Adel Elgaber
email: aelgaber@femto-st.fr

Julien Bernard
email: julien.bernard@femto-st.fr

Yacouba Ouattara
email: youattar@femto-st.fr

Energy-free Security in Wireless Sensor Networks

Keywords: Wireless sensor networks, security, compression, energy

Wireless sensor networks are often deployed in open and uncontrolled environments that make them more vulnerable to security attacks. Cryptographic algorithms can be used to protect the data collected by the sensors against an intruder. The cost in terms of energy to provide enough security can be quite large as these algorithms may be very complex. As communication is the main energy consumer, a way to save energy is to use data compression. We propose to measure the impact of the well-known DES algorithm on the energy consumption for various number of rounds and then, we show that energy-free security may be possible. We combine a cryptographic algorithm with a compression algorithm and show through a model that a node can provide security without consuming more energy. The only counterpart is the time for ciphering and compressing. We get some results from experiments on energy consumption of cryptographic and compression algorithms and establish the level of security that can be achieved in various cases, from a single node to a random network.

I. INTRODUCTION

A wireless sensor network (WSN) is a specific ad-hoc network with a large number of nodes that have limited energy. These networks are used for collecting information about natural phenomena and other applications. As they are generally deployed in open and uncontrolled environments, wireless sensor networks are more vulnerable to security attacks [START_REF] Vu | Wireless sensor network architecture and its security challenges[END_REF][2][3] [START_REF] Gaubatz | State of the art in ultralow power public key cryptography for wireless sensor networks[END_REF].

Many cryptographic protocols have been proposed to deal with security issues [START_REF] Martins | Security in wireless sensor networks: a survey of attacks and countermeasures[END_REF][5] [START_REF] Perrig | SPINS: Security protocols for sensor networks[END_REF]. They rely on cryptographic primitives like public key cryptographic algorithms or secret key cryptographic algorithms or cryptographic hash functions. The impact on energy consumption of some of these primitives has already been evaluated [START_REF] Lee | The price of security in wireless sensor networks[END_REF].

As communication is the main energy consumer, a way to save energy in wireless sensor network is to use data compression [START_REF] Kimura | A survey on data compression in wireless sensor networks[END_REF]. Again, the impact on energy consumption of compression algorithms has been evaluated [START_REF] Capo-Chichi | K-RLE: A new data compression algorithm for wireless sensor network[END_REF][10] [START_REF] Sadler | Data compression algorithms for energyconstrained devices in delay tolerant networks[END_REF].

In this paper, we first propose to measure the impact of the well-known Data Encryption Standard (DES) algorithm on the energy consumption of a MSP430-based node. Then, our main contribution is to show that energy-free security is possible. We combine a cryptographic algorithm with a compression algorithm and show through a model that a node can provide security without consuming more energy. The only counterpart is the time for ciphering and compressing involving CPU cycles, which is largely less consuming than communication.

In section II, we analyze the related work regarding security and compression algorithms in wireless sensor networks. Then, in section III, we provide experimental results for DES on a MSP430 based node from the Senslab plaform. In section IV, we give an energy consumption model for cryptographic algorithms and compression algorithms and in section V, we show that it may be possible to have energy-free security. In section VI, we use a linear network to achieve better security considering the energy of the whole network. Finally, in section VII, we show that, even on random networks, it is possible to have strong energy-free security with high probability.

II. RELATED WORK

Regarding security in general, there are two main families of cryptographic algorithms: public key cryptographic algorithms like RSA or ElGamal and secret key cryptographic algorithms like DES or Advanced Encryption Standard (AES). The advantages of public key cryptography is the availability of authentication and key exchange mechanisms. Public key cryptography is secure and reliable as it is based on strong mathematical theorems. Meanwhile it needs complex arithmetical and logical operations. Strong public key cryptography can affect the lifetime of a node [START_REF] Piotrowski | How public key cryptography influences wireless sensor node lifetime[END_REF] [START_REF] Gaubatz | State of the art in ultralow power public key cryptography for wireless sensor networks[END_REF].

The other solution is to use secret key cryptography. Secret key cryptography relies on simple operations like bit shifting and basic bitwise logical operations (or, and, xor) that can easily be adapted to sensor nodes. Lee et al [START_REF] Lee | The price of security in wireless sensor networks[END_REF] considered some well-known cryptographic algorithms (AES, RC5, Skipjack, XXTEA) and studied the influence of some parameters (number of rounds, size of the key) on the energy consumption of MicaZ and TelosB sensor nodes. In particular, they show that the energy consumption of RC5 encryption increases linearly with the number of rounds.

In a wireless sensor node, communication is the main energy consumer. An idea to save energy is to apply a compression algorithm before sending data so that the energy used for compression is counterbalanced by the energy saved for communication [8][11]. Capo et al. [START_REF] Capo-Chichi | K-RLE: A new data compression algorithm for wireless sensor network[END_REF] used a MSP430-based node and measured the consumption of different compression algorithms: S-LZW, Run-Length Encoding (RLE) and K-RLE.

III. EXPERIMENTS WITH DES ON MSP430

A. Data Encryption Standard (DES)

DES is a symmetric key cryptographic algorithm that was standardized in 1977 and has been used widely since then. DES is based on a Feistel scheme with a 56-bit key and 16 rounds. Each round consists in a fixed set of four operations called Expansion, Key mixing, Substitution and Permutation that operates on 64-bit blocks that are divided in two 32bit half-blocks [START_REF]FIPS 46-3, Data Encryption Standard (DES)[END_REF]. In our experiments, we use a custom implementation of DES in C that can be tuned to reduce the number of rounds.

B. The Senslab testbed

The Senslab platform [START_REF] Burin Des Roziers | Using SensLAB as a first class scientific tool for large scale wireless sensor network experiments[END_REF] is an experimental platform for wireless sensor networks. Its aim is to automate the deployment, test, and monitoring of wireless sensor network applications. Each of the four sites of the platform has 256 MSP430-based nodes that can be used to make tests. Each node can be monitored with several probes. The frequency of the measures can be chosen for each experiment. At the end of the experiment, the measures are stored in a simple file for each node.

In our experiment, we used the Senslab platform and we measured the power consumption of a node that was compressing some data with the DES algorithm. The frequency of the measures was set to 100ms.

C. Experiment description

The experiment consists in measuring the power consumed by the DES algorithm on a MSP430-based node of the Senslab platform. For each experiment, we used random input data of λ = 64 bits. The number of rounds ranges over the values 2, 4, 8 and 16. Each experiment is repeated five times and the average value is given.

D. Results

Table I shows the results obtained in the previous experiment with various number of rounds.

Figure 1 shows the same results on a plot. We observe that the relation between energy consumption and the number of rounds is linear, of the form: E = E 0 + r × E r where r is the number of rounds. E 0 is a constant that represents the energy for constant operations in the algorithm, i.e., operations that do not depend on the number of rounds : initial and final permutation, permuted choice 1 (PC1) in the key schedule. E r is the energy per round. Applying a linear regression on the data of table I, we find E 0 = 19.149µJ and E r = 1.348µJ/round. The estimation of this linear regression is also shown on figure 1.

As a conclusion of this experiment, we found a linear relationship between energy consumption and the number of rounds for DES. It is of the same kind as the one found in [START_REF] Lee | The price of security in wireless sensor networks[END_REF] for RC5. These uncorrelated results can lead to a general model for encryption with symmetric cryptographic algorithm. This model is described in the next section.

IV. MODELING COMPRESSION AND ENCRYPTION IN WSN

A. Modeling encryption

As seen before, we can model the energy consumption of encryption as:

E enc (λ, r) = λ × (E enc 0 + r × E enc r ) (1)
In addition to the previous experiment, we introduce λ, the length of the input data, in our model. There should be another constant factor that does not depend on the length of the input data, but we assume this constant factor is negligible. For example, in DES, the only operation that does not depend on the number of rounds and the length of the input data is PC1, which is a very simple operation compared to the rest of the algorithm.

Table II shows the values of E enc 0 and E enc r for various algorithms. The DES values are computed from our experiment. The RC5 values are computed from figure 3 in [START_REF] Lee | The price of security in wireless sensor networks[END_REF]. We took the values of the TelosB node as it is a MSP430-based node, and we summed the "setup" phase and "encryption" phase to have the full algorithm. We assumed the length of data to be λ = 8 bytes = 64 bits, by comparing this figure with figure 5 in [START_REF] Lee | The price of security in wireless sensor networks[END_REF] and taking into account that the word size was divided by two (16 versus 32 respectively).

Figure 2 shows the energy consumption of RC5 that was computed from [START_REF] Lee | The price of security in wireless sensor networks[END_REF] and the estimation that we did for this algorithm.

RC5 and DES have a similar constant term E enc 0 whereas the energy per round E enc r is much higher for RC5 than for DES with a factor greater than 8. As both algorithms are based on the same basic bitwise operations, this difference can be explained by the implementation and the quality of the measures. The important point is that table II gives us a good idea of the order of energy consumption of a symmetric cryptography algorithm.

B. Modeling compression

Now, we model compression in the same manner as encryption. We use the results of [START_REF] Capo-Chichi | K-RLE: A new data compression algorithm for wireless sensor network[END_REF] to model the energy consumption of compression as:

E comp (λ) = λ × E comp 0 (2)
We assume that the energy consumption for compression algorithms is only proportional to the length of the input data. Generally, compression algorithms do not have a setup phase, they only take decisions according to the input data.

Table III shows the values of E comp 0 and the compression ratio α for various algorithms: S-LZW [START_REF] Sadler | Data compression algorithms for energyconstrained devices in delay tolerant networks[END_REF], RLE and K-RLE. S-LZW and RLE are lossless compression algorithms while K-RLE is a lossly compression algorithm. All measures are taken from [START_REF] Capo-Chichi | K-RLE: A new data compression algorithm for wireless sensor network[END_REF] on the same sets of data of length λ = 500 bytes = 4000 bits.

C. Modeling communication

As we want to compare different scenarios with the simple scenario of just sending the data, we need a communication model. We take the communication model from [START_REF] Heinzelman | Energyefficient communication protocol for wireless microsensor networks[END_REF]:

E trans (λ) = λ × (E trans 0 + × d 2 ) (3)
In this model, E trans 0 is the electrical energy and is set to 50nJ/bit, is the transmit amplifier and is set to 100pJ/m 2 /bit, and d is the distance to the receiving node.

V. ANALYSIS OF DIFFERENT SCENARIOS WITH COMPRESSION AND ENCRYPTION

In this section, we examine several scenarios using compression and encryption and the models described in (1), ( 2) and (3).

• Scenario T: in this scenario, we only consider the transmission of λ bits of input data.

• Scenario CT: in this scenario, we consider the compression of λ bits of input data with a compression ratio of α that is then transmitted.

• Scenario ET: in this scenario, we consider the encryption of λ bits of input data that is then transmitted.

• Scenario CET: in this scenario, we consider the compression of λ bits of input data with a compression ratio of α that is then encrypted and transmitted.

There is no need for a fifth scenario with encryption followed by compression and then by transmission as it would consume more energy than scenario CET.

Our goal is to show that scenario CET can consume as much energy as scenario T which would provide energy-free security.

A. Energy for the different scenarios

The energy consumption for scenario T is given by:

E T (λ) = E trans (λ) = λ × (E trans 0 + × d 2 ) (4)
The energy consumption for scenario CT is given by:

E CT (λ) = E comp (λ) + E trans ((1 -α) × λ) = λ × (E comp 0 + (1 -α) × (E trans 0 + × d 2 )) (5)
The energy consumption for scenario ET is given by:

E ET (λ, r) = E enc (λ, r) + E trans (λ) = λ × (E enc 0 + r × E enc r + E trans 0 + × d 2 ) (6)
The energy consumption for scenario CET is given by:

E CET (λ, r) = E comp (λ) + E enc ((1 -α).λ, r) + E trans ((1 -α).λ) = λ × (E comp 0 + (1 -α) × (E enc 0 + r × E enc r + E trans 0 + × d 2 )) (7)
Table IV shows the energy consumptions with λ = 64 bits, r = 8 rounds and d = 25m in the different scenarios. λ = 64 bits is a typical size for a physical scalar data like temperature. r = 8 rounds is rather weak for DES and RC5. d = 25m is a typical distance in wireless sensor networks. We observe that, as expected, with any choice of algorithms, scenario CT consumes less energy than scenario T that consumes less energy than scenario CET that consumes less energy than scenario ET.

Table V shows the energy consumptions with λ = 64 bits, r = 16 rounds and d = 75m in the different scenarios. In this case, the number of rounds is r = 16, which is the maximum for DES and which is nearly the recommended number of rounds for RC5. The distance has been extended to 75m. We note that the energy consumption of scenario CET is less than the energy consumption of scenario T, with the DES algorithm combined with S-LZW or K-RLE. In the other case, the compression algorithm does not have a good enough compression ratio (RLE), or the encryption algorithm consumes so much that it cannot be counterbalanced by compression (RC5). These figures, with realistic parameters, show that it is possible to have energy-free security but we need a more precise condition.

B. Energy-free security

In this section, we try to precise the previous result, i.e., we try to compute the maximum number of rounds r max for various values of d and the given compression algorithms. The following theorem gives the computation of r max : Theorem 1: Security is free if and only if:

r ≤ α × (E trans 0 + × d 2 ) -(1 -α) × E enc 0 -E comp 0 (1 -α) × E enc r =rmax(α,d)
The proof is straightforward, it directly comes from ( 4) and ( 7) with the condition that E CET (λ, r) ≤ E T (λ). We note that the condition does not depend anymore on the length of the data which is normal because, in each scenario, the energy is proportional to the length of the data.

Table VI shows r max (α, d) for the three compression algorithms and the DES algorithm, with d varying from 25m to 100m. We observe that for distance of 25m, security can not be free whatever the compression algorithm is, i.e., r max (α, d) < 0. The RLE algorithm do not compress enough and can never provide free security.

The results for S-LZW and K-RLE are quite close. For a distance of 50m, the maximum number of rounds is 1 and 3 respectively, which provides no security at all as there exists some easy known attacks on DES. For a distance of 75m, as already seen, the full DES with 16 rounds can be used for free with both compression algorithms. For a distance of 100m, Triple-DES, that has 48 rounds and provides strong security, can be used for free in the case of K-RLE. to 100m. RC5 consumes more energy than DES and the results for RC5 are not very good. Even for a distance of 100m, the maximum number of rounds is 4 and 5 for S-LZW and K-RLE respectively, which does not provide any security. The solution in this case is to optimize the implementation of RC5 or to find a better compression algorithm.

VI. ANALYSIS WITH A LINEAR NETWORK

In this section, we try to improve the previous results considering a linear network. Our idea is that the energy consumed on the sending node can be counterbalanced globally over the network by the savings of the other nodes, due to the size of the compressed data. This could improve the security of the data while still competing with scenario T.

A. Model

We use a linear network of n + 1 nodes, the first node that generates data and n relays in the multi-hop communication to the base station. Each node only communicates with its closest neighbors that are at distance d. The last node communicates with the base station that is at distance d too.

On this linear network, we examine the global energy in scenario T and scenario CET. In each scenario, the data is sent by the first node, then received n times and transmitted n times until the base station.

We still use the communication model from [START_REF] Heinzelman | Energyefficient communication protocol for wireless microsensor networks[END_REF] for receiving:

E recv (λ) = λ × E recv 0 (8) E recv 0
is the electrical energy and is set to 50nJ/bit. The energy consumption for scenario T with a linear network is given by:

E T net (λ, n) = E trans (λ) + n × (E recv (λ) + E trans (λ)) = λ × (n × E recv 0 + (n + 1) × (E trans 0 + × d 2 )) (9)
The energy consumption for scenario CET with a linear network is given by:

E CET net (λ, r, n) = E CET (λ, r) + n × (E recv ((1 -α).λ) + E trans ((1 -α).λ)) = λ × (E comp 0 + (1 -α) × (E enc 0 + r × E enc r + n × E recv 0 + (n + 1) × (E trans 0 + × d 2 ))) (10)
+ .d 2 ) -(1 -α).E enc 0 -E comp 0 (1 -α) × E enc r =r net max (α,d,n)

B. Results

Table VIII shows r net max (α, d, n) for the three compression algorithms and the DES algorithm, with d = 25m and n varying from 1 to 15. We observe that with only one-hop before the base station, security is free with the K-RLE compression algorithm, even if the number of rounds provides very weak security. For n ≥ 5, the maximum number of rounds for S-LZW and K-RLE exceeds the number of round for DES, and for n ≥ 10, it exceeds the number of rounds for Triple-DES. This shows that very strong security can be achieved for free on a wide network.

Table IX shows r net max (α, d, n) for the three compression algorithms and the RC5 algorithm, with d = 25m and n varying from 1 to 15. In this case, the situation is better than in the experiment with a single node, but the level of security is not as strong as the level for DES. For n ≥ 15, the maximum number of rounds is 14 and 16 for S-LZW and K-RLE respectively, which a little less than the recommended 18-20 rounds for good security. Once again, the implementation of the algorithm must be improved in order to achieve better results.

VII. ANALYSIS WITH RANDOM NETWORKS

In this section, we compute r max on random networks. We focus on DES and K-RLE as it's the best combination of a compression algorithm and an encryption algorithm that we have.

A. Experiment

The difficulty in this experiment is to choose an application to make the measures. We decided to test a simple routing application on a square area of width w. The network is composed of n nodes uniformly distributed on the area. Two nodes can communicate if their distance is less than 0.4×w so that there is, on average, half of the nodes in the neighborhood of each node, whatever the width of the area. Among the n nodes, 20 nodes are chosen to be sources of messages of size λ = 64bits. Those messages are routed to a sink which is placed at the coordinates (0.9 × w, 0.9 × w) thanks to a shortest path algorithm which takes into account the square of the distance between each node (as the energy for transmitting a message is proportional to the square of the distance). Then, each source sends a message to the sink along the chosen path.

To compute r max for a given network, we first compute the energy E T that is consumed for scenario T. Then, we compute the energy that is consumed for scenario CET with r = 0 rounds, which is necessarily less than the E T (scenario CET with r = 0 rounds is similar to scenario CT). Then, the number of rounds is increased until the energy is more that E T which means we have reached r max . This experiment is repeated on 1000 different random network for each value of (n, w).

B. Results

Figure 3 shows the distribution of r max for n =50 nodes and w = 50m. We observe that this distribution is not symmetric and is quite wide so that the computation of an average is not very relevant. That's why we decided to compute the first decile of the measures, i.e., the value of r max that divide the data set in 10% of low values and 90% of high values. In this case, the first decile is 45 which means that for a random network with our simple routing application, taking DES with 45 rounds (nearly Triple DES) and K-RLE is energyfree with a probability of 0.9. This result is not a surprise as it can be compared to the results of table VIII. The range of the network is a little shorter in the case of random networks (20m), but the average number of hops from the sources to the sink must be high enough so that the energy for encryption is counterbalanced by the savings along the paths.

Table X shows the computation of the first decile for many values of (n, w). This table shows that in any case, it is possible to have strong energy-free security on a random network. We observe that, for a fixed w, r max increases sublinearly w.r.t. n. Adding more nodes on the area make paths shorter, but not short enough so that the gain in energy can bring many more rounds. We also observe that for a fixed n, r max increases over-linearly w.r.t. w. In this case, the distances between nodes is increased and the gain in energy can be used to do many more rounds.

VIII. CONCLUSION

We show that it is possible to provide strong and free security thanks to a careful choice of compression algorithm and cryptographic algorithm. We provide models for the energy consumption of compression algorithms and encryption algorithms. Our models are derived from experiments done by ourselves or found in the literature, with a MSP430-based node.

It would be interesting to implement these algorithms on real nodes and check that security is really free. The consumption of the transmission comes from a model that is not derived from real experiments so an extension of this work could be to verify this model with various radio chips.

Figure 1 .

 1 Figure 1. Energy consumption of DES with various number of rounds

Figure 2 .

 2 Figure2. Energy consumption of RC5 with various number of rounds[START_REF] Lee | The price of security in wireless sensor networks[END_REF]

Figure 3 .

 3 Figure 3. Distribution of rmax for n =50 nodes and w = 50m

TABLE I .

 I

	ENERGY CONSUMPTION OF DES WITH VARIOUS NUMBER
		OF ROUNDS
	Rounds	2	4	8	16
	Energy (µJ)	21.48	24.75	30.27	40.55

TABLE II

 II

	.	VALUES OF E ENC 0	AND E ENC r	FOR VARIOUS ENCRYPTION
			ALGORITHMS
		Algorithm	E enc 0 (nJ/bit)	E enc r (nJ/round/bit)
		DES	299.2	21.06
		RC5 [7]	336.4	173.28

TABLE III

 III

	.	VALUES OF E COMP 0	AND COMPRESSION RATIO α FOR
		VARIOUS COMPRESSION ALGORITHMS
		Algorithm	E comp 0	(nJ/bit)	α
		RLE	1.325	17%
		S-LZW		5.6	53%
		K-RLE	2.575	56%

TABLE IV .

 IV ENERGY CONSUMPTIONS (IN µJ) WITH λ = 64 BITS, r = 8 ROUNDS AND d = 25M

		Algorithms	E T	E CT	E ET	E CET
		DES + RLE	7.200	6.061	37.132	30.904
	DES + S-LZW	7.200	3.742	37.132	17.810
	DES + K-RLE	7.200	3.333	37.132	16.503
		RC5 + RLE	7.200	6.061	117.449	97.567
	RC5 + S-LZW	7.200	3.742	117.449	55.559
	RC5 + K-RLE	7.200	3.333	117.449	51.842
	TABLE V.	ENERGY CONSUMPTIONS (IN µJ) WITH λ = 64 BITS,
		r = 16 ROUNDS AND d = 75M
	Algorithms	E T	E CT	E ET	E CET
	DES + RLE	39.200	32.621	79.914	66.414
	DES + S-LZW	39.200	18.782	79.914	37.918
	DES + K-RLE	39.200	17.413	79.914	35.327
	RC5 + RLE	39.200	32.621	238.168	197.765
	RC5 + S-LZW	39.200	18.782	238.168	112.298
	RC5 + K-RLE	39.200	17.413	238.168	104.959

Table VII

 VII

shows r max (α, d) for the three compression algorithms and the RC5 algorithm, with d varying from 25m

TABLE VI

 VI

	.	rmax(α, d) WITH THE DES ALGORITHM
	Algorithms	25m	50m	75m	100m
	RLE	-	-	-	-
	S-LZW	-	1.291	18.024	41.450
	K-RLE	-	3.645	22.531	48.970

TABLE VII

 VII

	.	rmax(α, d) WITH THE RC5 ALGORITHM
	Algorithms	25m	50m	75m	100m
	RLE		-	-	-	-
	S-LZW		-	-	1.976	4.823
	K-RLE		-	0.228	2.524	5.737

TABLE VIII .

 VIII

		r NET max (α, 25, n) WITH THE DES ALGORITHM AND A
			LINEAR NETWORK
	Algorithms	n = 1	n = 2	n = 5	n = 10	n = 15
	RLE	-	-	-	2.615	10.517
	S-LZW	-	8.653	34.756	78.262	121.767
	K-RLE	2.134	11.955	41.416	90.518	139.620

TABLE IX

 IX

		.		r NET max (α, 25, n) WITH THE RC5 ALGORITHM AND A
					LINEAR NETWORK
		Algorithms	n = 1	n = 2	n = 5	n = 10	n = 15
		RLE		-	-	-	0.103	1.064
		S-LZW		-	0.837	4.010	9.297	14.585
		K-RLE		0.045	1.238	4.819	10.787	16.754
	Now we can state an extension of theorem 1 for a linear
	network and compute r net max (α, d, n):
	Theorem 2: Security is free in a linear network of n + 1
	nodes if and only if:
	r ≤	n.α.E recv 0	+ (n + 1).α.(E trans 0

TABLE X .

 X FIRST DECILE OF rmax FOR A NETWORK OF n NODES ON A SQUARE AREA OF WIDTH w

	n	w	50m	100m	200m	300m	400m
	50		45	52	72	102	141
	100		76	84	97	119	148
	150		98	108	121	138	163
	200		101	125	137	152	173


N Olivi-Tran
email: nathalie.olivi-tran@umontpellier.fr

M Cannat

A possible explanation of the nature of dark matter by Newtonian mechanics

Keywords:

We calculate the trajectory of a moving object on earth and the trajectory of the earth around the sun in a fourdimensional space (x, y, z, t). We see that the trajectory has a mass: this is what is called dark matter which mass depends on the mass of the moving object. Indeed the deformation of space, the track, deviates light as we are at the hypersurface of the universe, and that is the method of measurement of the 'mass' of the universe.

INTRODUCTION

I make here the hypothesis that our three dimensional universe with physical laws is embedded in a four-dimensional space where no physical law exist [START_REF] Olivi-Tran | The FLRW cosmological model revisited: Relation on the local time with the local curvature and consequences on the Heisenberg uncertainty principle[END_REF][START_REF] Olivi-Tran | What if our three dimensional curved universe was embedded in four dimensional space? Consequences on the EPR paradox[END_REF]. Time is a function of the radius of curvature of our universe and this leads to the acceleration of the expansion of our universe [START_REF] Olivi | Tran Is the rate of expansion of our universe really accelerating?[END_REF].

The model of dark matter that I present here is based on the fluctuations of the hypersurface of our universe. Our universe is three dimensional and curved, therefore it may be embedded in a four dimensional space with no curvature. The hyper-surface of our universe is superimposed to our present universe. But this hyper-surface is not 'smooth', this means that it may contain fluctuations. These fluctuations are due to real matter (due to the changes of curvature due to gravitation) but also to dark matter (or at least 'are' dark matter). Indeed, the local changes in the curvature of our universe may lead to what we call dark matter. This dark matter, although it is only a local deformation of the hyper-surface of our universe, may deviate radiations and can account for gravitational effects [START_REF] Trimble | Existence and nature of dark matter in the universe[END_REF].

TRAJECTORIES

In Newtonian mechanics

ma = mg ( 1
)
where a is the acceleration, m the mass of the moving object and g the gravitational acceleration on a given point on earth. As said before, our universe is threedimensional and curved and it can be embedded in a fourdimensional euclidean space. So we integrate equation (1) over x, y, z and t even if this corresponds not exactly to a euclidean space because t is not linearly proportional to a space dimension [START_REF] Olivi-Tran | The FLRW cosmological model revisited: Relation on the local time with the local curvature and consequences on the Heisenberg uncertainty principle[END_REF][START_REF] Olivi-Tran | What if our three dimensional curved universe was embedded in four dimensional space? Consequences on the EPR paradox[END_REF][START_REF] Olivi-Tran | Dimensional analysis of Einstein's fields equations[END_REF][START_REF] Olivi | Tran Is the rate of expansion of our universe really accelerating?[END_REF].

∞ ∞ madxdydzdt = (2) ∞ ∞ mgdxdydzdt (3)
and

∞ ∞ mvdxdydzdt = (4) ∞ ∞ mg(xi + yj + zk + t) + ( 5
)
mv 0 + (x 0 i + y 0 j + z 0 k + t 0 )dxdydzdt (
y 0 yj + z 0 zk + t 0 t) (10)
where v 0 is the initial velocity and x 0 , y 0 , z 0 , t 0 its starting point. So the trajectory of the object is a second order equation and not exactly a parabola.

For the trajectory of earth around the sun we have, in Newtonian mechanics:

ma = GM m r 2 (x r i + y r j + z r k) ( 11
)
where G is the gravitational constant in Newtonian mechanics, r is the distance between the sun and earth at time t 0 ,x r i + y r j + z r k the vector directed from earth to the sun at time

t 0 .. ∞ ∞ madxdydzdt = (12) ∞ ∞ GM m r 2 (x r i + y r j + z r k)dxdydzdt (13)
and

∞ ∞ mvdxdydzdt = (14) ∞ ∞ ( GM m r 2 (x r i + y r j + z r k)(xi + yj + zk + t) + (15) mv 0 + (x 0 i + y 0 j + z 0 k + t 0 ))dxdydzdt (16)
and

m(xi + yj + zk) = (17) GM m r 2 (x r i + y r j + z r k)(x 2 /2i + y 2 /2j + z 2 /2k + t 2 /2) + (18) mv 0 (xi + yj + zk + t) + (x 0 xi + y 0 yj + z 0 zk + t 0 t) (19)
This is the trajectory of earth around the sun.You may see that the trajectory has a mass m: this is dark matter; this is mass without the presence of an object or more precisely deformation of space due to the former presence of an object.

Acknowledgments: We acknowledge interesting discussions with H.C. and N.G.

CONCLUSION

Dark matter is in fact the track of the planets in the solar system, and of objects in the universe. This explains why we do not 'see' all the mass present in the universe. Indeed the present methods use the trajectory of electromagnetic waves and particles coming from space to measure the mass of the universe and these are deviated by the track of real mass.