Gilles Roudière
email: gilles.roudiere@laas.fr

Philippe Owezarski
email: philippe.owezarski@laas.fr

Evaluating the Impact of Traffic Sampling on AATAC's DDoS Detection

As Distributed Denial of Service (DDoS) attack are still a severe threat for the Internet stakeholders, they should be detected with efficient tools meeting industrial requirements. In a previous paper, we introduced the AATAC detector, which showed its ability to accurately detect DDoS attacks in real time on full traffic, while being able to cope with the several constraints due to an industrial operation. However, in a realistic scenario, network monitoring is done using sampled traffic. Such sampling may impact the detection accuracy or the pertinence of produced results. Consequently, in this paper, we evaluate AATAC over sampled traffic. We use five different count-based or time-based sampling techniques, and show that AATAC's resources consumption is in general greatly reduced with little to no impact on the detection accuracy. Obtained results are succinctly compared with those from FastNetMon, an opensource threshold-based DDoS detector.

I. INTRODUCTION

Distributed Denial of Service (DDoS) attacks consist in numerous hosts sending fake requests towards a victim's network. Those requests exhaust the victim's system resources, either its available computing power, networking bandwidth or even some protocol implementation specific resources. Thus, they can cause up to the system's total inability to handle legitimate clients' requests, and thus should be detected. However, appropriate detectors have to meet the industry requirements (real-time operation, accuracy, autonomy...) As today's bandwidths are very large, most actual monitoring systems sample the traffic in order to reduce the size of the data to be processed, limiting the computational resources used for monitoring purposes. 1 Consequently, a realistic DDoS detector should be efficient even when processing sampled traffic.

In a previous paper we introduced AATAC (Autonomous Algorithm for Traffic Anomaly Characterization), a fully autonomous anomaly detector that focuses on DDoS attacks. We proved, on some full traffic traces, that it can perform an efficient detection while consuming a very limited amount of resources. However, to be used in an industrial context, AATAC should also be able to perform efficiently over sampled traces. This is why in this paper we evaluate the AATAC algorithm over sampled traffic. We use several types of sampling techniques and rates, the aim being to understand which sampling techniques would be the most appropriate for maximizing the DDoS detection using AATAC. The results are compared with those from FastNetMon, the only publicly available DDoS detector we manage to install on our testbed.

The remaining of this paper is organized as follows. In section II we introduce other detectors built to operate over sampled traces. Our detector AATAC is briefly introduced in section III, while its evaluation over sampled traces is discussed in sections IV and V. Section VI concludes the paper and discusses possible future works.

II. RELATED WORKS

In this section, we present related works that study the impact of sampling on several monitoring tools, but mainly focus on the network anomaly detection.

Several related approaches rely on a flow-based sampling. Bartos et al. [START_REF] Bartos | Optimizing flow sampling for network anomaly detection[END_REF] study the impact of sampling on network anomaly detection. They propose an adaptive flow-level sampling technique that manages to limit the impact of sampling on the performance of a network behavioural anomaly detector. Other flow-based sampling techniques are presented by Jadidi et al. [START_REF] Jadidi | A Probabilistic Sampling Method for Efficient Flow-based Analysis[END_REF] or Andriolakis [START_REF] Androulidakis | Intelligent flow-based sampling for effective network anomaly detection[END_REF]. The problem with such approaches is that extracting flows from the traffic is already a complex task that has to deal with already packet sampled traffic. The main bottleneck for monitoring system is definitely situated at the packet level, even for systems that provide flow based information. Thus, we instead focus on packet sampling techniques as they are the most likely to significantly impact the computational cost reduction.

A framework to evaluate the impact of packet sampling over several various tools is presented in [START_REF] Silva | A Modular Sampling Framework for Flexible Traffic Analysis[END_REF]. The author discusses the very generic performances of each sampling algorithm. They propose a set of metrics that allow the evaluation of each technique's ability to produce a sampled traffic that efficiently represents the original one. Their results are really generic and do not target the network anomaly detection problem specifically, they thus might not apply in our situation. An extension of this study is presented in [START_REF] Silva | Analysing traffic flows through sampling: A comparative study[END_REF].

Jun et al. [START_REF] Jun | DDoS Attack Detection Using Flow Entropy and Packet Sampling on Huge Networks[END_REF] propose an adaptive sampling technique that intends to keep the amount of sampled traffic below a maximum inspection capability. They use the SDN technology to distribute the sampling over several switches. They perform their evaluation using Snort [START_REF] Roesch | Snort: Lightweight Intrusion Detection for Networks[END_REF] and Suricata [START_REF] Suricata | [END_REF], two detectors based on Deep Packet Inspection. As such techniques focus on the packets content, instead of more encompassing statistical features of the traffic, the technique might not suit all detection algorithms (including AATAC).

In an older paper, Brauckhoff et al. [START_REF] Brauckhoff | Impact of packet sampling on anomaly detection metrics[END_REF] evaluate the impact of sampling over anomaly detection metrics. Their evaluation uses traces containing the Blaster worm to evaluate several detection techniques at various sampling rates. With a detector they propose, and thanks to an entropy-based summarization technique, authors achieve a good detection independently from how high is the sampling rate. The paper also shows that flow-based detection is more impacted by the packet sampling than packet-based detection.

III. AATAC ALGORITHM OVERVIEW

AATAC is a balanced solution to the DDoS detection problem. It provides a real-time detection with low computing resources while still producing pertinent and eloquent results. It is a fully autonomous detector relying on unsupervised machine learning techniques, requiring very few configuration or updates. In this section we present only a brief overview of the algorithm. For more technical details, please refer to the original publication [START_REF] Roudière | A Lightweight Snapshot-Based DDoS Detector[END_REF].

AATAC's processing is illustrated on Figure 1. It is split into two distinct parts: a continuous and a discrete one. The first one quickly handles instances and maintain a data structure representing the traffic in real time. The second one uses this data structure to store, at a regular interval, a snapshot of the traffic. Those snapshots are then stored and used to detect anomalies in the traffic. As those snapshots can be plotted into a set of two-dimensional graphs, they provide the network administrator with a dynamic and pertinent view of the traffic properties when the anomaly occurs.

A. Continuous processing

To build a representation of the traffic at any time, the continuous processing relies on a statistical analysis on the traffic. Its processing is inspired from D-Stream [START_REF] Chen | Density-Based Clustering for Real-Time Stream Data[END_REF], a gridbased clustering algorithm.

To characterize the traffic properties, AATAC uses grids. Those grids are either grouped as histograms, to characterize a traffic feature distribution, or alone, to characterize a global feature of the traffic. Whenever AATAC processes a traffic instance (here a traffic flow and its associated characteristics), AATAC determines a set of grids it falls in. It then increases a density value assigned each of those grids (any instance might be weighted depending on the monitored feature). This processing is done in an incremental manner, which speeds up the processing and limits the computational resources required.

Over time, the density of each grid exponentially decays towards 0. How fast the densities decays is determined by the decay factor (λ). For reasons made explicit in [START_REF] Roudière | A Lightweight Snapshot-Based DDoS Detector[END_REF], λ is set using the R parameter, defined as R = λ ∆T , where ∆T is the parameter defining the time interval between two snapshot creations.

This incremental weighting technique inherently gives much more weight to recent instances in the model than to the older ones. Thus, at a given instant, all grids together form a shortterm complete characterization of the traffic.

B. Discrete processing

The discrete processing is executed at a regular interval ∆T , it can be split into three steps. First, it updates the continuous processing data structure, it then creates a traffic snapshot and finally detects anomalous behaviours.

As the continuous processing is incremental, the whole set of grids' densities might not create a consistent characterization at a given instant. The grids' densities are thus updated to be consistent with each other. Also, to avoid the number of stored grids to overgrow in the memory, several grids having a density lower than a really low value D l are removed from the data structure.

For performance reasons, AATAC's data structure cannot be stored as is. Grids organized as histograms are thus simplified, and transformed into histogram prototypes, a lighter and faster to process data structure. The traffic snapshot is created by storing all histogram prototypes along with the up-todate densities characterizing global traffic features. This set constitutes the snapshot features.

Finally in the anomaly detection phase, AATAC compares the lastly created snapshot to the set of N last snapshots. This is done using the k Nearest-Neighbour (kNN) algorithm applied for each analyzed snapshot feature. kNN produces an outlierness score that AATAC uses to determine how different is the lastly created snapshot from the previously created ones. Note that the scores are normalized.

The final anomaly detection is performed by detecting if the kNN score goes over a given threshold for any snapshot feature. If so, an alarm is raised. As the histogram prototypes and the global densities can be plotted, this alarm is given to the network administrator along with a set of graphs allowing him the have a pertinent and dynamic view of the traffic features when the anomaly occurs.

IV. EVALUATING THE IMPACT OF SAMPLING ON THE DDOS DETECTION

In this section, we describe the testbed we used to evaluate AATAC's performances over sampled traffic.

A. Sampling techniques

Basically, a packet sampling algorithm aims at selecting whether or not a packet should be sampled for further analysis. To evaluate AATAC's performances in various situations, we used five different sampling techniques with several sampling rate: systematic count-based, probabilistic, 1-out-of-N, systematic time-based and random time-based. Those techniques can be split into two categories: count-based or time-based. They cover the full range of common techniques for traffic sampling.

1) Count-based selection techniques: Count-based selection techniques give all packets the same probability to get sampled, independently from their content or arrival time. Statistically, this implies selecting one packet out of N .

The first approach we use is the systematic approach. It consists in systematically selecting a packet every N packets. This approach is pretty straightforward to implement and gives an accurate representation of the traffic in most situations. A traffic-wide feature's density Fig. 1: An overview of the AATAC algorithm However, it might suffer from a bias if the monitored features exhibit a periodic behaviour.

The random probabilistic sampling runs a random test whenever a packet arrives and selects it with a given probability p. As it is does not require an analysis of the packet's content, our approach uses a uniform probabilistic sampling that selects each packet with the same probability.

The last random count-based technique we use is the nout-of-N sampling. Every N packets, this technique chooses a random set of n indexes between 0 and N -1. Those indexes correspond to the indexes of the n packets that will be selected among the next N packets. In most cases, n is set to 1. The 1-out-of-N sampling is easier to implement while showing similar performances as any other n values.

2) Time-based techniques: Time-based techniques operate by defining a set of dates when a packet selection should be triggered. They generally do that by computing a waiting time interval between each selection. Unlike count-based sampling, time-based sampling does not produce an output that is proportional to the input in terms of number of packets. They instead smooth the variations of the traffic, which might not suit all detection techniques.

If the time between two selections is constant, such approach is said systematic. Like the count-based systematic approach, this technique might suffer from a bias if the traffic exhibits a periodic behaviour.

Other time-based techniques might randomly select the time to wait between two triggers. The technique we use generates an inter-selection interval following an exponential distribution.

3) Other techniques: Various other techniques have been proposed in the literature. For example, adaptive sampling techniques adapt the sampling rate depending on the load of the system (CPU, memory, bandwidth...). Other techniques, said hybrid, use a combination of two other techniques. While being interesting, those techniques do not suit our requirements as they introduce several other parameters that would have made difficult a generalization of the results. Also, such techniques are in fact not common in an industrial context. They require either a knowledge database, or setting up a feedback loop from a performance monitoring system to the sampling hardware, which is not easy to set up and maintain.

B. The testbed

To evaluate the performances of AATAC, we implemented it using C++. The experiments are run on a standard computer, featuring a 3.00GHz Intel Xeon CPU (E5-2623 v3). It features 8 cores but our implementation does not handle multithreading.

In the context of the ONTIC project, we built the Syn-thONTS labelled dataset to evaluate our algorithm. It contains a one hour long, payload-free and anonymized real traces captured at the entrance of a large cloud service provider. In an emulated network, we generated 12 realistic attacks and inserted them into the dataset. This set mainly includes DDoS attacks, such as fraggle attacks, smurf attacks, Syn flooding and UDP flooding attacks. The dataset is publicly available on the website of the ONTIC project [START_REF] Ontic | [END_REF].

We sampled the traces thanks to a custom tool called PCAPsampler [START_REF] Pcapsampler | [END_REF]. For count-based algorithms, we used the sampling rate of one packet out of 500, 1000, 2000, 5000 or 10000 packets. For time-based algorithms, we picked one packet every 1, 0.1, 0.01 or 0.001 seconds. As the average packet rate is around 67k packets per-seconds in the dataset, this corresponds to values between picking one packet out of 67000, to one out of 67. According to our industrial partners, a sampling rate of one out of 2000 is a common value used on operation. Those value ranges were thus chosen considering this information, but still wide enough for significantly covering different sampling rates.

Regarding the parameters of AATAC, we used the parameters exhibiting the better performances in our previous evaluation [START_REF] Roudière | A Lightweight Snapshot-Based DDoS Detector[END_REF]. Thus, we set N = 500, ∆T = 1 s and R = 0.9. To adapt to the sampling rate, the computed instances weights are multiplied according to the sampling rate.

While our implementation of AATAC is able to directly operate from pcap files, this is not possible with FastNetMon. We thus had to replay the whole set of traces (the 1h long traces multiplied by the 12 attacks to evaluate) to evaluate its accuracy. As this requires some time, we only evaluated FastNetMon's performances with the 1-out-of-N sampling algorithm considering a sampling rate of one packet out of 2000.

C. Evaluation methodology

To measure the accuracy of the detector, we run both implementations over the sampled traces. We consider an alarm as a true positive if it is raised while the anomaly is occurring, a false positive if it is not. To analyse the results, we use the well known Receiver Operating Characteristic (ROC) curve. However, as they suffer from the base-rate fallacy [START_REF] Bar-Hillel | The Base-Rate Fallacy In Probability Judgments[END_REF], we complete our evaluation by plotting the IDS operation curve of the detector. This method proposed by Nasr et al. [START_REF] Nasr | Performance Analysis of Wireless Intrusion Detection Systems[END_REF], consists in plotting the positive predictive value (PPV ) along with the false positive rate. Each plotted curve is compared to the zero reference curve (ZRC ), the operation curve of an ideal detector that should detect all anomalies while producing an increasing number of false positives.

The intrusion detection effectiveness (E ID ∈ [0, 1]), is then extracted from this curve. It corresponds to the normalized area between the actual IDS operation curve and the ZRC for a FPR between 0 and a maximum acceptable false positive rate T F P . The lower the E ID , the more effective the detector.

To evaluate how sampling impacts the computational resources required to run the algorithm, we measure how much time our implementation spends into either the continuous or the discrete parts of the algorithm. For the continuous part, we plot the time required to process one second of traffic, and for the discrete part, we measure the time required to create a single snapshot.

V. RESULTS

A. Detection accuracy

Figure 2 shows the ROC curves we obtain for the five sampling algorithms. All count-based sampling techniques achieve good results, but the 1-out-of-N technique seems to produce better and more stable results than other ones. From the time-based sampling techniques, the systematic sampling achieves the best results. For high sampling rates, the systematic time-based approach seems to produce better results than the unsampled traffic while for lower sampling rates, with one packet every 100ms or 1s, the results are not as good. Indeed, those sampling rates correspond to one packet every 6700 or 67000 packets. The count-based sampling algorithms produce better results for similar sampling rates, as they detect more attacks for equivalent false-positive rate. As we can see from the curves, once the false-positive value reaches a given value, AATAC seems to perform better with the sampled traffic than with the unsampled one. As stated before, we multiplied the weight of instances depending on the sampling rate. This has increased the detector's sensitivity.

That being said, the results are more significant with the lowest values of the FPR. Indeed, as we run a test every second, even a FPR of 0.001 means raising a false positive every 17 minutes, which is not acceptable in a real situation. With very low false positive rates, such as 10 -4 , AATAC performs better with unsampled traffic or high sampling rates. The IDS curves in Figure 3 confirm this analysis. Considering the PPV values for acceptable false-positive rate (below T F P = 10 -3 ), AATAC's with unsampled traffic IDS operation curves stays above the other ones. However, regardless of the sampling algorithm, the efficiency decrease stays very low. This is shown in Table I: the Intrusion Detection Effectiveness only increases by a very low amount when the sampling rate reduces. This table also confirms that the 1-out-of-N technique has the most stable results, especially for T F P = 10 -3 .

From those results, time-based sampling stands out. With high sampling rates, AATAC achieves a detection that is exceptionally good, looking better than with unsampled traffic. This may be explained by the fact that time-based sampling tends to smooth out the short term variations of the traffic, reducing the FPR. Also, AATAC relies on densities that depend on the number of packet received per second. Making this rate constant makes the detection less dependent on the packet rate itself, and more on other properties of the traffic. This has probably enhanced the detection.

Regarding the impact of sampling over FastNetMon accuracy, our evaluation showed that FastNetMon was completely unable to perform its detection over sampled traffic. Indeed, with standard sampling rate of one packet out of 2000, FastNetMon generated 7 false positives in our evaluation, while detecting no attacks out of 12. Decreasing the threshold only increased the number of false positives generated by Fast-NetMon. The detector's inability to detect DDoS is probably due to the fact it operates at an IP address level. Indeed, at such sampling rates, the amount of data corresponding to a specific IP address is too low for the detector to take a pertinent decision. This implies raising more false positive.

As a conclusion, we can see that AATAC performs an accurate detection independently for the sampling technique used. It even performs better at high sampling rates with timebased sampling. The sampling rate has a very limited impact on the quality of AATAC's detection, which is not the case for FastNetMon, which was totally unable to operate with sampled traffic at a standard sampling rate.

B. Computing power requirements

The calculated processing time of AATAC for each sampling algorithm are depicted on Figure 4. To process the traffic in real time, the continuous processing was designed to process the traffic with a linear complexity regarding the number of instances. This is confirmed on Figures 4a and4b. As the X axis corresponds to the inverse of how many packets are processed per second, the processing time follows a x → 1 x pattern.

Regarding the discrete processing, we can see on Figures 4c and4d that the sampling has little impact on the time required to create a snapshot. Indeed there is no specific reason why the treatment should be more complex with a snapshot built from a sampled traffic than with unsampled ones. However, we can see a slight increase of the processing time when few packets per second are sampled. This is probably due to the fact that a stronger sampling implies that the impact of each instance has to be higher on the model. This thus produces a more varying characterization of the traffic which might create more complex histogram prototypes. This impact is however minimal.

We can also see from the several curves that only the sampling rate has an impact on the processing time, while the sampling technique has none.

This evaluation proved that using AATAC with sampled traffic is possible and allows a significant resource consumption reduction, proportional to the sampling rate.

VI. CONCLUSION

In this paper, we evaluated the AATAC DDoS detector's performances over sampled traffic. For sampling rates going up to 1 packet out 2000, AATAC has shown an almost unchanged detection accuracy while benefiting from a substantial reduction of the required computational resources. From the several sampling techniques used, count-based sampling techniques produce results less dependent on the sampling rate (especially for the 1-out-of-N technique). However, at very high sampling rates, time-based sampling techniques tend to produce better results, as they smooth the short-term variations of the traffic. For further studies, it thus would be interesting to study a new sampling technique. A ideal one would probably use an adaptive mechanism, combining both time-based sampling for high sampling rates and count-based for lower ones.

  Fig. 2: ROC curves for the several sampling algorithms and multiple sampling rates.

Fig. 4 :

 4 Fig. 3: IDS operation curves for the several sampling algorithms and multiple sampling rates.

TABLE I :

 I Intrusion Detection Effectiveness for all 2 sampling techniques and several values of T F P .

This processing is often performed on the switches/routers whose main task is to switch/route packets, and are then largely loaded by this main task. Other tasks must then consume resources parsimoniously.


Marc Bruyère

Eder Gianni Antichi

L Fernandes

Rémy Lapeyrade

Steve Uhlig

Philippe Owezarski

Andrew W Moore

Ignacio Castro

Marc Bruyere

Eder L Fernandes

Remy Lapeyrade

Rethinking IXPs' Architecture in the Age of SDN

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Internet eXchange Points (IXPs) are a central element of the Internet ecosystem: IXPs can carry huge traffic volumes and interconnect a multitude of networks of different types [START_REF] Ager | Anatomy of a Large European IXP[END_REF], [START_REF] Chatzis | On the Benefits of Using a Large IXP as an Internet Vantage Point[END_REF]. The fundamental service provided by IXPs is a Layer-2 neutral facility where heterogeneous networks exchange IP traffic. While IXPs are the ideal vehicle to extend the benefits promised by Software Defined Networking (SDN) to the interdomain level [START_REF] Stringer | Cardigan: SDN distributed routing fabric going live at an Internet exchange[END_REF], [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF], [START_REF] Kumar | A softwaredefined flexible inter-domain interconnect using onos[END_REF], reliability and scalability are essential aspects of an IXP that cannot be compromised by the introduction of SDN. Consequently, transforming IXPs from their legacy design into SDN-enabled fabrics is plagued with challenges. In particular, the impact of control channel disruptions or outages can cause severe disturbances, potentially affecting hundreds of networks and huge traffic volumes [START_REF] Vu | How RTT Between the Control and Data Plane on a SDN Network Impacts on the Perceived Performance[END_REF], [START_REF] He | Measuring Control Plane Latency in SDNenabled Switches[END_REF], [START_REF] Giotsas | Detecting Peering Infrastructure Outages in the Wild[END_REF]. Moreover, control plane failures in large scale deployed SDN networks outweigh largely data or management ones combined together [START_REF] Govindan | Evolve or Die: High-Availability Design Principles Drawn from Googles Network Infrastructure[END_REF].

In this paper we propose Umbrella, a novel approach to IXP fabric management that can be deployed in any IXP topology to reduce the risks of a fabric excessively dependent on the control plane. Umbrella leverages SDN programmability to tackle in the data plane part of the control traffic, removing the actual MAC learning mechanism in legacy IXP networks. Specifically, the broadcasted Address Resolution Protocol (ARP) traffic is directly programmed within the data plane. Umbrella also uses a Layer-2 encoded path to minimize resource consumption, management cost and extend scalability. This approach greatly simplifies the management of the fabric: the only role of the controller is supervising the network by leveraging its global knowledge. Umbrella complements previous SDN architectures for IXPs in two manners. First, Umbrella supports the correspondingly enriched (and more complex) IXP through a more reliable and scalable fabric, in a similar fashion as [START_REF] Antichi | ENDEAVOUR: A Scalable SDN Architecture for Real-World IXPs[END_REF] for iSDX [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF]. Second, Umbrella's implementation supports SDN-enabled IXP architectures beyond the single hop IXPs (iSDX only allows single-hop fabrics), ensuring its applicability to any IXP topology. We envision SDN-enabled IXPs that enhance the controller's role as an intelligent supervisor, rather than an active and dangerously critical decision element. Umbrella is the first step in this direction.

The main contributions of this paper are:

• We introduce the Umbrella architecture and show how it leverages SDN programmability within the data plane. • We show how Umbrella complements the current solutions for SDN-enabled IXPs [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF] and allows their implementation in multi-hop IXPs, while reducing the risk of data plane disruptions. • We present how to incrementally deploy Umbrella and demonstrate its practicality, by reporting on its deployment in two real IXPs: TouSIX and NSPIXP-3. • We open source the Umbrella implementation 1 , including the IXP-manager application that generates the Umbrella logic and BIRD (Bird Internet Routing Daemon) configuration for the Route Server (RS).

II. BACKGROUND AND MOTIVATION

This section introduces IXPs' architecture, their main components -ARP-sponges and Route Servers (RSs)-and presents a motivating example for the proposed SDN-enabled fabric.

A. The IXP environment

IXPs are interconnection fabrics where multiple networks (i.e, members) meet to exchange traffic, frequently in huge volumes [START_REF] Ager | Anatomy of a Large European IXP[END_REF]. IXPs are typically implemented as a simple Layer-2 broadcast domain to which member networks connect BGP-speaking routers and exchange IP traffic. The network of an IXP is composed of edge and core switches [START_REF] Ager | Anatomy of a Large European IXP[END_REF]. The former connects to the IXP member routers while the latter interconnects physical IXP locations and aggregates the IXP traffic. IXP topologies range from single-hop cores, i.e., one core switch (e.g., AMS-IX, DE-CIX) to multiple-hops cores (e.g., LINX, MSK-IX).

Legacy IXPs typically operate at the Ethernet level and are rarely involved in routing decisions. To exchange traffic, IXP members need to know each other physical address, i.e., MAC address, which they learn using the ARP protocol. However, ARP traffic in big IXP fabrics can be large enough to compromise low-capabilities routers [START_REF] Wessel | Effects of IPv4 and IPv6 address resolution on AMS-IX and the ARP Sponge[END_REF]. Due to congestion during ARP storms, re-establishing BGP connections might be necessary, causing severe disturbances at the IXP which can even result in connectivity disruptions [START_REF] Vu | How RTT Between the Control and Data Plane on a SDN Network Impacts on the Perceived Performance[END_REF], [START_REF] He | Measuring Control Plane Latency in SDNenabled Switches[END_REF]. The ARP traffic volume grows even higher during network outages [START_REF]FranceIX outage[END_REF], when many routers attempt to resolve the IP addresses of peers that are not available.

1) ARP-Sponge: The state-of-the-art solution to ARP storms is the ARP-Sponge 2 . An ARP-Sponge Server limits the ARP traffic if it exceeds a certain limit. When the number of ARP requests for an IP address reaches the threshold (e.g., because an interface is down an does not respond), the ARP-Sponge server sponges such IP address: the server replies with its own MAC address to the ARP requests of that node and from there on all the ARP traffic sent to that node is instead sent to the ARP Sponge server. When the ARP-Sponge server receives traffic from a sponged IP address, it ceases to sponge such IP address. Although the ARP-Sponge mechanism prevents the escalation of ARP traffic, it suffers from multiple limitations:

• Single point of failure: while several ARP-Sponge servers could run in parallel, this would add complexity. • ARP Sponges do not eliminate all unwanted ARP traffic.

• To determine whether an IP address is reachable again, ARP-Sponges rely on heuristics that require broadcasting or flooding, which consumes routers' resources. • The ARP-Sponge server might fail to notice that an interface is up again. The server periodically ARPs for sponged addresses, when an IP replies the server unsponges it. It may happen however that the device that replied comes back but the server fails to notice it. 2) Route Servers (RS): Originally, each BGP node connected to the fabric had to establish BGP peerings (using TCP connections) with every other IXP member to obtain information about the networks' prefixes reachable at the IXP and exchange traffic. As IXPs grew in size [START_REF] Cardona | IXP Traffic: A Macroscopic View[END_REF], this solution implied keeping too many BGP sessions, with the associated administrative overhead, operational burden, and the risk of pushing IXP members' routers to their limit. IXPs introduced Route Servers (RSs) to address this problem [START_REF] Richter | Peering at Peerings: On the Role of IXP Route Servers[END_REF]. RSs store all the incoming route information from IXP members and forward it without modification to the other members. Thanks to the RS, an IXP member can receive all the routing information available at the IXP through a single BGP session. In particular, BIRD 3 is an open-source Internet routing daemon for UNIX-like platforms and the most popular RS at IXPs (e.g., LINX or DE-CIX) [START_REF] Richter | Peering at Peerings: On the Role of IXP Route Servers[END_REF].

B. The case for a stronger control and data plane separation

Previous work has shown how OpenFlow (OF) [START_REF] Mckeown | OpenFlow: Enabling Innovation in Campus Networks[END_REF] could be deployed at the exchange [START_REF] Boteanu | Minimizing ARP traffic in the AMS-IX switching platform using OpenFlow[END_REF], [START_REF] Pepelnjak | Could IXPs Use OpenFlow to Scale?[END_REF], [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF] using an IXP 2 ARP-sponge manual: http://ams-ix.net/downloads/arpsponge 3 http://bird.network.cz/ fabric with a central controller for all the peering routers at the IXP. In such architecture, the SDN controller would be co-located with the RS to ensure that the SDN and BGP control planes can talk to each other with a minimal delay [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF]. Despite advantages such as richer policies, one challenge remains: data plane issues may affect control plane messages, leading to a slow or unresponsive control plane, further aggravating the effect on the data plane. The critical problem resides in the centralized ARP-proxy: delays in the control channel might lead to all the connection oriented mechanisms (i.e., BGP, TCP) failing. For example, if the ARP messages of a peering router suffer delays to reach the SDN controller inside the IXP fabric, all the BGP sessions between such router and its peers would also suffer, forcing (in the worst case scenario) establishing new connections. Note that while a distributed ARP proxy could alleviate some of this issues, OF does not support such feature.

We show in Fig. 1 the disruption caused by ARP delays on the data plane or BGP sessions. We emulated the above SDN scenario of a central controller co-located with the RS on Mininet [START_REF] Handigol | Reproducible network experiments using container-based emulation[END_REF]. We instantiated virtual containers acting as peering routers: one for the RS and two more working as client hosts directly connected to one peering router each. We represented the IXP as a single open vSwitch coupled with a Ryu [START_REF]Ryu controller[END_REF] controller acting also as an ARP-proxy, as in [START_REF] Boteanu | Minimizing ARP traffic in the AMS-IX switching platform using OpenFlow[END_REF], [START_REF] Pepelnjak | Could IXPs Use OpenFlow to Scale?[END_REF], [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF]. Fig. 1 shows that, even a small delay of a few tens of milliseconds for ARP messages may trigger much larger disruptions on the data plane or BGP. Given the large volumes of traffic and IXPs' critical role, such disruptions are not acceptable [START_REF] Vu | How RTT Between the Control and Data Plane on a SDN Network Impacts on the Perceived Performance[END_REF], [START_REF] He | Measuring Control Plane Latency in SDNenabled Switches[END_REF]. To fully benefit from the advances brought by the existing literature [START_REF] Boteanu | Minimizing ARP traffic in the AMS-IX switching platform using OpenFlow[END_REF], [START_REF] Pepelnjak | Could IXPs Use OpenFlow to Scale?[END_REF], [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF], we advocate a stronger separation between the control and the data plane. Our approach is one possible solution towards this stronger separation.

III. A NEW SDN FABRIC FOR IXPS

We now present Umbrella: a new SDN fabric for IXPs, that focuses on the control-data plane dependencies to provide a robust, reliable, and scalable forwarding inside the IXP.

A. No broadcast traffic

IXPs apply strict rules [START_REF]Allowed Traffic Types[END_REF], [START_REF] Hughes | Internet Exchange Point Wishlist[END_REF] to limit the side effects of a Layer-2 shared broadcast domain, e.g., the MAC address of the router with which the member connects to the IXP must be known in advance. Only then the IXP will allocate an ethernet port on the edge switch and configure a MAC filtering Access Control List (ACL) with that MAC address [START_REF]IXP Technical Requirements[END_REF]. The location of all the member's routers is thus known to the IXP. Accordingly, Umbrella eliminates the need of location discovery mechanisms based on broadcast packets, i.e., ARP request or IPv6 neighbor discovery. Umbrella makes unnecessary the active ARP-proxy daemon proposed in previous SDN-enabled IXP solutions [START_REF] Boteanu | Minimizing ARP traffic in the AMS-IX switching platform using OpenFlow[END_REF], [START_REF] Pepelnjak | Could IXPs Use OpenFlow to Scale?[END_REF], [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF]. Note that while [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF] does not rely in broadcast-based discovery mechanisms, it still heavily depends on the ARP-proxy for mapping virtual to actual MAC and IP addresses. Umbrella makes on-the-fly translation of broadcast packets into unicast by exploiting the OF ability to rewrite the destination MAC address of a frame matching a rule [START_REF]OpenFlow Switch Specification[END_REF].

We propose a label-oriented forwarding mechanism to reduce the number of rules inside the core of the IXP fabric. Umbrella edge switches explicitly write the per-hop port destinations into the destination MAC field of the packet. The first byte of the MAC address represents the output port to be used by the core switch. With Umbrella, the number of flow table entries per core switch will scale with the number of active physical ports in the switch itself. This is important to guarantee the fabric scalability. While Umbrella's encoding scheme is currently limited to 256 output ports per hop, more bits in the port encoding (thus mapping more physical ports) can be used. We now explain how Umbrella works using the IXP topology in Fig. 2. The path to connect router-a to router-b through core-a traverses all the following ports: 1 and 2 in edge-2, 1 and 3 in core-a and 2 and 1 in edge-3. When the router-a sends an ARP request (i.e., broadcast message), the switch edge-2 receives the frame, rewrites the destination MAC address with the corresponding output ports of the path, 03:01:00:00:00:00, and forwards it to the core switch. Note that the output port of the first hop is not encoded: this port is directly set as an output action following the path encoding action.

Once the frame reaches the core (core-a), it is redirected to output port 3, and then to switch edge-3 (i.e., the forwarding in the core is based on the most significant byte). Finally, edge-3, before forwarding the frame through the output port indicated in the second byte of the MAC address (output port 1), rewrites that field with the real MAC address of router-b.

When the source and destination are directly connected to the same edge switch, no encoding is needed, and the the edge switch directly replaces the broadcast destination address with the target MAC destination address. In an IPv6 scenario, the OF match pattern indicated in the edge switch needs to be on the IPv6 ND target field of the incoming ICMPv6 Neighbor Solicitation packet [START_REF] Narten | Neighbor Discovery for IP version 6 (IPv6)[END_REF]. The matching table on the edge switch should maintain an association between IPv6 addresses and their location, as in the IPv4 case.

B. A label switching approach

Umbrella's forwarding mechanism allows reusing legacy switches in the core, limiting the burden (and costs) of upgrading the hardware. A core switch only needs to forward packets based on simple access filtering rules, whereas the edge switches need OF-like capabilities to rewrite the layer-2 destination field. While this approach is directly applicable to single-core IXP fabrics, it is not applicable to multiplehops fabrics. With a single hop, the core switch would expect the output port to be encoded in the most significant byte of the destination MAC address. In the multi-hop case, since a packet can traverse multiple core switches, a new encoding scheme is needed to distinguish the output ports at different core switches. This is a fairly common case in hypercube-like topologies (e.g., LINX, MSK-IX).

Adapting Umbrella to multi-hop IXPs is far from trivial. An encoding of the Layer-2 destination address where the most significant byte refers to the output port of the first core switch, the second byte to the second switch, and so on, is infeasible: unfortunately, a core switch might be the first or the second on the path depending on the route. Another solution could be using the input port of the frame in the forwarding rules installed in the core switches. With the input port, it is possible to locate the switch on the path and therefore look at the correct byte in the Layer-2 destination address. Alas, this approach may not work in arbitrary topologies. Moreover, this mechanism will lead to a rule explosion in the core, as the number of forwarding entries grows quadratically with the number of possible input ports. Instead, to deal with multi-hop IXPs, Umbrella leverages source routing in the following manner. Initially, the first edge switch selects the path. Then, an ordered list of output ports is encoded in the destination MAC address as a stack of labels. Finally, each core node processes the frame according to the value on the top of the stack and pops it before forwarding the frame. With this configuration each switch only needs to look at the most significant byte of the address, regardless of its position in the path toward the destination. Popping out from the MAC destination address, the last label used requires header rewriting capabilities, making this solution feasible only for OF-enabled core switches. In particular, every core switch must have two action tables: forwarding and copy-field 4 .

C. Umbrella and Route Servers

Umbrella handles the forwarding of BGP traffic for both the usual bilateral peerings and RSs. For bilateral BGP sessions, the TCP connection is treated as pure data plane traffic crossing the IXP, and the traffic is handled by the switch rules, without any control plane intervention. For the RSs, the BGP traffic entering the fabric is directed to the RS with a single rule at the edge switch, while the egress traffic is handled automatically through the existing rules on the edge switches.

D. Failure detection and recovery

Umbrella relies on well known OF features. In particular, Group Fast Failover is the OF 1.1 mechanism to react to link failures [START_REF] Sharma | Enabling Fast Failure Recovery in OpenFlow Networks[END_REF]. A fast failover group table can be configured to monitor the status of ports and interfaces and the switch forwarding actions, independently of the controller. Recovering from a data plane failure is more challenging. In this scenario an active probing of the data plane status from the controller is needed (for a detailed discussion see [START_REF] Kreutz | Software-defined networking: A comprehensive survey[END_REF]). In particular, the Umbrella controller can be instructed to implement the Local Link Discovery Protocol (LLDP) [START_REF] Sharma | Enabling Fast Failure Recovery in OpenFlow Networks[END_REF] or the Bidirectional Forwarding Detection (BFD) protocol [START_REF] Adrichem | Fast Recovery in Software-Defined Networks[END_REF]. Once a data plane failure has been detected, the Umbrella controller changes only the edge switch configuration with a fallback path.

IV. EXPERIMENTAL EVALUATION

In this section, we evaluate the Umbrella architecture. We first estimate the average number of flow rules needed at the edge if Umbrella was to be implemented at different large Layer-2 neutral IXPs (Section IV-A). Then we calculate the impact of the ARP proxy on the data plane performance and compare it with the Umbrella approach.

A. Flow table occupancy at the edge

Mapping the broadcast destination MAC address to the path inside the fabric requires additional flow rules at the edge of the fabric. This section provides a scalability analysis in terms of the required number of rules. From an ingress traffic point of view, Umbrella requires the OF switches at the edge to store 4 OF 1.5 specifications allow copying and rewriting part of a header field. three entries per peering router connected to the fabric. This is a necessary and sufficient condition to enable routing in any situation (i.e., IPv4 ARP, IPv6 Neighbor Solicitation and data plane traffic). All three entries share a common action in the rewriting and forwarding process due to the common treatment of Umbrella for location discovery and data plane traffic. From an egress traffic point of view, each edge switch needs to rewrite the MAC destination field of the received frame with the correct target. As the value to be inserted in the destination MAC field depends on the output port of the edge switch, the number of rules depends on the number of peering routers connected to the edge switch. The total number of rules for an edge switch is then the sum of the ingress and the egress ones. Fig. 3 shows the average number of rules per edge switch if Umbrella was to be implemented at different large layer-2 neutral IXPs. As the number of rules also depends on how many peering routers are connected to the fabric, we rely on PeeringDB to asses this extent [START_REF]PeeringDB[END_REF], [START_REF] Lodhi | Using peeringDB to Understand the Peering Ecosystem[END_REF].

The Umbrella architecture is applicable in today's IXPs, as shown by the relatively low number of flow entries required on Fig. 3. Indeed, today's OF-enabled switches already support flows ranging from a few thousands (e.g., Pica8 switches) to hundreds of thousands (e.g., Corsa and Noviflow switches) 5 .

B. Impact of control channel on the data plane performance

To estimate the impact of an unreliable control channel on the data plane performance, we compared Umbrella with the ARP-proxy application in a realistic scenario following a three-pronged approach. First, we studied the effect over the Round Trip Time (RTT) of ARPs (i.e., the time required to receive the ARP reply after an ARP request is sent) under artificially induced delays. Secondly, we carried a similar analysis by studying the effects of packet loss. Finally, we evaluated the impact of Umbrella and the ARP-proxy on the fabric's throughput when IXP members send traffic to their peering partners.

1) Experimental set-up:

To demonstrate the advantages of Umbrella and the drawbacks of approaches that rely on the control plane for implementing location discovery mechanisms, we reproduced an SDN IXP with 100 members and a RS controller managed either by Umbrella or an ARP-proxy application. The ARP-proxy and Umbrella are implemented as applications on top of the Ryu controller. Using Mininet [START_REF] Handigol | Reproducible network experiments using container-based emulation[END_REF], we emulated an IXP with a ring topology composed of three Open vSwitches, 102 Quagga BGP routers and one RS This setup mirrors the real TouSIX topology, though with more members. While this provides a representative benchmark, unfortunately, emulating larger scale networks is too resource intensive. In this topology, each peer privately peers with a single customer, and 1/3 of the participants peer openly at the RS. Accordingly, there is a total of 88 simultaneous BGP sessions in the network. Larger numbers of peering sessions at the RS resulted in very slow BGP convergence due to the high number of prefixes exchanged. The generation of ARP traffic in our experiments is like in a real IXP: ARP requests are triggered when a BGP connection is started and requests are issued until a reply is received. When the IXP members exchange traffic ARP messages are also sent across them. The experiments were performed in a machine with 16 Intel(R) Xeon(R) CPUs @ 3.00GHz and 32GB of RAM.

2) Impact of delays in the control channel: We now show in Fig. 4a the median RTT for the pair of all ARP requests/replies when the control channel experiences different delays. To give a better idea of the distribution, the whiskers indicate the median RTT +/-the Median Absolute Deviation (MAD). The plot shows that delays in the control plane translates into increasing RTTs of ARPs, which result in delayed BGP connections, ultimately undermining the ability of peers to exchange through the fabric.

For Umbrella, the case is rather different. Fig. 4b shows that, as expected by design, the control plane does not affect the ARP packets' RTTs. With Umbrella, the ARP requests are sent directly through the fabric without interaction with the central controller. The broadcast traffic is converted into unicast traffic and directly forwarded to the destination. It also eliminates the typical flooding in layer 2 networks when a switch has not learned yet the port associated to a MAC address, therefore avoiding bandwidth wasting.

3) Impact of packet loss on the control channel: Here, we study the impact of losses on the control channel on the RTT of the ARP packets. Fig. 5a and5b show the median RTT, in both the ARP-proxy approach and Umbrella, respectively. Fig. 5b stresses the strong separation between the control and data plane in Umbrella: there is almost absolute insensitivity between the packet loss rate on the control channel and the RTT of the ARP packets. In Fig. 5a, we show how the ARP-proxy solution again suffers more and more as the loss rate on the control plane increases. The ARP-proxy solution suffers because the TCP connection between the OF-enabled switch and the controller is sensitive to packet losses, hence notably increasing the response time to the ARP requests.

4) Impact on the data plane throughput: We now evaluate the potential impact of the control plane on the data plane throughput inside of the IXP fabric. Using the same setup from the previous experiments, peers now generate TCP packets using the iperf tool 6 and exchange traffic with the others routers present in the fabric. There are 88 iperf sessions, equal to the number of peering sessions. For this experiment, the links between the routers of the members and the switches of the IXP are limited to 50Mbps, as large traffic loads could overwhelm the emulator and result into a bottleneck. Consequently, the maximum throughput that could be observed is 4400Mbps. Differently from the previous two experiments, we introduce no added loss or delay to the control plane, instead we focus on the throughput that an Umbrella vs. and ARPproxy enabled IXP can deliver. Fig. 6 depicts the median throughput delivered by the IXP fabric for 10 trials, each executed for a period of 10 minutes. The whiskers indicate the median +/-the MAD. Because of the slow start of TCP, in the initial seconds of the experiment the throughput is low both for Umbrella and the ARP-proxy. As the experiment progresses and the TCP sessions increase their congestion window, Umbrella delivers a higher throughput than the ARP-proxy during the whole experiment. Furthermore, the upper whisker shows that Umbrella nearly reaches the maximum possible throughput. Considering the high level of performance that IXPs are expected to achieve, the results clearly show the advantages of Umbrella over control plane dependent solutions. V. EXPERIENCE WITH REAL DEPLOYMENTS In this section, we show our effort in bringing the research outcomes from Umbrella into operation. Specifically, we show how Umbrella has been introduced in TouSIX and NSPIXP-3 OSAKA.

Toulouse IXP. TouSIX is a non-profit neutral IXP organization founded in 2005 (Toulouse, France), providing an interconnected network infrastructure with four Points of Presence (PoPs) and connected with the FranceIX (Paris) and LyonIX (Lyon) IXPs. From May 2015 it is the first European IXP to fully leverage OF for its day-to-day operations. The TouSIX architecture is based on our Umbrella approach, providing us with operational experience of a real-world deployment.

The management of the legacy IXP was a complex task with problems resulting from wrong configurations done by the members (i.e., unwanted loops) and several broadcast storms. The lack of an appropriate monitoring infrastructure as well as the irregular incidence of these problems made the detection of these events hard. Umbrella was implemented to overcome these issues as well as to improve the stability and manageability of the whole infrastructure.

The new TouSIX OF-enabled topology has three PoPs with one OF-switch. Each PoP acts as an edge and there are no core switches. The three switches have been programmed according to the Umbrella principles and replace the previous legacy switches. A long testing period was required to validate the system configuration and its stability before going into production, with the first test campaign dating December 2014. At that time, only Pica8 switches had full hardware integration with the Open vSwitch agent, making the Pica8 P-3290 switches the natural choice. Unfortunately, the latest PicOS version available during the initial part of the tests did not allow installing rules matching the ipv6 nd target field. This option is necessary to enable the Umbrella scheme for IPv6 traffic (see Section IV-A). In addition, the software was flushing the entire flow table during hardware reboots or when the connection to the controller was lost: close collaboration with Pica8 solved these issues, triggering a new PicOS version (2.6).

The new OF-enabled fabric went live in May 2015. The OF Pica8 switches are placed on the top of the existing Cisco switches. While the former are being used for the data plane traffic, the latter have been kept to transport the control plane traffic. Thanks to the new switches, the transmission bandwidth has been improved by one order of magnitude on 2 out of 3 connections between edge switches. Member cables have migrated from the legacy Cisco switches to Pica8, with only a few seconds of service interruption per member. The Pica8 switches have been configured to not flush the flow table when the OF Agent looses connectivity with the OF controller. The deployment leverages Ryu [START_REF]Ryu controller[END_REF], the NTT Labs opensource controller, while parallel development is underway for the ON.Labs' ONOS controller [START_REF]ONOS[END_REF].

The Umbrella TouSIX fabric has been running seamlessly and with a reduced dependence on the network administrator. There is no more broadcast ARP traffic (see Table I), freeing the 10 members from receiving undesired traffic while allowing them to announce 399 IP prefixes. Umbrella reduced the total average volume of ARP traffic flowing through the fabric by 97%. Note that Umbrella solves here an additional problem: with this topology, a simple layer-2 approach combined with an ARP-proxy and an ARP-sponge would require of a spanning tree protocol to avoid loops. Table II shows the peak and average traffic of TouSIX and NSPIXP-3. TouSIX has a total of 14 nodes connected, including both routers and servers. The switches must include rules for the MAC destination forwarding, ARP target routing, and for the ICMPv6 ND target routing. As only IPv4 is on production at this stage, the total number of rules per switch is 28. The Umbrella design also frees capacity in the switches that have a CPU utilization with virtually no spikes, stable around the 8%. Differently, the TouIX switches did suffer spikes of CPU usage reaching the 100% of their capacity, which resulted from issues related to the convergence of different control plane protocols.

In terms of recovery time, when a switch with an empty table connects to the controller, installing the corresponding flow rules (i.e., the database polls and send rules to the switch) takes about 5.6s. and a hard reboot is about 64s. Note that due to the design of Umbrella, connecting a new router to TouSIX is a "plug & play" operation: the administrator automatically installs the corresponding flow rules for any "approved" router because the controller already knows its configuration. Consequently the time since an approved router is connected and it is fully operative is negligible. On the contrary, if a router is connected without approval of the administrator, such router will see all its traffic dropped. As the TouSIX-Manager operates the topology with proactive procedures, if the link in the control path is lost, all the rules already deployed on any switch remain. We tested the effect of turning off the TouSIX manager, to verify that traffic statistics where as usual without impacting the fabric operations.

NSPIXP-3 OSAKA. Launched by the WIDE project in 1997, NSPIXP-3 is the oldest IXP in Osaka (Japan). NSPIXP-3 has a single site, 10 nodes connected, and uses one switch with two VLANs, one for local peering and a second VLAN to interconnect the Osaka and Tokyo domain. In July 2017 a predeployment of Umbrella in combination with Faucet started. FAUCET [START_REF] Bailey | Faucet: Deploying SDN in the Enterprise[END_REF] is a very compact open source OF controller, enabling network operators to run networks the same way they do server clusters. The final migration to production has been achieved in December 2017. This deployment confirms the real-world applicability of Umbrella and demonstrates its success in almost eliminating the ARP traffic (see Table I).

VI. RELATED WORKS

Introducing OF at the IXP world is a recent idea, [START_REF] Gupta | SDX: A Software Defined Internet Exchange[END_REF], [START_REF] Gupta | iSDX: An Industrial-Scale Software Defined Internet Exchange Point[END_REF] developed a SDN-based eXchange point (SDX) to enable more expressive policies than legacy solutions while scaling to hundreds of participants while achieving sub-second convergence in response to configuration changes and routing updates. To achieve this, the multi-table version of the iSDX prototype considers a scenario where all the participants are connected through a single switch. In reality, however, there might be multiple hops within the IXP. Umbrella and iSDX are complementary designs: Umbrella, can support the iSDX architecture by directly forwarding the location discovery packets to each participant. In particular, after the decision process for the egress port performed by iSDX, the destination MAC address can be encoded using Umbrella. The packet is then delivered following the path encoded in the destination MAC. However, the proposed integration would require an additional OF switch to send ARP requests, for virtual next hops that would be handled by an ARP proxy. As Umbrella turns every broadcast ARP into unicast, the ARP requests can have their destination encoded within the path to the ARP Proxy. This would enhance flexibility, because the ARP proxy does not need to be connected to a specific switch in the IXP fabric 7 .

ENDEAVOUR [START_REF] Antichi | ENDEAVOUR: A Scalable SDN Architecture for Real-World IXPs[END_REF] leverages the Umbrella scheme to obtain an effective transport layer over a multi-hop topology, by removing all the side effects associated to broadcast traffic, and by enabling efficient access control over the fabric.

The Cardigan project [START_REF] Stringer | Cardigan: SDN distributed routing fabric going live at an Internet exchange[END_REF] implements a hardware based, default deny policy, capable of restricting traffic based on RPKI verification of routes advertised by devices connected to layer3 fabric. While this approach offers the required protection for a stable IXP, it is less suitable for IXPs that wish to remain neutral with regards to IP forwarding.

The ONOS CASToR [START_REF] Kumar | A softwaredefined flexible inter-domain interconnect using onos[END_REF], with its ARP Hygiene also relies on unicasting broadcast packets. However, the primary CASToR objective is to provide flexibility to operators to interconnect through a User Interface and APIs. CASToR can face scalability limitations as they use a single table.

While label switching techniques in combination with source routing has been used in the past, MAC-based routing in OF networks is a fairly new [START_REF] Hari | Path switching: Reduced-state flow handling in SDN using path information[END_REF], [START_REF] Agarwal | Shadow MACs: Scalable Label-switching for Commodity Ethernet[END_REF]. [START_REF] Schwabe | Using MAC Addresses As Efficient Routing Labels in Data Centers[END_REF] show that the destination MAC address can be used as a universal label in SDN environments and the ARP caches of hosts can be exploited as an ingress label table, shrinking the forwarding tables of network devices. [START_REF] Agarwal | Shadow MACs: Scalable Label-switching for Commodity Ethernet[END_REF] demonstrate that, using destination MAC addresses as opaque forwarding labels, an SDN controller can leverage large MAC forwarding tables to manage a plethora of fine-grained paths. Although these approaches have very nice properties for large-scale networks, they still rely on an ARP-proxy mechanism, which involves the limitations already discussed.

VII. CONCLUSION

Umbrella enhances IXP reliability, manageability and scalability. By handling the control traffic directly within the data plane, Umbrella reduces failures/disruptions and complements existing IXP architectures, enabling the deployability of existing SDX architectures in any IXP. We demonstrate the scalability of Umbrella practical and its applicability by reporting on two successful deployment in real IXPs. We see Umbrella as a first step towards SDN architectures less dependent on the control plane, supporting the controller in its role of an intelligent supervisor, rather than as an active and dangerously critical decision point.

Fig. 1 :

 1 Fig. 1: The dependency between control and data plane.

Fig. 2 :

 2 Fig. 2: Typical topology of a medium to large IXP.

Fig. 3 :

 3 Fig. 3: Average number of rules required at the edge.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Median RTT of ARP packets when the control channel suffers different delays (whiskers indicate the median +/-MAD).

Fig. 6 :

 6 Fig. 6: Median throughput in an IXP controlled by an ARPproxy or Umbrella (whiskers indicate the median +/-MAD).

TABLE I :

 I ARP traffic (packets/second), in the legacy & Umbrella IXPs at TouSIX & NSPIXP-3 Osaka.

	ARP traffic	TouSIX Legacy Umbrella	NSPIXP-3 Legacy Umbrella
	Max	(Pkt/s)	14.96	3	20	2
	Average (Pkt/s)	8.51	0.18	14	0.5
	Min.	(Pkt/s)	1.1	0	12	0

TABLE II :

 II Peak & average traffic (MegaBits per second) at TouSIX & NSPIXP-3 Osaka swicthes.

	Total traffic volume	TouIX	NSPIXP-3
	Peak (Mb/s)	9767	748
	Average (Mb/s)	48	229

http://github.com/umbrella-fabric/TouSIX-Manager

See http://pica8.org/blogs/?p=565, http://www.corsa.com/sdn-done-right/ and http://noviflow.com, respectively.

https://iperf.fr/

The ARP proxy function in iSDX is essential to compress the flows into Forwarding Equivalent Classes.

Acknowledgements

We thank J. Bailey (Google), C. Combes, H. Brunel and L. Guerby (TouSIX), S. Garrison (Pica8), M. Pels and Ariën Vijn (AMS-IX) for their very valuable inputs. This work is supported by the EARL EPSRC project (EP/P025374/1).

Marc Bruyere started his career in 1996 working for Club-Internet.fr, and for Cisco, Vivendi Universal, Credit Suisse First Boston, Airbus/Dimension Data, Force10 Networks, and Dell. He received is Ph.D degree from the LAAS CNRS, his thesis is about Open Source OpenFlow SDN for IXPs. He designed and deployed the first European OpenFlow IXP fabric for the TouIX. Now he is a PostDoc at the University of Tokyo.

Gianni


Marc Prévost

Marc Prévost Remainder

Padé Approximants

Marc Pr Évost

REMAINDER PADÉ APPROXIMANTS FOR THE HURWITZ ZETA FUNCTION

Keywords: 2000 Mathematics Subject Classification. Primary 41A21; Secondary 41A28, 11J72 Padé approximants, Zeta function, Bernoulli numbers. 1

, to construct approximations of the Hurwitz zeta function. We prove that these approximations are convergent on the positive real line. Applications to new rational approximations of ζ(2) and ζ(3) are provided.

Introduction

In [START_REF] Prévost | A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants[END_REF], we gave a new proof of the irrationality of ζ(2) = ∞ k=1 1/k 2 (and also of ζ(3) = ∞ k=1 1/k 3 ) based on an explicit construction of some Padé approximants of the remainder term

R 2 (1/n) = ∞ k=n 1/k 2 with R 2 (z) = ∞ k=0 z 2 (zk + 1) 2 .
More precisely, we have

ζ(2) = n-1 k=1 1/k 2 + R 2 (1/n) The function R 2 is meromorphic in C \ {0, -1, - 1 2 , - 1 3
, • • • } and so is not holomorphic at 0. However, R 2 is C ∞ at z = 0 and may be expressed as a Taylor series R2 (z) = ∞ k=0 B k z k+1 with a zero radius of convergence, B k being the kth Bernoulli number.

The idea was to compute an explicit representation of diagonal Padé approximant P n (z)/Q n (z) = [n/n] R2 (z) ∈ Q(z) of the series R2 (z) with a good estimation of the error term E n (z) = R 2 (z) -[n/n] R2 (z).

At last, we stated the relation ship

Q n (1/n)E n (1/n) = Q n (1/n)ζ(2) -Q n (1/n) n-1 k=1 1/k 2 -P n (1/n) = q n ζ(2) -p n
providing a sequence of rational approximation p n /q n and proving the irrationality of ζ(2).

The surprise was to retrieve the sequences used by Apéry [START_REF] Apéry | Irrationality of ζ(2) and ζ(3)[END_REF] for the same purpose.

The same method applied to the series ζ(3) = ∞ k=1 1/k 3 provides also the Apéry numbers given in [START_REF] Apéry | Irrationality of ζ(2) and ζ(3)[END_REF].

The term Remainder Padé Approximant (RPA) has been used later in a paper written with T. Rivoal [START_REF] Prévost | Remainder Padé approximants for the exponential function[END_REF] for the exponential function.

In this paper, we apply the same method and we prove that some sequence of RPA for Hurwitz zeta function is convergent on the real line.

For a ∈ C, (a) > 0 , the Hurwitz zeta function is defined as

ζ(s, a) = ∞ k=0 1 (k + a) s (1.1)
where the Dirichlet series on the right hand side of (1.1) is convergent for (s) > z s-1 e az 1 -e z dz, where C is is a loop around the negative real axis; it starts at -∞, encircles the origin once in the positive direction without enclosing any of the points z = ±2iπ, ±4iπ,. . . , and returns to -∞. Thus ζ(s, a) can be continued meromorphically to the whole complex s-plane except for a single pole at s = 1 with its residue 1.

If we write the formula (1.1) as :

ζ(s, a) = n-1 k=0 1 (k + a) s + ∞ k=0 1 (n + a + k) s (1.3)
and set

Ψ(s, x) := t 1-s ∞ k=0 t 1 + kt s then ζ(s, a) = n-1 k=0 1 (k + a) s + (n + a) 1-s Ψ s, 1 n + a . (1.4) Another expression of Ψ(s, t) is given in [13, chapt. XIII] by Ψ(s, t) = t 1-s Γ(s) ∞ 0 u s-1 e -u/t 1 -e -u du (1.5)
whose Taylor series is

Ψ(s, t) = ∞ k=0 B k k! (s) k-1 (-1) k t k , (1.6)
which is convergent only at t = 0, where (s) k is the Pochammer symbol ((s

) p := s(s + 1)(s + 2) • • • (s + p -1), (s) -1 = 1/(s -1)). For a given pair (m 1 , m 2 ), let [m 1 /m 2 ] Ψ(s,t) = Q m 1 (t
RP A(n, m 1 , m 2 ) := n-1 k=0 1 (k + a) s + (n + a) 1-s Q m 1 (1/(n + a)) P m 2 (1/(n + a)) . (1.7)
The crucial point is the convergence of this sequence (RP A(n, m 1 , m 2 )) when one or more of the parameters n, m 1 , m 2 tends to infinity. The convergence of Padé approximants is proved for many functions as meromorphic function, Stieltjes function,... [START_REF] Baker | Encyclopedia of Mathematics and its Applications[END_REF]. In this paper,when s is a real positive number, we prove that the function Ψ (s, t) is a Stieltjes function with respect to the variable t. First we need to remind the definition and some properties of Padé approximants.

Padé approximants

Let f be a formal series whose Taylor expansion about t = 0 is

f (t) = ∞ i=0 c i t i .
The Padé approximant [m 1 /m 2 ] f to f is a rational fraction Q m 1 / P m 2 whose denominator has degree m 2 and whose numerator has degree m 1 so that its expansion in ascending powers of t coincides with the expansion of f up to the degree m 1 + m 2 , i.e.

P m 2 (t)f (t) -Q m 1 (t) = O(t m 1 +m 2 +1 ).
(2.1)

The theory of Padé approximation is linked to the theory of orthogonal polynomials as following [START_REF] Brezinski | Padé-type approximation and general orthogonal polynomials[END_REF].

Let us define the linear functionals c and c (p) , p ∈ Z acting on P, space of polynomials by

c : P → R (or C) (2.2) x i → c, x i = c i i = 0, 1, 2, . . . (2.3) and c (p) , x i := c, x i+p = c i+p , i = 0, 1, 2, . . . (c i = 0, i < 0). (2.4)
Then the denominators of the Padé approximants [m 1 /m 2 ] f satisfy the following orthogonality property c (m 1 -m 2 +1) , x i P m 2 (x) = 0 i = 0, 1, 2, . . . , m 2 -1 where P m 2 (x) = x m 2 P m 2 (x -1 ) is the reversed polynomial.

Let us define the associated polynomials

R m 2 -1 (t) := c (m 1 -m 2 +1) , P m 2 (x) -P m 2 (t) x -t , R m 2 -1 ∈ P m 2 -1 , (2.5)
where c (m 1 -m 2 +1) acts on the variable x . Then

Q m 1 (t) = m 2 -m 1 i=0 c i t i P m 2 (t) + t m 1 -m 2 +1 R m 2 -1 (t) (2.6)
where

R m 2 -1 (t) = t m 2 -1 R m 2 -1 (t -1 ) .
If c admits an integral representation with a non decreasing function α, with bounded variation, such as

c i = L x i dα(x), (2.7)
then f is a Stieltjes function and the theory of Gaussian quadrature shows that the polynomials P n , orthogonal with respect to c, have all their roots in the support of the function α.

Moreover, if f satisfies the Carleman's criterion [3, p.240] then all paradiagonal sequences of ([m + J, m]) m Padé approximants with J ≥ -1 converge to f . The aim of the paper is to find, in the case of Hurwitz zeta function, the weight function dα depending on s and to prove that it is a positive function.

The error is defined by

f (t) -[m 1 /m 2 ] f (t) = t m 1 +m 2 +1 P 2 m 2 (t) c (m 1 -m 2 +1) P 2 m 2 (x) 1 -xt . (2.8)
The expression of the error above is understood as a formal one if c is only a formal linear functional [6, chapt. 3], but if c admits the integral representation (2.7) then the error rewrites as :

f (t) -[m 1 /m 2 ] f (t) = f (t) - Q m 1 (t) P m 2 (t) = t m 1 +m 2 +1 P 2 m 2 (t) L x m 1 -m 2 +1 P 2 m 2 (x) 1 -xt dα(x).
(2.9)

In the particular case m 1 = m 2 -1,

f (t) -[m 2 -1/m 2 ] f (t) = t 2m 2 P 2 m 2 (t) L P 2 m 2 (x) 1 -xt dα(x). (2.10) Note that [m 2 + p/m 2 ] f (t) = c 0 + c 1 t + • • • + c p t p + t p+1 [m 2 -1/m 2 ] fp (t) where f p (t) = ∞ i=0 c i+p+1 t i .

Statements of the results

In this section, we will prove that it is possible to compute the weight function underlying the coefficients

B k k! (s) k-1 (-1) k of the function Ψ(s, t) = ∞ k=0 B k k! (s) k-1 (-1) k t k .
The weight function w s will depend on s. For particular values of s (s > 0), w s is a positive weight function and thus Ψ(s, t) is a Stieltjes function depending on the variable t and the convergence of RPA will be proved for t = 1/(n + a). Theorem 1. For any complex numbers s, a, (s) > 0, (a) > 0 and any positive integer

m, m > (s) -1, then ζ(s, a) = 1 a s-1 1 s -1 + 1 2a + ∞ 0 1 a 2 + x 2 w s (x)dx (3.1)
where the weight w s is defined by

w s (x) := 2(-1) m x s Γ(s)Γ(m + 1 -s) ∞ x (t -x) m-s d m dt m 1 e 2πt -1
dt.

We will prove in the section 4 that this formula (3.1) is a consequence of Hermite's formula for the function ζ(s, a).

As proved by Touchard [START_REF] Touchard | Nombres exponentiels et nombres de Bernoulli[END_REF], Bernoulli numbers satisfy

B k = -i π 2 L x k dx sin 2 (πx) ,
where L is the line L := -1 2 + iR and thus can be viewed as moments of order k ≥ 0 of the positive weight function 1/sin 2 (πx) on L. Therefore, for s = 2, the coefficients (1.6) are the moments of a positive weight. For s = 3, these coefficients appear as their derivative (k + 1)B k . The derivative of 1/sin 2 (πx) is a weight function, symmetric around -1/2 whose support is also the line L. But for integer value of s greater than 4, it is no more true for all k ≥ 0. A much more difficult case is the case when s is real.

B k k! (s) k-1 = B k of
Nevertheless, from the previous Theorem, we obtain an integral representation of

B k k! (s) k-1 but for k ≥ 2.
Theorem 2. Integral representation of Bernoulli numbers.

If s is a complex number such that (s) > 0, then for k ∈ N,

B k+2 (s) k+1 (k + 2)! = (-1) k/2 +∞ 0 x k w s (x)dx (if k even) 0 (if k odd) (3.2) with w s (x) := 2(-1) m x s Γ(s)Γ(m + 1 -s) ∞ x (t -x) m-s d m dt m 1 e 2πt -1 dt (3.3)
where m is any integer satisfying m > (s) -1.

Proof We use the expansion of ζ(s, a) in terms of Bernoulli numbers [11, p.160]. We have

ζ(s, a) = a -s + n k=0 (s) k-1 B k k! a -k-s+1 + 1 Γ(s) ∞ 0 1 e t -1 - n k=0 B k k! t k-1 e -a t t s-1 dt
which is valid for (s) > -2n + 1, (a) > 0 and n a positive integer. By comparing this formula with (3.1), we get the formula (3.2).

Remark. The positivity of the weight function is important because it guarantees the convergence of Padé approximants. In the case when s is a positive real number then w s is positive on its support. This gives the following main theorem.

Theorem 3. For every positive real number s, s = 1, for every complex number a, (a) > 0, for all integers n ≥ 0, p ≥ 1, the following sequence

RP A(n, m + p, m) := n-1 k=0 1 (k + a) s + (n + a) 1-s [m + p/m] Ψ(s,t) t=1/(n+a) . (3.4)
converges to ζ(s, a) when m tends to ∞.

Theorems (1 and 3) will be proved in Section 4.

Remarks. For n ∈ N, p = -1, Theorem 3 is proved only for s = 2, 3 in [9]. RP A(n, n-1, n) (resp. RP A(n, 2n -1, 2n)) are the Apéry's numbers for ζ(2) (resp. ζ(3)). If n = 0, formula (3.4) yields ∀s > 0, ∀a ∈ C, (a) > 0, ∀p ≥ 1, lim m→∞ a 1-s [m + p/m] Ψ(s,t) t=1/a = ζ(s, a).
Moreover, for the particular case n = 0, a = 1, RP A(0, m + p, m) is a rational function depending on the variable s and which converges to ζ(s, 1) = ζ(s) when m tends to ∞ (by Theorem 3).

For example, for p = 1, RP A(0, 2, 1) = (s+2)(s+3) 12(s-1) , RP A(0, 3, 2) =

(s+2)(s 2 +12s+31) 2(s-1)(s 2 +3s+62) , RP A(0, 4, 3) = (s+2)(s+4)(3s 2 +88s+405) 120(s-1)(s 2 +7s+54)
, RP A(0, 5, 4) = (s+2)(s+4)(s 5 +124s 4 +2644s 3 +23730s 2 +92939s+122130) 2(s-1)(s 6 +101s 5 +1755s 4 +21415s 3 +153884s 2 +657564s+977040) RP A(0, 6, 5) = (s+2)(s+4)(s+6)(s 5 +746s 4 +28162s 3 +498112s 2 +3612925s+8457750) 84(s-1)(s 6 +429s 5 +10387s 4 +134511s 3 +1044772s 2 +4891020s+9666000)

If n ≥ 1 and a = 1, then RP A(n, m+p, m) is no more a rational fraction since it contains powers of (n + a) with s as exponent:

RP A(1, 2, 1) = 2 1-s (s 2 +11s+36) 48(s-1) + 1, RP A(1, 3, 2) = 2 -s-1 (s 3 +26s 2 +231s+726) (s-1)(s 2 +3s+242) + 1, RP A(1, 4, 3) = 2 -s-4 (3s 4 +166s 3 +2937s 2 +22334s+64800) 15(s-1)(s 2 +7s+180) + 1, RP A(2, 2, 1) = 3 1-s (s 2 +17s+90) 108(s-1) + 2 -s + 1, RP A(2, 3, 2) = 3 -s (s 3 +38s 2 +527s+2710) 2(s-1)(s 2 +3s+542) + 2 -s + 1.
Nevertheless, if s is an integer, a = 1 and p ≥ 1, we obtain a sequence of rational approximations of ζ(s) by computing RP A(n, m + p, m), depending on the two integers parameters n and m.

Proof of Theorems

4.1. Proof of Theorem 1. We start from Hermite's formula for ζ(s, a), (a) > 0, which is a consequence of Plana's summation formula ( [START_REF] Srivastava | Zeta and q-Zeta functions and associated series and integrals[END_REF], p.158):

ζ(s, a) = 1 2 a -s + a 1-s s -1 + 2 ∞ 0 (a 2 + y 2 ) -s/2 sin s arctan y a dy e 2πy -1 . ( 4.1)
Note that this integral converges for every complex number s = 1.

Let us define

I s := t 0 x s (t -x) -s a 2 + x 2 dx (t ≥ 0, 0 < (s) < 1
). The proof is based on the formula [8, p. 142]: we have for (α) > 0 and (β) > 0,

1 0 u α-1 (1 -u) β-1 (u + z) α+β du = B (α, β) (1 + z) -α z -β (z ∈ C, |ph(z)| < π) (4.2)
where B (α, β) =

1 0 u α-1 (1 -u) β-1 du = Γ (α) Γ (β) Γ (α + β) .
With the change of variable x = tu, we have

I s = 1 0 tu s (1 -u) -s a 2 + t 2 u 2 du = i 2 1 0 u s-1 (1 -u) -s 1 a + itu - 1 a -itu du.
By applying formula (4.2) with α = s, β = 1 -s and z = ± a it (|ph(z)| < π since (a) > 0) we get

I s = Γ(s)Γ(1 -s) a s-1 2i (a -it) -s -(a + it) -s .
Let us set t = a tan θ. We obtain

I s = Γ(s)Γ(1 -s)a -1 cos s (θ) sin(sθ) = Γ(s)Γ(1 -s)a s-1 (a 2 + t 2 ) -s/2 sin s arctan t a .
Thus, the following identity

1 Γ(s)Γ(1 -s) t 0 x s (t -x) -s a 2 + x 2 dx = a s-1 (a 2 + t 2 ) -s/2 sin s arctan t a (4.3)
holds for all real t ≥ 0 and complex s, 0 < (s) < 1. Now, for s complex satisfying (s) > 0 and m an integer such that m > (s) -1, we claim that the following formula, sin(s arctan t/a)

(a 2 + t 2 ) s/2 = a 1-s Γ(s)Γ(m + 1 -s) d m dt m t 0 x s (t -x) m-s 1 + x 2 dx (4.4)
is satisfied. First, it is true for m = 0 (4.3). Let us assume that (4.4) is true for m then using

d m+1 dt m+1 t 0 x s (t -x) m+1-s 1 + x 2 dx = d m dt m d dt t 0 x s (t -x) m+1-s 1 + x 2 dx = d m dt m t 0 (m + 1 -s) x s (t -x) m-s 1 + x 2 dx ,
we obtain formula (4.4) by using the identity Γ(m + 2 -s) = (m + 1 -s)Γ(m + 1 -s).

To prove Theorem 1, we replace sin(s arctan(t/a)) (a 2 + t 2 ) s/2 (formula 4.4) in Hermite's formula (4.1), and we apply integration by parts m times:

ζ(s, a) = a -s 2 + a 1-s s-1 + 2 a 1-s Γ(s)Γ(m+1-s) ∞ 0 d m dt m t 0 x s (t-x) m-s a 2 + x 2 dx dt e 2πt -1 = a -s 2 + a 1-s s-1 + 2(-1) m a 1-s Γ(s)Γ(m+1-s) ∞ 0 t 0 x s (t-x) m-s a 2 +x 2 dx d m dt m 1 e 2πt -1 dt = a -s 2 + a 1-s s -1 + 2(-1) m a 1-s Γ(s)Γ(m+1-s) ∞ 0 x s dx a 2 +x 2 ∞ x (t-x) m-s d m dt m 1 e 2πt -1
dt.

The permutation in the last integral is valid since the function

x s a 2 +x 2 (t -x) m-s d m dt m 1 e 2πt -1 is less than t m a 2 d m dt m 1 e 2πt -1 which is integrable in [0, ∞[. Remark.
If we consider a = 1, we get an integral representation of the Riemann zeta function:

ζ(s) = 1 2 + 1 s -1 + ∞ 0 1 1 + x 2 w s (x)dx (4.5) with w s (x) := 2(-1) m x s Γ(s)Γ(m + 1 -s) ∞ x (t -x) m-s d m dt m 1 e 2πt -1 dt (4.6)
where m is an integer satisfying m > (s) -1. So, if s is an integer, we can consider m = s and

ζ(s) = 1 2 + 1 (s -1) + 2(-1) s-1 Γ(s) ∞ 0 x s 1 + x 2 d s-1 dx s-1 1 e 2πx -1 dx (4.7) = 1 2 + 1 (s -1) + 2 Γ(s) ∞ 0 1 e 2πx -1 d s-1 dx s-1 x s 1 + x 2 dx (4.8)
which yields

ζ(2) = 3 2 + π ∞ 0 x 2 1 + x 2 1 sinh 2 πx dx = 3 2 + ∞ 0 1 e 2πx -1 4x (1 + x 2 ) 2 dx (4.9) ζ(3) = 1 + π 2 ∞ 0 x 3 1 + x 2 cosh πx sinh 3 πx dx = 1 - ∞ 0 1 e 2πy -1 2x(x 2 -3) (1 + x 2 ) 3 dx (4.10) ζ(4) = 5 6 + π 3 3 ∞ 0 x 4 1 + x 2 2 + cosh(2πx) sinh 4 πx dx = 5 6 - ∞ 0 1 e 2πx -1 8x(x 2 -1) (1 + x 2 ) 4 dx. (4.11)
4.2. Proof of Theorem 3. We consider the function Ψ(s, t) given in (1.6) written as

Ψ(s, t) = B 0 s -1 -B 1 t + t 2 Ψ 2 (s, t) (4.12)
where

Ψ 2 (s, t) := ∞ k=0 B k+2 (k + 2)! (s) k+1 (-1) k t k .
Thus the Padé approximant [m + p/m] Ψ(s,t) satisfies

[m + p/m] Ψ(s,t) = B 0 s -1 -B 1 t + t 2 [m + p -2, m] Ψ 2 (s,t) .
For s a positive real number, the weight w s is positive (Theorem 2) and so Ψ 2 (s, t) is a Stieltjes function 5.1. Case s = 2. In this case, the weight in the expression (3.1) is defined as

Ψ 2 (s, t) = ∞ 0 1 1 + x 2 t 2 w s (x)dx. Its coefficients c k := B k+2 (k + 1)! (s) k+1 (-
w 2 (x) := 2x 2 ∞ x d 2 dt 2 1 e 2πt -1 dt = πx 2 sinh 2 (πx) . (5.1)
Another expression of w 2 (x) is

w 2 (x) = 1 π |Γ(1 + ix)Γ(1 -ix)| 2 .
Generalizing a result by Carlitz [START_REF] Carlitz | Bernoulli and Euler numbers and orthogonal polynomials[END_REF], Askey and Wilson [START_REF] Askey | A set of hypergeometric orthogonal polynomials[END_REF] gave an explicit expression for the orthogonal polynomial P m with respect to the weight function w 2 :

P m (x) = (m + 1)(m + 2) 3 F 2   -m, m + 3, 1 -x ; 1 2, 2   (5.2) = m k=0 m + 1 k + 1 m + k + 2 k + 1 x -1 k , (5.3
)

and iR P n (x)P m (x) πx 2 sin 2 πx dx =    0 (if n = m), (-1) n 2(n + 1)(n + 2) 2n + 3 (if n = m).
The roots of the polynomials P n are located on the imaginary axis since the weight πx 2 sin 2 πx is positive on this line (see 2.7). The three-term recurrence relation is

P m+1 (x) = 2(2m + 3) (m + 1)(m + 2)
x P m (x) + P m-1 (x) (5.4) with initial conditions: P -1 (x) = 0, P 0 (x) = 2.

The Padé approximants are expressed with orthogonal polynomials and their associated polynomials (see 2.5).

Before computing these associated polynomials, we need some suitable modified moments, i.e., the moments of the binomial x -1 k .

Let us define the linear functional c (s) acting on the space of polynomials as

c (s) , x j := B j j! (s) j-1 (-1) j , j ∈ N,
and x 2 c (s) by x 2 c (s) , x j := c (s) , x j+2 . The following modified moments [9, p.228] satisfy c (2) , x k = (-1) k k + 1 .

(5.5)

The identity

x 2 x -1 k = (k + 1)(k + 2) x k + 2 + (k + 1) 2 x k + 1
leads to the following formula for modified moments for the functional x 2 c (2) :

x 2 c (2) , x -1 k = (-1) k (k + 1) (k + 2)(k + 3) . (5.6)
The associated polynomials R m-1 (2.5) are defined by

R m-1 (t) := x 2 c (2) , P m (x) -P m (t) x -t
where the variable is x. From the expression (5.3) for P m , we get the following formula for

R m-1 R m-1 (t) = m k=0 m + 1 k + 1 m + k + 2 k + 1 x 2 c (2) , x -1 k - t -1 k x -t . (5.7)
Using the expression of the polynomial

x -1 k - t -1 k x -t on the Newton basis with abscissae {0, 1, 2 • • • , k -1} x -1 k - t -1 k x -t = t -1 k k i=1 x -1 i -1 i t -1 i ,
we can write a compact formula for R m-1 :

R m-1 (t) = m k=0 m + 1 k + 1 m + k + 2 k + 1 t -1 k k i=1 (-1) i-1 t -1 i (i + 1)(i + 2)
.

So, we get the [m + 1/m] Padé approximant to the function Ψ(2, t)

[m + 1/m] ψ(2,t) = B 0 -B 1 t + t 2 R m-1 (t) P m (t) = B 0 -B 1 t + t R m-1 (1/t) P m (1/t)
and also an approximation of ζ(2, a):

ζ(2, a) ≈ n-1 k=0 1 (k + a) 2 + 1 n + a [m + 1/m] Ψ(2,t) | t=1/(n+a) = n-1 k=0 1 (k + a) 2 + 1 n + a + 1 2(n + a) 2 + 1 (n + a) 2 ε m (a) =: v n,m (a) u n,m (a) = RP A(n, m + 1, m)
where

ε m (a) = m k=0 m + 1 k + 1 m + k + 2 k + 1 k i=1   n + a -i -1 k -i   (-1) i-1   k i   (i+1)(i+2) m k=0 m + 1 k + 1 m + k + 2 k + 1 n + a -1 k .

Irrationality of ζ(2)

A consequence of the previous formula is another proof of the irrationality of ζ(2).

Indeed, if d k := LCM [1, 2, • • • , k], then [9, Lemma 3.1]
d k i k i ∈ N.
Consequently, the numerator of ε m (1) multiplied by d 2 m+2 is an integer and for all integers n, m, d

2 m+2 v n,m (1) ∈ N, d 2 m+2 u n,m (1)
∈ N. The error (formula (2.10)) applied to the function Ψ(2, t) rewrites as

Ψ(2, t) -[m + 1/m] Ψ(2,t) (t) = t 2m P 2 m (t) iR P 2 m (x) 1 -xt x 2 sin 2 (πx) dx
and the error term satisfies

ζ(2) - v n,m (1)
u n,m (1)
≤ 1 P 2 m (n + 1) iR P 2 m (x) 1 -x/(n + 1)
x 2 sin 2 (πx) dx (5.8)

≤ 1 P 2 m (n + 1) iR P 2 m (x)
x 2 sin 2 (πx) dx (5.9)

≤ 1 P 2 m (n + 1) 2(m + 1)(m + 2) π(2m + 3) . (5.10)
Note that u n,m (1) = P m (n + 1). Now, we consider r ∈ Q such that m = rn ∈ N. Applying the Stirling formula to the expression (5.3) for orthogonal polynomials P n , we get lim

n (P rn (n + 1) 1/n = max t∈[0,1] (r + t) r+t t 3t (r -t) r-t (1 -t) 1-t = (r + σ(r)) r (1 -σ(r))(r -σ(r)) r =: ρ(r) (5.11)
where σ(r)

= -r 2 + √ r 4 +4r 2 2
is a zero of t 2 + r 2 t -r 2 = 0. So, using the prime number theorem

d n = e n+o(n) , lim sup n ζ(2)d 2 rn+2 u n,rn (1) -d 2 rn+2 v n,rn (1) 1/n ≤ lim n (d 2 rn+2 u n,rn (1)
) 1/n lim n 1 (P rn (n + 1) 2/n = e 2 max(r,1) /ρ(r).

The inequation e 2 max(r,1) /ρ(r) < 1 is satisfied for r ∈]0.

w 3 (x) := - x 3 3 ∞ x d 3 dt 3 1 e 2πt -1 dt = π 2 x 3 cosh πx 3 sinh 3 (πx) .
(5.12)

Another expression of x 2 w 3 (x) is

x 2 w 3 (x) = 1 12π |Γ(1 + ix)| 8 |Γ(2ix)| 2 .
This weight has been investigated by Wilson [START_REF] Wilson | Some hypergeometric orthogonal polynomials[END_REF]. The orthogonal polynomial P n satisfies:

P m (x 2 ) = (m + 1)(m + 2) 4 F 3   -m, m + 3, 1 -x, 1 + x ; 1 2, 2, 2   (5.13)
= m k=0 m + 1 k + 1 m + k + 2 k + 1 x -1 k x + k k /(k + 1) (5.14)
and the orthonality relation is iR P n (x 2 )P m (x 2 ) π 2 3

x 5 cos πx

sin 3 πx dx =    0 (if n = m), 1 3 (-1) n (n + 1)(n + 2) 2n + 3 (if n = m).
The weight x 5 cos πx sin 3 πx is positive on the line iR thus, the roots of the polynomials P n are located on the imaginary axis.

We set Π m (x) := P m (x 2 ). The three-term recurrence relation is

Π m+1 (x) = 2(2m + 3) (m + 1)(m + 2) 2 x 2 + 2m + 3 m + 2 Π m (x) - m + 1 m + 2 Π m-1 (x) (5.15)
with initial conditions: Π -1 = 0, Π 0 = 2.

In the previous subsection, we have defined the linear functional c (s) acting on the space of polynomials as c (s) , x j := B j j! (s) j-1 (-1) j , j ∈ N, and x 4 c (s) by x 4 c (s) , x j := c (s) , x j+4 .

Before computing the associated polynomials, we need the moments of x -1 k

x + k k and x -1 k + 1

x + k k .

Lemma 1. Let us define the polynomials ν 2j+1 (x) and ν 2j (x) by

ν 2j+1 (x) = x -1 j + 1 x + j j ν 2j (x) = x -1 j x + j j .
Then,

x 4 c (3) , ν 2k+1 (x) = (-1) k+1 (k + 1) 2 2(k + 3)(k + 2)
(5.16)

x 4 c (3) , ν 2k (x) = (-1) k (k + 1) 2 2(k + 3)(k + 2)
(5.17)

Proof. c (2) satisfies c (2) , x j = B j (-1) j . We denote c (2) by c (2) , x j := c (2) , -jx j-1 .

From the definition of c (3) , c (3) , x j = 1 2 B j (j + 1)(-1) j , we have 2) .

c (3) = - 1 2 xc (
In [START_REF] Prévost | A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants[END_REF] p. 230, it is proved that -c (2) , N 2k (x) = 0, -c (2) , N 2k+1 (x) = (-1) k (k+1) 2 where

N 0 (x) = 1, N 2k (x) = x k x + k k , N 2k+1 (x) = x k + 1 x + k k . Now, x 4 c (3) , ν 2k+1 (x) = -1 2 x 5 c (2) , ν 2k+1 (x) = 1 2 -c (2) , x 5 ν 2k+1 (x) .
The proof of (5.16) follows from the expansion

x 5 ν 2k+1 (x) = αN 2k+6 (x) + βN 2k+5 (x) + γN 2k+4 (x) + δN 2k+3 (x) + µN 2k+2 (x) -2µN 2k+1 (x), with α = (k + 1)(k + 2) 2 (k + 3) 2 , β = -2(k + 1)(k + 2) 3 (k + 3), γ = (k + 1)(k + 2) 2 (2k 2 + 6k + 5), δ = -(k + 1)(k + 2)(2k + 3)(2k 2 + 6k + 5), µ = (k + 1) 5 .
For (5.17), we use the expansion

x 5 ν 2k (x) = (k+1) 2 (k+2) 2 (k+3)N 2k+5 (x)+(k+1) 2 (k+2)(2k 2 +6k+5)N 2k+3 (x)+(k+1) 5 N 2k+1 (x).
Now, the associated polynomials Θ m-1 (2.5) are defined as

Θ m-1 (t) := x 4 c (3) , Π m (x) -Π m (t) x -t
where the variable is x. From the expression (5.14) for Π m , we get the following formula for Θ m-1 ∈ P 2m-2 :

Θ m-1 (t) = m k=0 m+1 k+1 m+k+2 k + 1 1 k + 1 x 4 c (3) , x-1 k x+k k - t-1 k t+k k x -t . (5.18)
Using the expression of the polynomial

x -1 k x + k k - t -1 k t + k k x -t on the New- ton basis with abscissae {0, 1, -1, 2, -2 • • • , k, -k} [9, formula 55], i.e. x -1 k x + k k - t -1 k t + k k x -t = 2k-1 j=0 ν 2k (t) ν j+1 (t) ν j (x) [(j + 2)/2]
.

and Lemma 1, 3) , ν 2j (x) = (-1) j (j + 1) 2 2(j + 3)(j + 2) ,

x 4 c (3) , ν 2j+1 (x) = -x 4 c (
we can write a compact formula for Θ m-1 :

Θ m-1 (t)= m k=0 m + 1 k + 1 m + k + 2 k + 1 1 k + 1 2k-1 j=0 ν 2k (t) ν j+1 (t) x 4 c (3) , ν j (x) = m k=0 m+1 k+1 m+k+2 k+1 ν 2k (t) k+1 k p=1 t t-1 p t+p p (-1) p 2(p+1)(p+2)
.

So, for m ≥ 2, we get the [2m + 2/2m] Padé approximant to the function Ψ(3, t)

[2m + 2/2m] Ψ(3,t) = B 0 2 -B 1 t + 3 2 t 2 B 2 + t 5 Θ m-1 (t) Π m (t) = B 0 2 -B 1 t + 3 2 t 2 B 2 + t 4 Θ m-1 (1/t) Π m (1/t) ,
and an approximation of ζ(3, a) :

ζ(3, a) ≈ n-1 k=0 1 (k + a) 3 + 1 2(n + a) 2 + 1 2(n + a) 3 + 1 4(n + a) 4 + 1 (n + a) 5 m (a) =: f n,m (a) g n,m (a) where m (a) = Θ m-1 (n + a) Π m (n + a) = m k=0 m + 1 k + 1 m + k + 2 k + 1 n+a k+1 k p=1   n + a -p -1 k -p     n + a + k k -p   (-1) p   k p   2 2(p+1)(p+2) m k=0 m + 1 k + 1 m + k + 2 k + 1 n + a -1 k n + a + k k /(k + 1)
.

Irrationality of ζ(3)

The irrationality of ζ(3) has been proved by Apery in a celebrated paper [START_REF] Apéry | Irrationality of ζ(2) and ζ(3)[END_REF]. A little later, a particular straightforward and elegant proof of this irrationality was given by Beukers [START_REF] Beukers | A note on the irrationality of ζ(2) and ζ(3)[END_REF]. The author gave another proof in [START_REF] Prévost | A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants[END_REF] (r + t) r+t (1 + t) 1+t t 3t (r -t) r-t (1 -t) Numerically, it seems that this property is valid for all s > 0, s = 1 and more generally for all complex numbers s, s = 1.

  1 and uniformly convergent in any finite region where (s) ≥ 1 + δ, with δ > 0. It defines an analytic function for (s) > 1. The Riemann ζ function is ζ(s, 1). An integral representation of ζ(s, a) is given by ζ(s, a) 1 e -ax 1 -e -x dx (s) > 1, (a) > 0, (1.2) where Γ(s) = ∞ 0 y s-1 e -y dy is the gamma function. If (s) ≤ 1, the Hurwitz zeta function can be defined by the equation ζ(s, a) = Γ(1 -s) 1 2πi C

For p = 1 ,

 1 s = 2 or p = 2, s = 3 , we will give in this section the formal expression of RP A(n, m + p, m) and the expression of the error. If a = 1, then the rational approximations RP A(n, rn + 1, rn) to ζ(2) and RP A(n, rn + 2, rn) to ζ(3) with some suitable value of r lead to another proofs of the Apéry's theorem. The case for s ≥ 4 remains an open problem.

5 . 2 .

 52 74, 1.53[. The irrationality of ζ(2) follows from the following limit lim n ζ(2)d 2 rn+2 u n,rn (1) -d 2 rn+2 v n,rn (1) = 0. Case s = 3. For s = 3, the weight in the expression (3.1) is

3 .

 3 using the RPA. Actually, RP A(m, 2m -1, 2m) provides exactly the Apery's numbers for ζ(3). In the following, we show that the previous approximation also proves irrationality of ζ(3) with RPA's depending on other parameters.Actually,∀n, a ∈ N, d 3 m+1 Θ m-1 (n + a) ∈ Z, d 3 m+1 f n,m (n + a) ∈ Zand d 3 m+1 g n,m (n + a) ∈ Z.Note that g n,m (n + a) = Π m (n + a). The proof is similar than in the previous subsection. We consider the case a = 1. The error (formula (2.10)) applied to the function Ψ(3, t) becomes Ψ(3, t) -[2m + 2/2m] Ψ(3,t) Now, we consider r ∈ Q such that m = rn ∈ N. Using the Stirling formula in the expression of the orthogonal polynomials (5.14), lim n (Π rn (n + 1)) 1/n = max t∈[0,1]

3

 3 µ(r))(r + µ(r)) r (1 -µ(r))(r -µ(r)) r =: η(r)where µ(r) = r √ 1 + r 2 is a zero of (1 -t 2 )(r 2 -t 2 ) = t 4 . rn+1 g n,rn (1) -d 3 rn+1 f n,rn (1) 1/n ≤ lim n (d 3 rn+1 g n,rn (1)) 1/n lim n 1 (g n,rn (1)) 2/n = e 3 max(r,1) /η(r)The inequation e 3 max(r,1) /η(r) < 1 is satisfied for r ∈]0.74, 1.36[. The irrationality of ζ(3) then follows from the following limitlim n ζ(3)d 3 rn+1 g n,rn (1) -d 3 rn+1 f n,rn(1)= 0.ConclusionIn section 3, for a = 1, we have proved that for all fixed integers n, p ≥ 1, and for each s > 0, s = 1, then the sequence(RP A(n, m + p, m)) m converges to ζ(s).If m and n are linked such that m = rn, then the convergence of (RP A(n, rn + p, rn)) n is proved for s = 2 and the rate of convergence is ρ(r) (see(5.11)): lim sup n (|ζ(s) -RP A(n, rn + p, rn))| 1/n ≤ ρ(r).

  )/ P m 2 (t) be a Padé approximant to the function Ψ.

	Then by replacing the remainder term Ψ s,	1 n + a	in (1.4), by the Padé approximant
	[m 1 /m

2 ] Ψ(s,t) , computed at t = 1/(n + a), we obtain the RPA approximant