99	200.82	270.14
	ECBC, m = 2	4%	7%	10%	14%	15%	15%
	ECBC, m4	2%	4%	7%	11%	12%	13%
	ECBC, m = 5	2%	4%	6%	10%	11%	13%
	ECBC, m = 8	2%	3%	5%	9%	10%	12%
	ECBC, m = 10	2%	3%	4%	8%	10%	12%
	ECBC, m = 20	2%	3%	4%	8%	9%	11%
	ECBC, m = 40	3%	4%	4%	9%	9%	11%
	ECBC, m = 80	3%	4%	5%	10%	11%	12%
	ECBC, median estimation	2%	3%	5%	9%	10%	11%
	ECMC, m = 2	3%	17%	34%	76%	95%	141%
	ECMC, m = 4	2%	6%	14%	41%	53%	83%
	ECMC, m = 5	2%	3%	11%	33%	44%	72%
	ECMC, m = 8	3%	2%	5%	19%	27%	46%
	ECMC, m = 10	2%	2%	3%	14%	21%	38%
	ECMC, m = 20	2%	3%	3%	7%	10%	22%
	ECMC, m = 40	2%	3%	4%	7%	8%	15%
	ECMC, m = 80	3%	4%	5%	8%	10%	13%
	ECMC, median estimation	2%	2%	4%	17%	24%	41%
	Gaussian copula	1%	2%	2%	3%	4%	6%
	Survival Clayton copula	2%	2%	3%	9%	12%	20%
	Clayton copula	3%	7%	9%	13%	14%	14%
	Empirical copula	5%	6%	9%	16%	22%	35%

Table 6 .

 6 RMSE in % of the exact value for the Gaussian lognormal model with correlation ρ = 0.1 for all pairs, in dimension 25, for a sample size n = 80.

		Quantile Quantile Quantile Quantile Quantile Quantile
		80%	90%	95%	99%	99.5%	99.9%
	Near exact value	93.94	111.70	129.88	176.26	201.28	270.35
	ECBC, m = 2	3%	7%	10%	14%	14%	14%
	ECBC, m = 4	1%	4%	6%	10%	12%	13%
	ECBC, m = 5	1%	3%	5%	10%	11%	12%
	ECBC, m = 8	1%	2%	4%	8%	9%	12%
	ECBC, m = 10	1%	2%	3%	7%	8%	11%
	ECBC, m = 20	2%	2%	3%	6%	7%	10%
	ECBC, m = 25	2%	2%	3%	6%	7%	10%
	ECBC, m = 40	2%	2%	3%	6%	7%	10%
	ECBC, m = 50	2%	3%	3%	6%	8%	10%
	ECBC, m = 100	2%	3%	4%	6%	9%	10%
	ECBC, m = 200	2%	3%	4%	7%	9%	11%
	ECBC, median estimation	1%	2%	3%	6%	8%	10%
	ECMC, m = 2	3%	18%	34%	76%	97%	142%
	ECMC, m = 4	2%	6%	15%	40%	53%	85%
	ECMC, m = 5	2%	4%	11%	33%	43%	70%
	ECMC, m = 8	2%	1%	5%	20%	28%	47%
	ECMC, m = 10	2%	1%	3%	15%	21%	37%
	ECMC, m = 20	2%	2%	2%	6%	10%	24%
	ECMC, m = 25	2%	2%	3%	6%	9%	18%
	ECMC, m = 40	2%	2%	3%	5%	6%	11%
	ECMC, m = 50	2%	2%	3%	5%	6%	11%
	ECMC, m = 100	2%	3%	4%	6%	7%	12%
	ECMC, m = 200	2%	3%	4%	6%	8%	11%
	ECMC, median estimation	1%	2%	3%	7%	11%	25%
	Gaussian copula	1%	1%	2%	3%	3%	6%
	Survival Calyton copula	2%	1%	3%	9%	13%	20%
	Clayton copula	3%	6%	9%	13%	14%	15%
	Empirical copula	3%	4%	6%	10%	13%	29%
	Table

Acknowledgements

We are grateful to anonymous referees and to the editor for many interesting remarks who helped to improve the article. This work has been partially supported by the MultiRisk LEFE-MANU project and the research chair Actuariat responsable sponsored by Generali.

Appendix A. Algorithm

Let us describe how to simulate from a checkerboard or checkermin copula with sample x 1 = (x 11 , . . . , x 1d ), . . . , x n = (x n1 , . . . , x nd ) and partition I m : Using the rank marginals, transform the sample of the copula in the pseudosample u 1 , . . . u n , where

where R ij is the rank of x ij amongst (x 1j , . . . , x nj ). Notice that the coordinates of u i belong to the set {0, 1/n, . . . , (n -1)/n}.

(1) Choose randomly and uniformly one vector u i from the pseudosample {u 1 , . . . , u n }.

(2) Let ũ = (ũ 1 , . . . , ũd ) be the vector with coordinates given by ũj = u ij • m m .

I.e, each coordinate ũj in the vector ũ is the largest rational of the form k/m less than or equal to u ij . (3) Simulate an element v ∈ [0, 1] d according to an independent or comonotone copula, depending if we want to simulate from a checkerboard or a checkermin copula respectively. (4) Then, the vector z = ũ + v/m is a simulation from the required checkerboard or checkermin copula.


Lilian Bossuet
email: lilian.bossuet@univ-st-etienne.fr

Pierre Bayon

Viktor Fischer
email: fischer@univ-st-etienne.fr

Electromagnetic Transmission of Intellectual Property Data to Protect FPGA Designs

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The microelectronics industry is faced with increased costs of production of integrated circuits (ICs). This increase is due to the costly technology refinement and the increasing complexity of systems (e.g. the transition from 32 nm to 28 nm technology has been accompanied by a 40% increase in the manufacturing costs of wafers 300 mm in diameter and by a 30% increase in the manufacturing costs of 450 mm wafers). For several years, this led to a sharp increase in the number of companies that do not have the means to produce IC (fabless companies) and to the relocation of production. ICs manufactured today are produced with a high added value in a highly competitive industry. In addition, the time-to-market is increasingly tight. This has made expensive devices the target of counterfeiting, cloning, illegal copy, theft and malicious hardware insertion (such as hardware Trojans) [START_REF] Rostami | Hardware Security: Threat Models and Metrics[END_REF], [START_REF] Colombier | Survey of hardware protection of design data for integrated circuits and intellectual properties[END_REF].

The threat model of IC and IP

The counterfeiting of ICs has become a major problem in recent years [START_REF] Ke | Counterfeit electronics: A rising threat in the semiconductor manufacturing industry[END_REF]. For example, the number of counterfeit electronic circuits seized by U.S. Customs between 2001 and 2011 has been multiplied by around 700 [START_REF] Gorman | Counterfeit Chips on the Rise[END_REF]. Between 2007 and 2010, U.S. Customs confiscated 5.6 million counterfeit electronic products [START_REF] Agma | Alliance for Gray Markets and Counterfeit Adatement[END_REF]. Overall, counterfeiting is estimated to account for about 7% of the semiconductor market [START_REF] Pecht | Bogus! Electronic manufacturing and consumers confront a rising tide of counterfeit electronics[END_REF], which represents a loss of around US$ 22 billion in 2014 for the lawful industry.

Fig. 1 is a simplified diagram of the life cycle of an IC from its design by a fabless designer to its recycling. This cycle includes many threats to the designer's intellectual property: netlist theft, mask theft, chip over-production (overbuilding), theft of the untested device, discarded device, reverse engineering, device counterfeiting, cloning, relabeled-repackaged-falsified "like new device", and hardware Trojan insertion. The threat model in Fig. 1 focuses on IC manufacturing and does not include the specific case of IP design and licensing. Indeed, for digital circuit design the reuse of IP is more and more important due to prohibitive cost of ASIC design, but the IP business suffers from a lack a security due to the intrinsic form of IPs sales and exchanges. Figure 2 presents a dedicated threat model focused on an IP life cycle. Many dedicated threats target the IP life cycle and result to revenues losses for the IP designers [START_REF] Rostami | Hardware Security: Threat Models and Metrics[END_REF], [START_REF] Colombier | Survey of hardware protection of design data for integrated circuits and intellectual properties[END_REF]. The IP threat model includes illegal re-use, illegal sales, cloning (illegal copy) of the IP. The extent of threats targeting IPs is linked to the type of IP: soft IPs (typically hardware description language files), firm IPs (synthesized netlist), and hard IPs (FPGA bitstream or physical layout).

Fig. 2. Simplified life cycle of an IP with its threat model

We propose a way to counter theft, illegal copy, cloning and counterfeiting of ICs and IPs by designing a salutary hardware (salware) [START_REF] Bossuet | SALWARE: Salutary Hardware to design Trusted IC[END_REF]. The term salware is the

IP Designer Distribution

Hardware IP One of the solutions for the IP designers to protect their intellectual property is to be able to detect the presence of a copy of an IP embedded in a digital device by using IP identification. Works on IP watermarking and IP fingerprinting try to provide the IP identification service. But, most of the time the published solutions are not practical mainly because of the complexity of the watermarking/fingerprinting verification scheme [START_REF] Legal | Automatic low-cost IP watermarking technique based on output mark insertion[END_REF], [START_REF] Marchand | IP watermark verification based on power consumption analysis[END_REF]. Efficient IP identification scheme needs to be contactless, rapid and ultra-lightweight. Up to now, these three characteristics are not available in the state-of-the-art. To meet these requirements, in this chapter we propose an ultra-lightweight binary frequency shift keying (BFSK) transmitter to forward IP identity (that could be generated for example by a feedback shift-register or a physical unclonable function [10]) discreetly using an electromagnetic channel. Such circuit is usually called "spy circuitry". Using the electromagnetic channel, it is possible to contactless check the presence of an IP inside a digital device. A preliminary version of this work was presented during the conference VLSI-SOC 2015 [11].

Salware vs. malware

In the area of security, the techniques used to attack and to defend have always been similar and the means designed for attacks can sometimes be used for protection. Our strategy in investigating the means of attack and malicious hardware is to develop new efficient salware.

Small, barely detectable hardware Trojans can disable part of a device or allow information leakage without degrading system performance [START_REF] Bossuet | Contactless transmission of intellectual property data to protect FPGAs designs[END_REF], [START_REF] Karri | Trustworthy Hardware: Identifying and Classifying Hardware Trojans[END_REF]. The same characteristics are required to design efficient salware, which is our objective. Embedding a Trojan inside an IP to protect the IP during its time-limited evaluation by the client was recently proposed in [START_REF] Tehranipoor | A Survey of Hardware Trojan Taxonomy and Detection[END_REF]. This work modifies the FSM of the IP with the aim of disrupting its normal behavior. In this way, the IP vendor can define the "expiry date" of the FSM control and disable it. In fact, this application uses a Trojan like time-based activation mechanism [START_REF] Bossuet | Contactless transmission of intellectual property data to protect FPGAs designs[END_REF], [START_REF] Karri | Trustworthy Hardware: Identifying and Classifying Hardware Trojans[END_REF]. Other activation mechanisms can be used to disrupt the IP in the case of illegal use such as an expired hardware license or an illegal copy. For example, a Trojan-like salware can use a PUF response to conditionally block an IP execution. Such a physicalcondition-based activation makes it possible to link an IP to the hardware (hardware-linked license).

Another well-known threat in cryptographic engineering is side channel attacks [START_REF] Narasimhan | Hardware IP Protection During Evaluation Using Embedded Sequential Trojan[END_REF], [START_REF] Kocher | Differential Power Analysis[END_REF]. Most of the dynamic characteristics of both hardware and software implementations of cryptographic primitives can be used for side channel analysis: computation time, power consumption, electromagnetic radiation, optical radiation, even the sound produced during computation. However, the techniques used for side channel analysis can be used to implement a salware block: e.g. for reading intellectual property data from the device or for device authentication (watermark checking). Some published works propose spy circuitry using side channels to identify the embedded intellectual property. For example in [START_REF] Kamoun | Experimental Implementation of 2ODPA attacks on AES design with flash-based FPGA Technology[END_REF], the thermal channel representing a contactless communication was used to transfer information from an embedded tag to a remote receiver. However the embedded thermal tag used in this commercial solution requires a relatively large area (255 Spartan-3 slices). In [START_REF] Marsh | Protecting designs with a passive thermal tag[END_REF], the authors propose using two shift registers to generate a recognizable signature-dependent power consumption pattern to reveal the IP signature. Power consumption was also used in [START_REF] Ziener | Power Signature Watermarking of IP Cores for FPGAs[END_REF] to communicate the IP watermark data using classical differential power analysis (DPA [START_REF] Narasimhan | Hardware IP Protection During Evaluation Using Embedded Sequential Trojan[END_REF]). To reinforce such work, the authors of [START_REF] Becker | Side-channel based watermarks for integrated circuits[END_REF] propose using the power supply signal of an IP as a physical hash function for fingerprinting.

As we mentioned above, hardware Trojans can be designed to change the operation of the infected device, but also to silently leak information. Hardware Trojans can use side channels to forward secret information such as a symmetric cipher key [START_REF] Kerckhof | Intellectual Property Protection for FPGA Designs with Soft Physical Hash Functions: First Experimental Results[END_REF] from cryptographic hardware implementation [START_REF] Bossuet | Architectures of flexible symmetric key crypto engines -a survey: from hardware coprocessor to multi-crypto-processor system on chip[END_REF], [START_REF] Lin | Trojan Side-Channels: Lightweight hardware Trojans through Side-Channel Engineering[END_REF], even when secure key management is used [START_REF] Kutzner | TROJANUS: An Ultra-Lightweight Side-Channel Leakage Generator for FPGAs[END_REF]. Hardware Trojan is also used to cause or amplify side-channel leakage of cryptographic hardware [START_REF] Gaspar | HCrypt: A Novel Reconfigurable Crypto-processor with Secured Key Management[END_REF]. Note that using side channels to detect a hardware Trojan has also been the subject of several studies [START_REF] Gallais | Hardware Trojans for Inducing or Amplifying Side-Channel Leakage of Cryptographic Software[END_REF], [START_REF] Du | Self-referencing: A Scalable Side-Channel Approach for Hardware Trojan Detection[END_REF], [START_REF] Narasimhan | Hardware Trojan Detection by Multiple-Parameter Side-Channel Analysis[END_REF].

However, designing salware with a Trojan-like hardware could present a new opportunity to protect IC and IP. In this paper, we propose a, Trojan-like, IC/IP information provider that is discreet, contactless, ultra-lightweight, and with a high bitrate. It uses an electromagnetic side channel to transmit useful information.

Except [START_REF] Kamoun | Experimental Implementation of 2ODPA attacks on AES design with flash-based FPGA Technology[END_REF], all the related works use power consumption as a communication channel which is not contactless. Unlike the proposed solution, all the related works are not lightweight and rapid as the section 5 of this chapter will show.

EM communication of IP data

Principle

Previous works on the electromagnetic attacks targeting true random number generators (TRNGs) showed that electromagnetic radiation can be used very efficiently for both active (fault injection [START_REF] Rad | A Sensitivity Analysis of Power Signal Methods for Detecting Hardware Trojans Under Real Process and Environmental Conditions[END_REF]) and passive (side channel analysis [START_REF] Bayon | Contactless Electromagnetic Active Attack on Ring Oscillator Based True Random Number Generator[END_REF]) attacks. Compared to power analysis, the attacker measuring the near-field electromagnetic emissions can obtain additional partial information about the device, since, unlike measurement of power consumption, electromagnetic radiation can be measured locally. One of the main advantages of this side channel is that it is impossible to hide the leak concerning electromagnetic radiation by using a global countermeasure. Moreover the electromagnetic test bench is not expensive (less than US$ 10K without an oscilloscope, which is the most expensive component). Last but not least, a spectral analysis of the electromagnetic radiation provides information on the oscillating structure such as a ring-oscillator [START_REF] Bayon | Contactless Electromagnetic Active Attack on Ring Oscillator Based True Random Number Generator[END_REF]. For all these reasons, we use the electromagnetic channel for our IC/IP identification scheme. To this end, we designed an ultra-lightweight BFSK transmitter.

As mentioned above, salware and malware can be based on similar principles. The same is true for the proposed BFSK principle, which can be used to design both salware (i.e. IP identity transmitter) and malware (i.e. stolen data transmitter driven by a hardware Trojan), as illustrated in Figure 3. There are two differences between using the BFSK as salware or malware. First, IP identification is activated outside the device by an ID checker, while the Trojan is activated internally. For example, the Trojan can be activated by a specific event (e.g. specific input sequence, internal data value, system state) or by pre-defined timing (e.g. a specific number of clock cycles) [START_REF] Bossuet | Contactless transmission of intellectual property data to protect FPGAs designs[END_REF], [START_REF] Karri | Trustworthy Hardware: Identifying and Classifying Hardware Trojans[END_REF]. Second, the enable signal of the BFSK transmitter is provided outside the salware: it is the same signal as that used to activate the IP identification. For malware, the BFSK transmitter's enable signal is driven internally by the hardware Trojan control logic. In this case, the Trojan activates the enable signal when it is ready to send the stolen data. Note that an enable signal is required in both applications to reduce the power consumed by the ring oscillator. Moreover, a permanently activated transmitter could be detected more easily by a spectral analysis of electromagnetic emanations of the device and could also cause local heating and premature aging of the chip.

Ultra-lightweight digital BFSK transmitter

Electromagnetic radiation is an efficient side channel since, unlike measurement of power consumption, electromagnetic radiation can be measured locally. For this reason, we use the electromagnetic channel for our IP identification scheme. To this end, we designed an ultra-lightweight BFSK transmitter which could be activated outside the device by an ID checker or internally by a specific event (e.g. specific input sequence, internal data value, system state). Note that an enable signal is required to reduce the power consumed by the ring oscillator. Moreover, a permanently activated transmitter could be detected more easily by a spectral analysis of electromagnetic emanations of the device and could also cause local heating and premature aging of the chip.

BFSK is one of the common modulation schemes used in digital communication.

The binary data are sent using a sinusoidal carrier at two frequency tones f 0 and f 1 , representing high ('1') and low ('0') logic levels. The binary data arriving at the transmitter input at certain bitrates determine the commutation of the tones at the transmitter output. The proposed BFSK transmitter uses a dedicated configurable ring-oscillator, as shown in Fig. 4. The configurable ring-oscillator is designed using one multiplexor, N+K delay elements, and a feedback chain controlled by a NAND gate for activation of transmission to reduce power consumption. Actually, the transmitter is used only during a short time when the enable signal is high, and it consumes power only during this small piece of time. The power consumption of this transmitter is thus completely negligible.

Input data controls the multiplexor, as shown in Fig. 4. When input data is low, the ring oscillator uses N delays and its oscillation frequency is f 0 . When input data is high, the ring oscillator uses N+K delays and its oscillation frequency is f 1 .

Since the ring oscillator's oscillation frequency decreases with an increase in the number of delay elements, frequency f 0 is higher than frequency f 1 . These two frequencies have to be selected according to the bandwidth of the electromagnetic analysis platform, which is used to acquire and measure the transmitted signal.

The bandwidth of our test bench, which is described in Section 3, was limited to 100 MHz and 1 GHz by the low-noise amplifier. The number of tiles used by the BFSK transmitter is very low, i.e. from 3 tiles (N=0, K=1) to 11 tiles (N=4, K=5). These values are equivalent to less than 0.022% and less than 0.080% of the total number of tiles included in the targeted 600K-gate FUSION FPGA, respectively. This very small number of tiles is very promising for good dissimulation of the BFSK transmitter inside the sea of gates/tiles. The number of FUSION tiles required by the BFSK transmitter is given by the following equation (i).

Number_FTiles = N + K + 2 (i)

In order to estimate the number of resources needed for implementation with Xilinx SRAM FPGA or Altera SRAM FPGA, Table I gives the number of 4-input look-up-tables (LUT4) used by the BFSK transmitter with such FPGAs. The number of LUT4 required by the BFSK transmitter is given by the following equation (ii).

Number_LUT4 = N + K + 1 (ii)
To evaluate the logical resources needed by the BFSK transmitter in ASIC implementations, the right hand column in Table I gives the number of equivalent gates (EG) in the transmitter. The gate count was estimated using the Virtual Silicon standard cell library based on the UMC L180 0.18 µm 1P6M Logic process (UMCL18G212T3 [START_REF] Bayon | EM leakage analysis on True Random Number Generator: Frequency and localization retrieval method[END_REF]). The delay gates are replaced by more efficient standard NOT gates. The gate count of a standard NOT gate is 0.67 EG, and that of the standard multiplexor, 2.33 EG. The standard NAND gate uses 1 EG. So the number of gates of the whole BFSK transmitter ranges from 4.67 EG (N = 0, K = 1) to 10.03 EG (N = 4, K = 5). Note that one flip-flop requires between 5.33 EG and 12.33 EG to store a single bit [START_REF] Bayon | EM leakage analysis on True Random Number Generator: Frequency and localization retrieval method[END_REF].

Such a transmitter is clearly ultra-lightweight in both FPGA and ASIC implementations. The small logical resources requirement of the proposed spy circuitry makes reverse engineering it harder, although not impossible [START_REF]0.18 μm VIP Standard Cell Library Tape Out Ready[END_REF]. Even with recent CMOS technologies, the attacker can reverse engineer ICs using a scanning electron microscope and an automatic tool for circuitry extraction [START_REF]0.18 μm VIP Standard Cell Library Tape Out Ready[END_REF], [START_REF] Torrance | The state-of-the-art in semiconductor reverse engineering[END_REF]. Nevertheless, the smaller the piece of hardware used for BFSK transmitter the harder it is to detect during reverse engineering. Detection of the transmitter using standard Trojan detection methods [START_REF] Subramanyan | Reverse Engineering Digital Circuits Using Structural and Functional Analyses[END_REF], [START_REF] Agrawal | Trojan Detection using IC Fingerprinting[END_REF] is not feasible because the transmitter does not change the data path of the circuit and because of the ultra-low signal-tonoise ratio on the electromagnetic channel, as shown in our experimental results below (Section 4).

Experimental results

The electromagnetic radiation of the device was evaluated using the near-field electromagnetic analysis test bench described in [START_REF] Bayon | Contactless Electromagnetic Active Attack on Ring Oscillator Based True Random Number Generator[END_REF]. The border between the far field and the near field can be considered to be about 23 mm from the device, depending on the hardware concerned. The most important part of the test bench is the acquisition chain. It determines the signal to noise ratio and measurement precision.

The chain, as presented in Fig. 5, is composed of:

 A Langer magnetic probe with a frequency range of from 30 MHz to 3 GHz and a spatial resolution of approximately 500 µm.  A Miteq low-noise amplifier with a frequency range of from 100 MHz to 1 GHz.  A Tektronix real time signal analyzers RSA5106B with a frequency range from 1Hz to 6.2GHz [START_REF] Jin | Hardware Trojan Detection using Path Delay Fingerprint[END_REF].

As presented in Fig. 5, the device to be tested (the board) is fixed to a XYZ table with repeatability of movement of 1 µm. The test bench, including the acquisition chain, XYZ table, FPGA configuration and power supply variations, is controlled by a computer. This test bench was first developed for electromagnetic attacks of TRNGs [START_REF] Rad | A Sensitivity Analysis of Power Signal Methods for Detecting Hardware Trojans Under Real Process and Environmental Conditions[END_REF], and [START_REF] Bayon | Contactless Electromagnetic Active Attack on Ring Oscillator Based True Random Number Generator[END_REF].

The targeted FPGA for the experimental work is an Altera Cyclone III EP3C25 that uses a 65nm CMOS technology. It contains 24 624 four-inputs LUT and 608 256 RAM bits.

Fig. 5. Near-field electromagnetic analysis test bench

Electromagnetic analysis of IC is contactless, local, and can be spatial or/and temporal. This last point makes it possible to perform frequency analysis of the electromagnetic emanation. In the your bench the spectral range is limited to 100 MHz and 1 GHz. Standard devices aimed at direct BFSK demodulation cannot be used for these relatively high frequencies. Available integrated BFSK demodulators are limited to a few dozen megahertz. For this reason, we developed a dedicated BFSK demodulation scheme for our needs, in which a spectral analysis of the low noise amplifier output (a component of the test bench) is performed to measure the f 0 and f 1 spectral contribution. The transmitted high (low) level is detected when f 1 spectral contribution is higher (lower) than that of f 0 .

For the coherent demodulation of the electromagnetic radiation, we propose a slippery window spectral analysis. Indeed, overall spectral analysis masks the effects of the no stationarity of the signal and therefore provides no information about its temporal evolution. Slippery window spectral analysis is a threedimensional representation of the signal: amplitude, frequency, and time. It requires two quantities Fw, the width of the FFT window frame and the difference Δτ between two frames. For our experiment, we chose Fw equal to 16 384 points (2 14 -point FFT) and Δτ equal to 100 points. For each frame, the FFT provides the software demodulator with the amplitude of signals f 0 and f 1 which enables the demodulator to distinguish between a transmitted '1' or '0'.

To illustrate data transmission from the circuit via the EM channel, we used a shift register that stored the following 16-bit sequence: "0101000111110011". The clock frequency of the shift register is 1 MHz. When the enable signal of the transmitter is given, the sequence is sent cyclically to the BFSK transmitter, which transmits it via the electromagnetic channel. The following gives the result of the coherent demodulation obtained at a 1 Mbps bit rate, which served as a proof of concept.

Fig. 6 and Fig. 7 present the temporal evolution of the spectral analysis (amplitude) of the BFSK transmitter's electromagnetic emission when N = 6 and K = 10, which corresponds to the following frequencies: f 0 = 289 MHz (Fig. 6) and f 1 = 119 MHz (Fig. 7). Notice also that we placed a small antenna in the close vicinity of the ring. With N = 6 and K = 10 the BFSK transmitter uses only 17 four-inputs LUT of the FPGA that represents 0.065% of the available logical resources of the used Altera FPGA for theses experimental results. For the direct coherent demodulation of the electromagnetic radiation, we propose to use a slippery window spectral analysis in order to obtain a spectral cartography. Indeed, overall spectral analysis masks the effects of the non-stationarity of the signal and therefore provides no information about its temporal evolution. Slippery window spectral analysis provides a three-dimensional representation of the signal (spectral cartography): amplitude, frequency, and time. The used Tektronix real time signal analyzers [START_REF] Jin | Hardware Trojan Detection using Path Delay Fingerprint[END_REF] allows us to obtain directly the spectral cartography with direct reading of the patent that contains the transmitted data sequence. Without knowledge of the BFSK parameters, the electromagnetic transmission cannot be easily detected because it cannot be distinguished from spectral noise. The signal-to-noise ratio of the BFSK transmission is -135 dB for a 1 GHz bandwidth. Such an ultra-low SNR represents efficient protection against unwanted BFSK transmitter detection via a side channel. However, knowing the N and K parameters, the BFSK designer can calibrate the demodulation (determine the two frequencies) by electromagnetic analysis of the ring oscillators based on the differential spectral analysis as described in [START_REF] Bayon | Contactless Electromagnetic Active Attack on Ring Oscillator Based True Random Number Generator[END_REF].

Second version of the ultra-lightweight digital EM transmitter

Now, based on the previous experimental results we propose an enhance EM transmitter. Indeed, by using the Tektronix real time signal analyzers [START_REF] Jin | Hardware Trojan Detection using Path Delay Fingerprint[END_REF], we are able to clearly distinguish the transmitter data sequence by analyzing the spectral cartography at only one frequency. Originally, we use two frequencies in order to simplify the demodulation [11].

The new version of the transmitter is based on the use of only one frequency with an on-off controllable ring-oscillator. The binary data are sent using only one sinusoidal carrier at f 0 representing high ('1') logic level. A low ('0') logic level is obtained without transmit sinusoidal carrier (i.e. the ring-oscillator is off). The proposed EM transmitter uses an on-off controllable configurable ring-oscillator, as shown in Fig. 9. The on-off controllable ring-oscillator is designed using one multiplexor and K delay elements, and a feedback chain controlled by a two NAND gate for activation of transmission. The logical resource required for this transmitter is really low because it uses only K 4-inputs LUT with Altera and Xilinx FPGAs. The previously presented (Fig. 5) experimental setup is used to test this transmitter. The parameter K is fixed to 6 to set the frequency f 0 to 309 MHz. The Fig. 10 presents the measured spectral cartographies center to f 0 with four data rates from 1 Mbps to 16 Mbps (we used the same data sequence that for the Fig. 8). This figure shows that it is possible to demodulate the transmitted sequence for 1 Mbps and 2 Mbps data rates without signal processing. It is harder to directly demodulate the data for 4 Mbps and 16 Mbps data rates, but they could be demodulated by using signal processing. These results clearly validate the proof of concept of the proposed EM transmitter. Nevertheless, the provided results (Fig. 10) are given when the EM transmitter works alone inside the FPGA. In order to test the behavior of the EM transmitter when it is embedded in a larger system we add two large IPs in the FPGA. These two IPs are an AES cipher and a AES decipher [START_REF] Kerckhof | Intellectual Property Protection for FPGA Designs with Soft Physical Hash Functions: First Experimental Results[END_REF]. These two IPs require 1 772 LUT4 (4.76 % of the targeted Altera FPGA) when the EM transmitter requires only 6 LUT4 (0.025 % of the targeted Altera FPGA). The transmitter ratio between the two IPs and the EM transmitter is equal to 295.3. Fig. 11 presents the floorplan of the FPGA after configuration. Notice that we have forced the placement of the EM transmitter in order to place it in the center of the full system.

In this case, it is always possible to demodulate the data sequence with 1 Mbps data rate. Nevertheless, it could be necessary to precisely place the EM probe over the device under test. Fig. 12 illustrates the modification of the spectral cartography at f 0 function of the positions of the probe. Four positions, P #0 , P #1 , P #2 and P #3 are tested on the same horizontal axis. The space between each position is a 3 mm gap. On Fig. 12, it appears that the data sequence is always visible, but du to other spectral contributions that come from the two other IPs inside the device, it is more difficult to directly demodulate the sequence for the positions P #0 and P #1 .

The position P #2 gives the best result. This research of the best position is really fast for the designer of the system because he knows the precise position of the EM transmitter inside the chip.

Comparison with state of the art spy circuitries using a sidechannel

Table II compares the implementation of the proposed ultra-lightweight BFSK transmitter with other recently published state of the art methods. Table II gives the spy circuitry application (App.) for each reference; this may be IP protection (IPP) or hardware Trojan (HT) or both (for the presented work, PW [11]). In addition, Table II gives the year of publication (YoP), the side channel used, the hardware resources required only for the leakage generator (for example we do not take the hardware used for IP watermark generation or the Trojan's payload into account). Unfortunately, the principles compared do not use the same amount of hardware resources. For the sake of correctness, we give the implementation results as they are presented in the referenced papers. Nevertheless, the implementation bitrate of these previously published works can be roughly compared with our proposed solution. Based on published data, we computed the bitrate of all the proposals by using the number of clock cycles needed to send information via the side channel. For all the references presented in this table, the bitrate was computed using a 1 MHz frequency for data synchronization (same frequency is used during the experimental works presented previously).

As can be seen in Table II, the proposed work reaches the highest bitrate. The reason for such a good performance is first that we use a spectral analysis of the local electromagnetic leak based on a simple frequency modulation. Except for [START_REF] Kamoun | Experimental Implementation of 2ODPA attacks on AES design with flash-based FPGA Technology[END_REF], all the other solutions use a global measurement of power consumption, which reduces the signal-to-noise ratio of the information leaked via the side channel. Our proposal is clearly the smallest spy circuitry ever published. Although solutions based on circular shift-registers are well adapted to last generation FPGA families, since the 16-bit shift registers can be designed using only one look-up table, they are not suitable for ASIC technologies. Currently, an ASIC implementation of a 16-bit shift register requires 16 flip-flops whereas the solution we propose occupies an area equivalent to only one D-flip-flop.

In this chapter, we present the proposed spy circuitry for IP protection, but it can also be used for hardware Trojan. Most of the other proposals could also be used for both applications. Note that in 2012, Kasper et al. proposed to use the work initiated in [START_REF] Bossuet | Architectures of flexible symmetric key crypto engines -a survey: from hardware coprocessor to multi-crypto-processor system on chip[END_REF] for hardware Trojan or IP watermarking implementation [START_REF]Spectrum Analyzers Datasheet[END_REF]. However, by using electromagnetic emanation and a configurable ring oscillator, the proposed solution is the most convincing for industrial applications (e.g., those aimed at IP protection) because of its very small area and high bitrate. 6 Industrial scenarios using the proposed IP protection

According to the previous section, in comparison with other works, our proposal goes clearly towards using a spy circuit in an industrial context for IP protection. Two industrial scenarios are presented in the following. The first scenario is the identification of embedded IP in the supply chain. This identification is used in order to be sure to don't use counterfeiting (fake) devices.

It is therefore crucial and strategic to be able to detect counterfeit IC as soon as possible in the supply chain (this is particularly crucial for military and space grad devices). Fig. 13 shows a possible framework to manage the device under test (control the enable signal) and check the IP identification by using an EM probe, an amplifier and a dedicated acquisition system including a spectral analysis and the proposed demodulation method. Due to the high bit rate of the proposition solution the identification of the ID requires less than 500 µs (with 1 Mbps bit rate). This counterfeiting detection could be completed by other physical (invasive or not) and optical inspection [START_REF] Kasper | Side Channels as Building Blocks[END_REF]. The second scenario occurs when an IP designers would like to verify the illegal presence of its IP inside a device (ASIC or FPGA). In this case the proposed transmitter provides to the identification scheme a data like a PUF [START_REF] Tehranipoor | Counterfeit Integrated Circuits -Detection and Avoidance[END_REF] or a watermarking. Watermarking is a technique of steganography which provides the ownership of an IC (or an IP) by checking for the presence of hidden information called the watermark [START_REF] Legal | Automatic low-cost IP watermarking technique based on output mark insertion[END_REF], [START_REF] Marchand | IP watermark verification based on power consumption analysis[END_REF]. Most of the watermarking methods proposed in the literature need a complex verification scheme. Nevertheless it is possible to use power consumption as proposed in [START_REF] Marchand | IP watermark verification based on power consumption analysis[END_REF] but it is easy and cheap to use global countermeasure in order to mask the power consumption due to the watermark [START_REF] Bossuet | A PUF based on a transient effect ring oscillator and insensitive to locking phenomenon[END_REF].

Using electromagnetic emanation in this scenario is better because as electromagnetic emanation is local it is really hard to mask it by using a global countermeasure. Moreover, in this paper we have shown that due to the SNR of BFSK signal, it is unrealistic to try to detect it without the precise knowledge of the used frequencies for data transmission.

Electromagnetic probe

Amplifier

No identified IP

Identified IP REF : F18BC9D

Validity check ok

Conclusion

IP protection has become crucial topics for hardware security due to the lack of trust in IP market. In this chapter, we have presented two ultra-lightweight transmitters of IP identity using the electromagnetic side channel. Based on a configurable ring oscillator, our first solution exploits a BFSK signal to transmit information by way of the electromagnetic channel. Our second version is simpler; it is based on an on-off ring oscillator to transmit information with only one frequency. By performing a slippery window spectral analysis of the near field electromagnetic emanations captured locally over the transmitter circuitry, the proposed transmission achieves a high bitrate (experimentally at less 1 Mbps), which has not been achieved before. Moreover, the transmitter occupies very small area less than the requirement of a small D-flip-flop. Such a small requirement of logical resources makes reverse engineering of the chip very difficult and detection of the transmitter using standard Trojan detection methods is not feasible.

Fig. 1 .

 1 Fig. 1. Simplified life cycle of an IC from a fabless designer to device recycling and the associated threat model.

Fig. 3 .

 3 Fig. 3. Electromagnetic transmission of data (i.e. IP identification data or stolen secret data by a hardware Trojan such as the secret key for symmetric cipher).

Fig. 4 .

 4 Fig. 4. Architecture of the ultra-lightweight digital BFSK transmitter based on a configurable ring oscillator. The proposed BFSK transmitter was implemented in Microsemi FUSION flash based FPGA (130 nm CMOS technology) containing 600K logic gates (M7AFS600). The device contains 13 824 tiles, each tile can be used to implement one D-flip-flop or one configurable multiplexor-based logic block implementing any 3-input logic function.The configurable number of delays in the ring oscillator of the proposed BFSK transmitter makes it possible to select precisely the two frequencies f 0 and f 1 using parameters N and K. Table I lists the ring oscillator frequencies and the number of Fusion tiles used by the BFSK transmitter for five values of N and K, with N ranging from 0 to 4, and K ranging from 1 to 5. According to TableI, f 0 can be chosen between 119 MHz (N=4) and 385 MHz (N=0) and f 1 can be chosen between 70 MHz (N=4, K=5) and 280 MHz (N=0, K=1). The exact value of f 0 depends on the number of delay elements, but also on the placement and routing of the transmitter. For the values N and K listed in TableI, the frequency variation was less than 1.7% (the maximum frequency deviation in Table I is 2 MHz when N = 4).

Fig. 6 .

 6 Fig. 6. Amplitude vs time evolution of the spectral analysis at f 0 = 289 MHz.

  Fig 8 shows the spectral cartographies obtained at f 0 = 289 MHz and f 1 = 119 MHz.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Amplitude vs time evolution of the spectral analysis at f 0 = 119 MHz.

Fig. 9 .

 9 Fig. 9. Architecture of the ultra-lightweight digital EM transmitter based on an on-off controllable ring oscillator.

Fig. 10 .

 10 Fig. 10. Spectral cartographies center (red trace) on f 0 = 309 MHz with 1Mbps data rate (a), 2 Mbps (b), 4 Mpbs (c) and 16 Mbps (d).

Fig. 11 .

 11 Fig. 11. Floorplan of the test system with one AES cipher, one AES decipher and the EM transmitter.

Fig. 12 .

 12 Fig. 12. Spectral cartographies center (red trace) on f 0 = 309 MHz with 1Mbps data rate for four different positions of the EM probe over the chip.

Fig. 13 .

 13 Fig. 13. Rapid and contactless IP identification in the supply chain by using EM transmission of IP' ID.

Legal Sales Layout / FPGA Bitstream Netlist Firmware IP .HDL Software IP Illegal Clone (Re-Use) Illegal Copy (Piracy) Illegal Sales Competitor IP Designer Reverse Engeenering IP Purchaser / SoC Integrator (not trusted)

IP IP IP IP Competitor'

s IP Hardware Trojan Designer Hardware Trojan Insertion opposite

  of malware (malicious hardware). While salware can use the same techniques, strategy and means as malware[START_REF] Bossuet | SALWARE: Salutary Hardware to design Trusted IC[END_REF], salware uses an embedded piece of hardware that is barely detectable, hard to circumvent, and is inserted in an IC or an IP to provide intellectual property information and/or to remotely activate the circuit or IP after its manufacture and during its use. IP watermarking, physical unclonable function (PUF) for IC authentication, remote activation, logic encryption, finite state machine (FSM) encryption, memory encryption, bus encryption, hardware metering, VHDL/Verilog obfuscation, bitstream encryption (for SRAM and Flash based FPGAs), are examples of the well-known salwares.

TABLE I .

 I Hardware implementation results of the BFSK transmitter

	N	K	f0 (MHz)	f1 (MHz)	Fusion Tiles	LUT4	EG
		1	385	280	3	2	4.67
		2	383	210	4	3	5.34
	0	3	384	151	5	4	6.01
		4	385	130	6	5	6.68
		5	381	111	7	6	7.35
		1	272	189	4	3	5.34
		2	272	156	5	4	6.01
	1	3	270	120	6	5	6.68
		4	271	106	7	6	7.35
		5	269	93	8	7	8.02
		1	168	144	5	4	6.01
		2	169	124	6	5	6.68
	2	3	169	100	7	6	7.35
		4	168	91	8	7	8.02
		5	168	79	9	8	8.69
		1	146	128	6	5	6.68
		2	147	112	7	4	7.35
	3	3	146	92	8	5	8.02
		4	145	84	9	6	8.69
		5	144	74	10	7	9.36
		1	123	110	7	6	7.35
		2	121	98	8	7	8.02
	4	3	122	83	9	8	8.69
		4	121	77	10	9	9.36
		5	119	70	11	10	10.03

Acknowledgment

The work presented in this paper was realized in the frame of the SALWARE project number ANR-13-JS03-0003 supported by the French "Agence Nationale de la Recherche"and by the French "Fondation de Recherche pour l'Aéronautique et l'Espace".