Valeria Loscrí

Nathalie Mitton

Emilio Compagnone

OpenCV WebCam Applications in an Arduino-based Rover ⋆

Keywords: Rover, OpenCV, WebCam applications

In this work we design and implement Arduino-based Rovers with characteristics of re-programmability, modularity in terms of type and number of components, communication capability, equipped with motion support and capability to exploit information both from the surrounding and from other wireless devices. These latter can be homogeneous devices (i.e. others similar rovers) and heterogeneous devices (i.e. laptops, smartphones, etc.). We propose a Behavioral Algorithm that is implemented on our devices in order to supply a proof-of-concept of the effectiveness of a Detection task. Specifically, we implement the "Object Detection" and "Face Recognition" techniques based on OpenCV and we detail the modifications necessary to work on distributed devices. We show the effectiveness of the controlled mobility concept in order to accomplish a task, both in a centralized way (i.e. driven by a central computer that assign the task) and in a totally distributed fashion, in cooperation with other Rovers. We also highlight the limitations of similar devices required to accomplish specific tasks and their potentiality.

Introduction

In the last few years, we have witnessed a technological development in robots, computing and communications, that has allowed the design of Network Robotic Systems (NRS) [START_REF] Sanfeliu | Robotics and Autonomous Systems[END_REF]. A NRS is composed by a robotic unit, able to communicate and cooperate with both other similar units and other interconnected devices. A very interesting feature of this type of systems is the possibility to "use" the same robots to accomplish different and complex tasks and the same set of robots may perform the same task in different conditions [START_REF] Lundh | Autonomous Functional Configuration of a Network Robot System[END_REF]. There have been increasing interests in deploying a team of robots able to cooperate and to self-coordinate, to fulfill also complex tasks such as target tracking [START_REF] Blzovics | Target Tracking and Surrounding with Swarm Robots[END_REF], disaster recovery [START_REF] Kuntze | SENEKA -Sensor Network with Mobile Robots for Disaster Management[END_REF], etc. NRS concept will marry in a very natural fashion with the concept of Swarm Robotics, where many robotic units are considered to cooperatively accomplish very complex tasks [START_REF] Nagi | Convolutional Support Vector Machines for quantifyingl the visual learning and recognition progress in swarm robotic systems[END_REF] [START_REF] Pinciroli | ARGoS: a Modular, Multi-Engine Simulator for Heterogeneous Swarm Robotics[END_REF]. Beni [START_REF] Beni | From swarm intelligence to swarm robotics[END_REF] provides a very effective and precise definition of a swarm of robots : "The group of robots is not just a group. It has some special characteristics, which are found in swarms of insects, that is, decentralized control, lack of synchronization, simple and (quasi) identical members". In this work, we overcome the concept of members that have to be similar by considering devices that are able to communicate with any interconnected node. In any case, we outline how the devices are able to cooperate and self-coordinate also without a central unit that drives them.

In this paper, we build an Arduino-based platform, that presents some important characteristics such as re-programmability, modularity in terms of type and number of components and communication capability. Our devices are also equipped with motion capability and are able to exploit the concept of controlled mobility, by reaching specific target locations. Our robots can leverage information both from the surrounding and from other interconnected devices (i.e. others similar robots, smartphones, laptops, etc.). Based on the acquired information, our devices will behave in a specific fashion, in order to accomplish a specific task assigned either by a central unit (e.g. a central computer where a human may ask the accomplishment of specific tasks) or by another robot. These actions are realized through a very simple Behavioral Algorithm that will be implemented onto the devices and will be tested in different scenarios, in order to verify the effectiveness and to highlight the critical aspect of the system. The robot will react based on external stimulus, acquired through a WebCam or communication equipment. In order to make the analysis of the acquired images feasible, we referred to a well-known vision tool, Open Source Computer Vision Library, OpenCV.

The main contributions of this work can be summarized as follows:

-Modifications of specific tools such as OpenCV libraries originally developed for more powerful devices (e.g. computer) to adapt them to distributed and constrained environments; -Development of a testbed on real Arduino-based platforms; -Proposal of a very simple Behavioral Algorithm to be implemented as a proof-of-concept of the developed tools and platform.

The rest of the paper is organized as follows. Section 2 describes the simple Behavioral Algorithm and the basic rules the robot accomplishes for searching target. Section 3 presents the tools we use to acquire data and how to exploit this information. Section 4 details how we modified existing tools in order to allow effective actions in a distributed and constrained environment. Section 5 presents the platform setup and implementation of the Behavioral Algorithm integrated with the modified libraries. Finally, we conclude this work in Section 6.

The Behavioral Algorithm

This section details our Behavior Algorithm for mobile robots. The main goal is to search for possible targets, identify them and thus, cover them by having robots reach them, in a distributed way. Each robot runs the same algorithm independently of the others and cooperate to fulfill a list of tasks, a list of targets to reach.

First of all, it is worth defining what a target is. In this context, a target represents an object with specific characteristics in terms of shape and/or color, that has been detected by the robot. The specific characteristics are defined by a human controller through a central computer, but our project can be easily modified, by including a dynamic definition of the target also defined by other inter-connected devices. Initially, each robot is assigned with a target. A same target can be assigned to several robots.

The algorithm is mainly composed of two phases: the Searching Phase and the Approaching Phase. During both phases, an underlying obstacle avoidance process is running. It is run on every robot when moving. More details about the obstacle avoidance implementation are given in Section 4. [START_REF] Blzovics | Target Tracking and Surrounding with Swarm Robots[END_REF]. The details of the Behavioral Algorithm are given in the following pseudo-code, on Algorithm 1. Our algorithm will terminate when all tasks assigned to the robot are completed.

The robot starts with the task on top of its list. It first enters the Searching Phase (Lines 19-37 in Algo. 1) which consists in locating the target. If the robot does not identify the pre-assigned target in its environment (in its vision field), it moves forward for an arbitrary time ∆t (e.g 1 or 2 sec.) by following a Random Way Path (RWP) [START_REF] Hyytiä | Random waypoint model in n-dimensional space[END_REF]. More precisely, it travels a prefixed distance then it stops and checks for the presence of the target within its 'new' environment. It then repeats this process while the target is not identified. Once the target is identified, the robot switches to the Approaching Phase (Lines 4-19 in Algo. 1). The robot simply heads to the direction of the target till either reaching it or realizing that the target is a "false positive", due to for example to lights. In order to detect these "false positives", a specific mechanism is set up to allow the robot to make a new search without being influenced by the previous results as described in Section 4.2. It then switches back to the Searching Phase. Once the target is reached, the robot advertises other robots though its communication channel. Upon reception of this message, other robots abort their task to move to the next one. This latter phase can be revisited as a dynamic change of the task, based on the input that one of them broadcasts.

3 Background: software and hardware used in this work

In this section, we give a brief description of the hardware and software tools later used in this paper for our different implementations.

Hardware

In order to evaluate our contributions, we use the following hardware platform. An Arduino module [1] is set on an aluminum robotic platform with 4 wheels. Arduino allows us to re-program our node and to make it able to interact with

7:

Move forward for ∆t at speed s.

8:

Listen to other robots.

9:

if Target reached by another robot then 10:

TaskCompleted← TRUE.

11:

else 12:

Scan environment.

13:

if Target identified then

14:

TargetFound←TRUE.

15:

else 16:

Change direction.

17:

end if

18:

end if

19:

end while {Approaching Phase}

20:

while (TargetFound = TRUE) AND (TaskCompleted = FALSE) do

21:

∆t ← Random()

22:

Move toward target for ∆t at speed s

23:

if Target is reached then

24:

TaskCompleted← TRUE.

25:

Advertise other robots.

26:

else

27:

Listen to other robots.

28:

if Target reached by another robot then

29:

TaskCompleted← TRUE.

30:

else

31:

Check false positive.

32:

if False positive identified then

33:

TargetFound← FALSE.

34:

end if

35:

end if

36:

end if

37:

end while

38:

end while 39: end while the external environment. Our Arduino module is a UNO rev3 which main characteristics are a microcontroller ATmega328, a 32B flash memory with a 16MHz clock1 that we extended with an IMU board, a motor board and a PING sensor.

Since our devices are equipped with motion capabilities, we apply on them an ultrasound sensor in order to avoid obstacles (See Section 4.3). Moreover, we consider a WebCam to exploit images for task accomplishment. We used Open Source Computer Vision Library, OpenCV [12] libraries together with Computer Vision in order to "extract" meaningful data from the images. A very important component of our devices is the Inertial Movement Unit (IMU), a platform with an accelerometer, magnetometer and gyroscope. The IMU makes the Rover navigable. In effect, there exist other "easier" and well-performing solutions to realize the "navigation" function, such as stepper [START_REF] Jinasena | Design of a Low-cost Autonomous Mobile Robot[END_REF], but the ratio cost/efficiency of our solution is better than the other available.

Since we chose a cheap micro controller, each rover has only limited computing capacity and is not able to run the tasks we address. Therefore, we embed a miniPC MK802II in our robots, that we modified in order to use a Linux platform instead of the Android available. The choice of this OS (Operating System) is related to the possibility to better control the platform with high level program languages, to increase the computational capability and to control external devices (i.e. those already mentioned), through an USB interface.

Wifi miniPC cardboard is used for peer-to-peer communications.

Software

In order to make our devices able to perform some specific tasks, we focused on Object Detection techniques. More specifically, we consider a Computer Vision library, named OpenCV [12]. OpenCV has been developed by Intel and supplies a set of high level functions to acquire images and computer vision in real time.

An important feature of OpenCV is that it allows the execution on different platforms (Linux, windows, etc). We mainly focus on two specific techniques:

-Face Recognition based on the Cascade Classifiers method; -Object Detection based on the HSV Model [5].

In the following, we detail these two techniques.

Object Detection via HSV Hue, Saturation and Value (HSV) consists in a re-deployment of points of the RGB Cartesian Space into a cylindric space.

The main goal of this operation is a more intuitive analysis of the colors in its components, since in the RGB, the analysis is very complex. In Figure 1 we show how the HSV model works. In the figure, in each cylinder, the angle around the central axis represents the tonality, the distance from the central axis is the saturation, and the distance from the basis of the cylinder is the value.

Figure 2 shows the difference from an optical point of view obtained when we apply the HSV model on an RGB model. The evaluation of this difference has been very important for the usage of the HSV model as Object Detection. In order to make possible the detection operation, it is necessary to consider a color filtering phase based on the function inRange of OpenCV.

This filtering phase is paramount in the Object Detection, since we consider the form-color combination. It is worth noticing that the exact values for color filtering are not known and for that it is necessary the use of a specific tool that allow the extraction of HSV values from an image (see Figure 3).

It is also important to notice that H channels range from 0 to 360 degrees and in OpenCV this range is comprised from 0 to 180 degrees. Through the function findContour in OpenCV, it is possible to detect elementary geometrical forms as squares, rectangles, circle, etc. and it is also possible to define proper own geometrical forms. In Figure 3 we show the result we obtained by considering as objective the form-color detection of a can. In order to be able to realize the goal of Face Recognition, we need to formulate it as a Machine Learning problem, where an initial training phase is mandatory. In OpenCV, it is also possible to find some configuration to detect specific parts of a face, stored in XML files. If something different is necessary, it is possible to train the Classifier by using a proper own set of images with the specific object that need to be detected. The training phase needs to be realized in a very accurate way, since many different factors need to be taken into account, such as the brightness, quality, color, etc. More specifically, the training phase we considered is called HaarTraining, [4]. This procedure requires a lot of computational resources, since the training phase requires the analysis of many images to extract the information necessary to create a classifier. The operation is based on some intermediate phases, based on the analysis of "negative" and "positive" images. Specifically, the "positive" images are those that contain inside the target that has to be identified. The "negative" images are those without the target to be identified. The term "cascade" is referred to the fact that the classifier is the result of various simple classifiers (phases), that are applied in a sequential way either to a frame or to an image. In practice, the output of a classifier is the input of the next classifier and so on. In Figure 4, we can observe the result of searching a human face inside an image, by applying the Cascade Classifiers method. It is thus very efficient.

Adaptation of Tools

This section presents how the tools described in Section 3 are modified to make the robots able to detect in an effective way a target object by considering information coming both from the surrounding (i.e. WebCam) and the other Rovers (by enabling communication among them). In this section, we will detail how the tools described above have been modified in order to be suitable with our distributed devices. It is worth noting that the libraries of OpenCV have been conceived for powerful centralized systems whereas we are considering distributed devices. We focus on two specific techniques:

-Object Detection based on HSV model; -Face Recognition based on Cascade Classifiers.

The first fundamental step, for both techniques, consists in the choice of the right filter to analyze images/frames. In OpenCV various filters are available such as Canny, Gaussian, Kalman, ConDensation, etc. In order to test the different filters on our Rovers, we modeled them as Python models and we implemented them on the devices.

HSV Object Detection

The Object Detection task consists in a first phase named Detection Phase. The frame coming from the WebCam passes through a filter and has to be opportunely resized, due to the scarce computational resources of our device. After the resizing, the frame will be converted from RGB to HSV. It is also necessary to determine the characteristic values of a color. This problem has been faced by realizing a suitable interface as shown in Figure 7.

Resizing an image makes the detection more difficult, but by iteratively applying a specific OpenCV filter that enlarges the image, we can overcome this issue. Unfortunately, the impurities of the color spectrum will be enlarged too. In order to solve this additional problem, we apply the GaussianBlur filter that makes the image more homogeneous. After these basic operations, the target definition algorithm can be integrated in the Behavioral Algorithm and is ready to be tested. The test phase shows an additional problem related to the "retrieving" of the right HSV values from the WebCam. This data generates a stream video that has to be sent to the GUI. By taking the limited resources of our device into account, we had to apply a MPEG encoding to the stream, whereas originally data was YUV. This encoding change alters the correct HSV values. It is also worth recalling that the HSV technique is based on the brightness and contrast concepts. In order to obtain a mechanism able to work in every environmental condition, we need powerful hardware, but we will show in Section 5, that in certain conditions (places with low light and with constant brightness), the algorithm implemented on our Rover works in a very effective way.

Cascade Classifiers Face Recognition

In order to realize the Face Recognition task, we formulate it as a Machine Learning problem and we consider a training phase named HaarTraining [START_REF] Lienhart | An Extended Set of Haar-like Features for Rapid Object Detection[END_REF]. In the training phase, many factors need to be taken into account in a very accurate way, such as brightness, quality of the image, color, etc. The HaarTraining procedure requires a lot of computational resources, since the training phase is based on the analysis of many images in order to have the necessary information to realize a classifier. Concerning this training phase, we can distinguish the analysis of "positive" images (that contains the target that has to be identified) and "negative" images (where the target is not detected). In order to perform the Face Recognition task, we consider a suitable OpenCV Classifier. Also in this case, as in the previous application of HSV, we need to manage the image by resizing it and we convert the RGB model to the GRAY model. This latter step is necessary in order to decrease the amount of information to process and to improve the speed of the analysis. The next step consists in the integration of the data into the Behavioral Algorithm. After this integration, we performed some tests and we noticed that the movements of the Rovers and data of the Task seemed misaligned. Once again, the scarce processing resources have made a correct processing of data frames, impossible. In order to fix this problem, we reduced the number of frames processed in the unit of time. Instead of considering a continuous data flow, we limited the number of frames to 15, for each operation. In practice, we analyzed 15 frames at the center, 15 frames at left and 15 frames at right. Moreover, we included a "release" phase for the Web-Cam after each acquiring, that implies the presence of "settling" frames, where the WebCam exploits its focus functions to define the image. In Section 5, we experimentally determine the number of frames useful for this purpose and that have to be discarded from the analysis of the image. Another kind of issue is related to the "false positive" detection, namely some object that is not a face is wrongly identified as a target. In order to detect false positives, we modified the Recognition Algorithm, by considering "trustable" a detection where at least 3 over 10 frames recognize an object as target.

In summary, we can claim that the tools we considered for managing multimedia data, opportunely modified in order to work in an effective way with the Rovers, are really effective for detection tasks.

Implementation

In this section we describe how the various phases of the Behavioral Algorithm have been implemented into our Rover by using our adapted tools. The main actions performed by our devices are detailed in Figure 5.

Obstacle detection and bypassing

As mentioned in Section 2, an obstacle detection and bypassing is underlying and run by every moving robot. Obstacle detection is performed through the use of the robot ultrasound sensors. At every time, the robot checks whether it detects an object by applying the mechanism described in Section 4.1. If this object (identified as an obstacle if not the target) is close, the robot has to bypass it. To do so, the robot just turns α • right. If the obstacle is still there, the robot turns 2 × α • left. If the obstacle is still there, the robot moves backward, turns α • right again and resumes its previous movement (either in searching or approaching phase).

Searching Phase After a preliminary check of the obstacle presence, the Rover is able to perform the Searching Phase, by acquiring the data frames in a suitable way. Data acquisition is performed as Whether the robot detects the target, it will start to reach it. The searching phase is performed through the processing of the WebCam frames. From the analysis of the frames, the device makes a comparison with the target, and will be able to establish whether the target has been found or not. Objets are detected by using adapted tool described in Section 4.1 and identified by using the face recognition tool as explained in Section 4.2.

Approaching Phase In this phase, the robot simply heads in direction of the target till either reaching it or realizing that the target is a "false positive". To detect "false positive", at every step, the robot simply runs the face recognition tools detailed in Section 4.2 again to check whether it confirms the detected object in the right target. The target is considered as reached based on the size Fig. 6: Target reachability. On the left side, the target is supposed unreached since the green diagonal length is under a threshold. On the right side, the target is considered to be close enough and reached. of objects on the WebCam pictures. The robot stops when the diagonal length of the identified target is larger than a threshold as shown on Figure 6 Once the target is reached, the robot needs to advertise its peers to allow them to abort their current task and move to the next one in the list. To do so, the robot sends a radio beacon to its neighbors.

Testbed Description and Results

This section describes the performance of the Behavioral Algorithm implemented on our Rovers in order to accomplish Detection and Covering Tasks.

We consider two scenarios: the first one is based on the use of the HSV technique and the last is a Face Recognition Task in which the Rovers are also able to communicate to each other in order to exchange information about the tasks.

HSV Scenario

In this scenario the task, robots are assigned the detection of a specific object, namely the Rover has to move to towards the target. In Figure 7 one can observe the generation of the HSV Values, by using the tools of the HSV Gui.

When the HSV Task starts, the Behavioral Algorithm allows the process of the Searching Phase and Approaching Phase and the Rover tries to accomplish the assigned task. In this work, we were able to test the effectiveness of the algorithm implemented, by considering an environment with low light and constant brightness conditions. In Figure 8 we can observe a snapshot of the experiments conducted. In order to test the effectiveness of the algorithm, we considered many different scenarios, by keeping similar brightness conditions, and the Rover was able to accomplish its assigned task in all the tests. Fig. 7: HSV Gui Interface. The orange object is the target assigned.

Face Recognition Scenario

In this scenario the target to be detected is a human face.

The goal of the test is the localization of the face and the first Rover that identifies and reaches the Target, sends an "alert" message to the others, in order to "attract" them. The other devices are required to dynamically change their task in this case. In Figure 9(a), we show the Laboratory room where the test has been realized.

We realized some tests to verify the exact number of frames used by the WebCam as "settling" frames, and we verified that 5 frames are necessary and have to be discarded. In practice, the first 5 frames cannot be analyzed.

In the first phase, the Rovers start the Target Searching Phase (see Figure 9(b)).

In the specific test we are considering, the Rover 1 is the first one that detects the target, and then, it starts the Approaching Phase as shown in the Figure 10(a), whereas Rovers 2 and 3 continue in their searching phase. When Rover 1 reaches the target, it broadcasts a message to Rovers 2 and 3 by switching on its red led, in order to be visually identified as a target (see Figure 10). When Rovers 2 and 3 receive the message, they are able to dynamically move to the next task and, their objective will be modified (see Figure 10(b)).

We repeated this kind of tests many times, by changing the initial position of the Rovers and by modifying the position of the target. We noticed, that the only problem related with this kind of experiments was an excessive luminosity. In fact, in this case the limited number of frames elaborated by the devices is not enough accurate.

Conclusions and Future Works

In this work we faced the real implementation of mobile devices based on Arduino platforms and equipped with many sensors and a WebCam. The devices are able to accomplish Detection Tasks of objects and faces. The target objects can be identified by exploiting the combination of shape/color. On the Rovers we implemented a Behavioral Algorithm, where the commands related with the detection have been integrated. In order to make effective the tools considered for the detection purpose, we opportunely modified them, considering the scarce resources available and the movement of the Rovers. We realized a proof-ofconcept of our Behavioral Algorithm and we have shown its effectiveness in different scenarios. The main issue is related with the brightness conditions of the environment and with rapid changing of the luminosity, but the use of more powerful computational devices allows to overcome easily this kind of matter. As future work, we will consider quad-core mini-pc, and we will test our Behavioral Algorithm in more variable conditions of luminosity.

Algorithm 1 2 :

 12 Behavioral Algorithm • Local variables: TargetFound = FALSE; TaskCompleted = FALSE; 1: while the task list is not empty do Move to next task on the list 3: while (TargetFound = FALSE) AND (TaskCompleted = FALSE) do 4: {Searching Phase} 5: while (TargetFound = FALSE) AND (TaskCompleted = FALSE) do 6: ∆t ← Random().

Fig. 1 :

 1 Fig. 1: HSV model.

Fig. 2 :

 2 Fig. 2: RGB vs HSV.

Fig. 3 :

 3 Fig. 3: Identification Process of a Can.

Fig. 4 :

 4 Fig. 4: Identification of an Human Face. @Lena.

Fig. 5 :

 5 Fig. 5: Rover actions Flowchart.

Fig. 8 :

 8 Fig. 8: The target is reached.

  (a) The room where the tests have been realized. (b) Searching Phase for the three Rovers. (c) Identification of the Target.

Fig. 9 :

 9 Fig. 9: The Phases to search and identify a target

  (a) The led is switched on. (b) The others two Rovers cooperate with the first one and reach it.

Fig. 10 :

 10 Fig. 10: Cooperation among the Rovers.

http://store.arduino.cc/index.php?main page=product info&products id=195&language=en

partially supported by the FP7 VITAL.


Florian Caullery
email: florian.caullery@univ-amu.fr

A divisibility criterion for exceptional APN functions

Keywords: 1991 Mathematics Subject Classification. Primary 11C08, 11T06, 11T71 vector Boolean functions, almost perfect nonlinear functions, algebraic surface, CCZ-equivalence, Gold function, Kasami function, exceptional number

We are interested in the functions from F 2 m to itself which are Almost Perfectly Nonlinear over infinitely many extensions of F 2 , namely, the exceptional APN functions. In particular, we study the case of the polynomial functions of degree 4e with e odd and we give a necessary condition on an associated multivariate polynomial for the function to be exceptional APN. We use this condition to confirm the conjecture of Aubry, McGuire and Rodier in some new cases.

Introduction

A vectorial Boolean function is a function f : F 2 m → F 2 m . It is well known that all those functions admit a polynomial representation. Throughout this paper, we will refer to f as a function as well as a polynomial. These objects arise in fields like cryptography and coding theory and are of particular interest in the study of block-ciphers using a substitution-permutation network (SP-network) since they can represent a Substition Box (S-Box). In 1990 Biham and Shamir introduced the differential cryptanalysis in [START_REF] Biham | Differential Cryptanalysis of DES-like Cryptosystems[END_REF]. The basic idea is to analyze how a difference between two inputs of an S-box will influence the difference between the two outputs. This attack was the motivation for Nyberg to introduce the notion of Almost Perfectly Nonlinear (APN) function [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF] which are the function providing the S-Boxes with best resistance against differential cryptanalysis. An APN function is a vectorial Boolean function such that ∀a = 0, b ∈ F 2 m there exist at most two solutions to the equation: f (x + a) + f (x) = b. A complete classification of APN function is an interesting open problem that has been widely studied by many authors. A first approach toward the classification was to consider only power functions and the studies was recently extended to polynomial functions (Carlet, Pott and al [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like crypto-systems[END_REF][START_REF] Edel | A new APN function which is not equivalent to a power mapping[END_REF][START_REF] Edel | A new almost perfect nonlinear function which is not quadratic[END_REF]) or polynomials on small fields (Dillon [START_REF] Dillon | APN Polynomials: An Update. Fq9[END_REF]). On the other hand, several authors (Berger, Canteaut, Charpin, Laigle-Chapuy [START_REF] Berger | On almost perfect nonlinear functions over F 2 n[END_REF], Byrne, McGuire [START_REF] Byrne | Quadratic Binomial APN Functions and Absolutely Irreducible Polynomials[END_REF] or Jedlicka [START_REF] Browning | An APN permutation in dimension six[END_REF]) showed that APN functions cannot exist in certain cases. Some also studied the APN functions on fields of odd characteristic (Leducq [19], Pott and al. [START_REF] Dobbertin | APN functions in odd characteristic[END_REF][START_REF] Poinsot | Non-Boolean almost perfect nonlinear functions on non-Abelian groups[END_REF], Ness, Helleseth [START_REF] Ness | Tor A new family of ternary almost perfect nonlinear mappings[END_REF] or Wang, Zha [START_REF] Zha | Power functions with low uniformity on odd characteristic nite elds[END_REF][START_REF] Zha | Almost perfect nonlinear power functions in odd characteristic[END_REF] ).

One way to face the problem of the classification is to consider the function APN over infinitely many extensions of F 2 , namely, the exceptional APN functions. The two best known classes of exceptional APN functions are the Gold functions: f (x) = x 2 i +1 and the Kasami functions f (x) = x 4 i -2 i +1 , both are APN whenever i and m are coprime. We will refer to 2 i + 1 and 4 i -2 i + 1 respectively as the Gold and Kasami exponent. Hernando and McGuire proved that those two functions are the only monomial exceptional APN functions [START_REF] Hernando | Gary Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions[END_REF]. It was the starting point for Aubry, McGuire and Rodier to formulate the following conjecture:

Conjecture 1 ([1]
). The only exceptional APN functions are, up to Carlet Charpin Zinoviev-equivalence (as defined below), the Gold and Kasami functions.

We provide the definition of the Carlet Charpin Zinoviev equivalence: Definition 1.1 ([8]). Two functions f and g are Carlet Charpin Zinoviev (CCZ-)equivalent if there exist a linear permutation between their graphs (i.e. the sets {x, f (x)} and {x, g(x)}).

It is worth pointing out that all the functions CCZ-equivalent to an APN function are also APN [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like crypto-systems[END_REF].

By means of a simple rewriting of the definition of APN function in terms of algebraic geometry, Rodier was able to prove that, if the projective closure of the surface X defined by the equation:

f (x) + f (y) + f (z) + f (x + y + z) (x + y)(y + z)(z + x) = 0
has an absolutely irreducible component defined over F 2 m , then f is not an exceptional APN function [START_REF] Rodier | Borne sur le degré des polynômes presque parfaitement non-linéaires[END_REF]. We will denote by X the projective closure of X.

From now on we let

q = 2 m , φ(x, y, z) = f (x) + f (y) + f (z) + f (x + y + z) (x + y)(y + z)(z + x) , φ i (x, y, z) = x i + y i + z i + (x + y + z) i (x + y)(y + z)(z + x) . and A = (x + y)(y + z)(z + x).
Writing f = d i=0 a i x i with d the degree of f , we have φ = d i=0 a i φ i and so the homogeneous equation of X is given by

ϕ(x, y, z, h) = d i=0 a i φ i h d-i .
The idea is to use the fact that if X ∩ H, where H is an hyperplane, has a reduced absolutely irreducible component then X has an absolutely irreducible component (see [START_REF] Aubry | A few more functions that are not APN infinitely often, Finite Fields Theory and applications[END_REF]). We wish to exploit this criterion to prove that the functions which are not CCZ-equivalent to a Gold or Kasami monomial are not exceptional APN. This approach enabled Aubry, McGuire and Rodier to state, for example, that there is no exceptional APN function of degree odd not a Gold or Kasami exponent as well as functions of degree 2e with e an odd number [START_REF] Aubry | A few more functions that are not APN infinitely often, Finite Fields Theory and applications[END_REF].

The next step was to study the polynomials of degree 4e. Under the assumption that φ e is absolutely irreducible, Rodier proved that an exceptional APN function should have its associated polynomial φ divisible by another polynomial with a specific form (see [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF]). In the same paper, he treated the case of exceptional APN function of degree 12. It was later showed in [START_REF] Caullery | The exceptional APN functions of degree 4e[END_REF] that there is no exceptional APN polynomial functions of degree 4e with e > 3 such that φ e is absolutely irreducible.

At this point, a natural question is: what happens when φ e , with e odd, is not absolutely irreducible? Using the symmetry in the variables x, y and z of the polynomial φ and the language of Weil divisors, we will determine all the possible divisors of the surface X. This result includes the main result of [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF] as a corollary and gives what I believe to be the limit of this kind of reasoning. With this tool, we will treat the smallest untreated example, namely e = 5 and confirm the correctness of the conjecture in this case.

The state of the art

Using the approach described in the introduction Aubry, McGuire and Rodier obtained the following results in [START_REF] Aubry | A few more functions that are not APN infinitely often, Finite Fields Theory and applications[END_REF].

Theorem 2.1 ( [START_REF] Aubry | A few more functions that are not APN infinitely often, Finite Fields Theory and applications[END_REF]). If the degree of the polynomial function f is odd and not an exceptional number then f is not an exceptional APN function.

Theorem 2.2 ([1]

). If the degree of the polynomial function f is 2e with e odd and if f contains a term of odd degree, then f is not an exceptional APN function.

There are some results in the case of Gold degree 2 i + 1:

Theorem 2.3 ([1]). Suppose f (x) = x 2 i +1 + g (x) where deg (g) ≤ 2 i-1 + 1. Let g (x) = 2 i-1 +1 j=0
a j x j . Suppose moreover that there exists a nonzero coefficient a j of g such that φ j (x, y, z) is absolutely irreducible. Then f is not an exceptional APN function.

This result has been extended by Delgado and Janwa in [START_REF] Delgado | On the Conjecture on APN Functions[END_REF] with the two following theorems:

Theorem 2.4 ([11]). For k ≥ 2, let f (x) = x 2 i +1 +h(x) ∈ F q where deg(h) ≡ 3 (mod 4) < 2 i + 1. Then f is not an exceptional APN function. and Theorem 2.5 ([11]). For k ≥ 2, let f (x) = x 2 i +1 + h(x) ∈ F q where deg(h) = d ≡ 1 (mod 4) < 2 i + 1. If φ 2 i +1 and φ d are relatively prime, then f is not an exceptional APN function.
There also exist a result for polynomials of Kasami degree 2 2i -2 i + 1:

Theorem 2.6 ([16]). Suppose f (x) = x 2 2i -2 i +1 +g (x) where deg (g) ≤ 2 2k-1 - 2 k-1 + 1. Let g (x) = 2 2k-1 -2 k-1 +1 j=0
a j x j . Suppose moreover that there exist a nonzero coefficient a j of g such that φ j (x, y, z) is absolutely irreducible. Then f is not an exceptional APN function.

Rodier proved the following results in [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF].

Theorem 2.7 ( [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF]). Let f : F q → F q be an exceptional APN function of degree 4e with e such that φ e is absolutely irreducible. Then the polynomials of the form

(x + y) (x + z) (y + z) + P, with P (x, y, z) = c 1 x 2 + y 2 + z 2 + c 4 (xy + xz + zy) + b 1 (x + y + z) + d 1 , for c 1 , c 4 , b 1 , d 1 ∈ F q 3 , divides φ.
Remark 2.8. This theorem is originally stated for e ≡ 3 (mod 4) but its proof is also valid with e such that φ e is absolutely irreducible (see [START_REF] Caullery | The exceptional APN functions of degree 4e[END_REF]).

There are more precise results for polynomials of degree 12.

Theorem 2.9 ( [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF]). If the degree of the polynomial f defined over F q is 12, then either f is not an exceptional APN function or f is CCZ-equivalent to the Gold function x 3 . Also, using the same approach, the present author proved the following: Theorem 2.10 ( [START_REF] Caullery | The exceptional APN functions of degree 4e[END_REF]). If the degree of the polynomial f defined over F q is 4e with e > 3 and such that φ e is absolutely irreducible, then f is not an exceptional APN function.

In particular, φ e is absolutely irreducible when e ≡ 3 (mod 4) (see lemma 4.4) so there is no exceptional APN function of degree 4e with e ≡ 3 (mod 4).

New Results

The main result of this paper is: Theorem 3.1. Let f : F q → F q be an exceptional APN function of degree 4e with e odd and let

φ(x, y, z) = f (x) + f (y) + f (z) + f (x + y + z) (x + y)(y + z)(z + x)
be its associated polynomial. Let σ be a generator of the Galois group Gal(F q 3 /F q ). One of these three conditions holds

(1) The polynomial φ is divisible by

(A + P (x, y, z)) (A + σ (P (x, y, z))) A + σ 2 (P (x, y, z)) ,
where P (x, y, z) is a symmetric polynomial of degree 2 defined over F q 3 . (2) The polynomial φ is divisible by

(Ψ(x, y, z) + L(x, y, z)) (AΨ(x, y, z) + R(x, y, z)) σ (AΨ + R(x, y, z)) σ 2 (AΨ(x, y, z) + (R(x, y, z)) ,
where Ψ(x, y, z) is a non absolutely irreducible symmetric factor of φ e defined over F q 3 but not over F q and R(x, y, z) and L(x, y, z) are symmetric polynomials of degree respectively less than deg(AΨ) and deg(Ψ) defined respectively over F q 3 and F q .

(3) The polynomial φ is divisible by Aψ 3 (x, y, z) + S(x, y, z) σ Aψ 3 (x, y, z) + S(x, y, z) σ 2 Aψ 3 (x, y, z) + S(x, y, z) , where ψ(x, y, z) is a square-free non absolutely irreducible symmetric factor of φ e defined over F q 3 such that ψ, σ(ψ) and σ 2 (ψ) are coprime. Remark 3.2. If φ e is absolutely irreducible, then we get directly theorem 2.9 as there is clearly no polynomial satisfying the conditions ( 2) and ( 3).

In section 6, we give a direct application of the last result to the case of polynomial APN function of degree 20.

Theorem 3.3. Let f : F 2 m → F 2 m be an exceptional APN function of degree d = 20.
Then m is odd and f is CCZ-equivalent to x 5 .

Preliminary lemmata

We will need the following lemmas: Lemma 4.1. ( [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF]) The kernel of the mapping ̟ : f → φ is made of q-affine polynomials.

Lemma 4.2. ( [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF]) The class of APN functions is invariant under the addition of q-affine polynomials.

In particular, this result means that we can restrict ourselves to polynomials without any term of degree a power of 2.

Lemma 4.3. ( [START_REF] Aubry | A few more functions that are not APN infinitely often, Finite Fields Theory and applications[END_REF]) Writing i = 2 j k we have:

φ i = A 2 j -1 φ 2 j k . In particular, φ i is reduced if i is odd. Proposition 1. ([ 18
]) The polynomial φ 2 i +1 decomposes into absolutely irreducible factors as follow:

φ 2 i +1 (x, y, z) = α∈F 2 i -F2 (x + αy + (α + 1) z) .
Lemma 4.4 ([18]). The polynomial φ e is absolutely irreducible if e ≡ 3 (mod 4).

Lemma 4.5 ([1]

). The polynomials φ e and A are coprime if and only if e is odd.

Our proof of theorem 3.1 relies on the two following propositions: Proposition 2 ( [START_REF] Rodier | Borne sur le degré des polynômes presque parfaitement non-linéaires[END_REF]). The surface X associated to an exceptional APN function does not contain any absolutely irreducible component defined over F q different from x + y = 0, y + z = 0 or z + x = 0.

Lemma 4.6 ([1]

). Let H be an hyperplane in P 3 (F q ). If the curve X ∩ H has a reduced absolutely irreducible component defined over F q then X has an absolutely irreducible component defined over F q .

Proof of theorem 3.1

The goal of this proof is to describe how an absolutely irreducible factors of φ should look like under the assumption that f is an exceptional APN function.

The key idea is to use lemma 4.6 along with the fact that the equation of the intersection of the surface X with the hyperplane infinity is known. For the sake of clarity, we will use the language of Weil divisors (see [START_REF] Shafarevich | Basic Algebraic Geometry[END_REF] for an introduction to Weil divisors) but one could directly translate this proof into terms of absolutely irreducible factors of polynomials.

Let f be an exceptional APN function of degree d = 4e. From proposition 2, its associated projective surface X does not contain any absolutely irreducible component defined over F q excepted perhaps x + y = 0, x + z = 0 or y + z = 0.

Let H ∞ be the plane at infinity in P 3 (F q ) (i.e. the plane of equation h = 0). By lemma 4.6, the intersection X ∩ H ∞ cannot contain any reduced absolutely irreducible component defined over F q different from x + y = 0, y + z = 0 or z + x = 0. From lemma 4.3 we have:

(5.1)

φ d = A 3 φ 4 e , meaning that X ∩ H ∞ is defined by the equation A 3 φ 4 e = 0.
Let D be the divisor associated to the hyperplane section X ∩ H ∞ . We denote by A 0 , A 1 and A 2 the divisors associated, respectively, to the section of the planes of equation x + y = 0, y + z = 0 and z + x = 0 with the plane H ∞ . Let p i be an absolutely irreducible factor of φ e . We will denote by C i the divisors associated to the section of the surface of equation p i (x, y, z) = 0 with the plane H ∞ . Then, from (5.1) and lemma 4.3:

D = 3(A 0 + A 1 + A 2 ) + 4 i C i .
Now let X 0 be an absolutely irreducible component of X which contains the line x + y = 0 in H ∞ . As we have supposed that f is an exceptional APN function, X 0 is defined over an extension of F q , say F q t . We choose t to be the smallest possible. Throughout this paper we will refer to σ as a generator of the Galois group Gal(F q t /F q ). We set X 0 to be the divisor associated to the section X 0 ∩ H ∞ , as X 0 is a component of X, X 0 is a subdivisor of D, and as X 0 contains the line x + y = 0 in H ∞ we have X 0 ≥ A 0 . Our goal is to find the possible forms for X 0 .

5.1. The case where X 0 ≥ 2A 0 .

In that case we have:

X 0 + X σ 0 ≥ 4A 0 .
But that is a contradiction since X 0 + X σ 0 must be a subdivisor of D and D contains only three times A 0 . 5.2. The case where X 0 contains only one time A 0 .

From the previous section, we know that X 0 is of the form A 0 + D 0 where D 0 is a subdivisor of D which does not contain A 0 . Thus there exists two other absolutely irreducible components of X, say X 1 and X 2 , with associated divisors respectively X 1 and X 2 , that contains only one time A 0 .

Let G be the Galois group Gal(F q t /F q ), since G fixes the line x + y = 0 in H ∞ , the group G acts on the X i and let us consider the orbit of X 0 under this action. If it contains just X 0 , then X 0 is defined over F q which is impossible from proposition 2. If it contains X 0 and X 1 then G fixes X 2 and X 2 is then defined over F q , that is again in contradiction with proposition 2. Finally, that means that it contains the three components. Then G acts transitively on these three components. Let G 1 the stabilizer of X 0 . Then the group G/G 1 is isomorphic to Z/3Z, and G 1 is the only subgroup of G of index 3. The same is true for the lines y +z = 0 and z +x = 0.

The case X

0 = A 0 + i n i C i .
First suppose that all the n i s are zero, hence X 0 = A 0 and then the equation of X 0 would be x + y + b = 0 with b ∈ F q t and b ∈ F q . In this case x + y + b would divide f (x) + f (y) + f (z) + f (x + y + z). As b ∈ F q , by the action of G, x + y + σ(b) would be a distinct plane containing the line x + y = 0 in H ∞ . As there are only three distinct components of X containing the line x + y = 0 in H ∞ and as t is minimal, this implies that t = 3. By symmetry of the variables x, y, z in the expression of 3 which is of the form given in 1 in theorem 3.1. Now suppose that there exist at least one nonzero n i . Thus we have:

f (x) + f (y) + f (z) + f (x + y + z), z + y + b and x + z + b divide also f (x)+f (y)+f (z)+f (x+y+z). Finally f (x)+f (y)+f (z)+f (x+y+z) is divisible by (x+y+b)(z +y+b)(x+z +b) = (x+y)(y+z)(z +x)+b(x 2 +y 2 +z 2 +xy+xz +zy)+b
X 1 = A 0 + i n i C σ i and X 2 = A 0 + i n i C σ 2 i .
Now suppose that X 0 is not invariant under the transposition (x, y), then the divisor

X 4 = A 0 + i n i C (x,y) i
is different from the precedents and j X j = 4A 0 + D 1 should be a subdivisor of D (we recall that φ is symmetric). That is a contradiction to the fact that D contains only three times A 0 and hence X 0 is invariant under (x, y).

Denote Y 0 (respectively Z 0 ) the image of X 0 by the permutation (x, y, z) (respectively (x, z, y)) and define

Y 1 = Y σ 0 and Y 2 = Y σ 2 0 . With the same argument as before, Y 0 should be invariant under (y, z), that is i n i C i is invariant under (x, z). Thus i n i C i (i.e. the product ψ = i p i (x, y, z) ni ) is symmetric.
For the sake of contradiction, suppose now that there exists an i and k such that n k and n i are nonzero and C k = C σ i . Hence, X 0 + X 1 + Y 0 + Y 1 + Z 0 contains at least five times C k which cannot happens since D contains it only four times. The same is true when we consider σ 2 . Now suppose that one of the n i , namely n k , is greater than 1. Then X 0 + Y 0 + Z 0 > A 0 + A 1 + A 2 + 6C k , but there is only four times C k in D because φ e is reduced (see lemma 4.3), so that is a contradiction and all the n i s are maximum 1.

To summarize, X 0 should be of the form A 0 + i n i C i where n i ≤ 1 and i n i C i is invariant under the action of the symmetry group and does not share any common component with i n i C σ i or i n i C σ 2 i . By the argument of [25, section 5.9] (see also 5.2.3 in the present paper), we get the condition (3) of theorem 3.1.

The case X

0 = A 0 + A 1 + i n i C i . If X 0 = A 0 + A 1 + i n i C i we get X 1 = A 0 + A 1 + i n i C σ i and X 2 = A 0 + A 1 + i n i C σ 2
i . With the notations above we also have

Y 0 = A 1 + A 2 + i n i C (x,y,z) i
. Now we just have to remark that the subdivisor of D,

X 0 + X 1 + X 2 + Y 0 is greater than 3A 0 + 4A 1 + A 2 .
That is impossible since D contains only three times A 1 . Hence X 0 cannot be of the form A 0 + A 1 . In the same way, we eliminate the case

X 0 = A 0 + A 2 + i n i C i . 5.2.3. The case X 0 = A 0 + A 1 + A 2 + i n i C i .
First suppose that the n i s are all zero. That is the case 5.9 in [START_REF] Rodier | Functions of degree 4e that are not APN infinitely often[END_REF], we copy the proof here for the sake of completeness. In this case, the equation of such X 0 is of the form (x+y)(x+z)(y+z)+P (x, y, z) where P is a polynomial of degree at most 2. Let σ be a generator of G. The equation of X 1 is (x+y)(x+z)(y +z)+σ(P )(x, y, z) and the equation of X 2 is (x+y)(x+z)(y+z)+σ 2 (P )(x, y, z). Since these polynomials are irreducible (we have supposed that X 0 is irreducible) and distinct, they are prime with each other. Therefore f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 0 + x 1 + x 2 ) is divisible by (5.2)

2 i=0 (x + y)(x + z)(y + z) + σ i (P )(x, y, z)
The equation of the curve X ∞ is ((x + y)(x + z)(y + z))

3 φ 4 e = 0 so we find that the product (5.2) can contain only three summands, hence σ 3 (P ) = P . Hence P is defined on F q 3 and X 0 also. The product (5.2) must be symmetric in the variables x, y, z, since if it were not, the image of the product (5.2) by some element of the symmetry group G of the 3 variables would be different, and also divide f (x) + f (y) + f (z) + f (x + y + z), therefore forcing the curve X ∞ to contain more than 3 time the line x + y = 0. If P is not symmetric in the variables x, y, z, then the orbit of P by the symmetry group G of the 3 variables would be contained in the set {P, σ(P ), σ 2 (P )} since the product (5.2) is symmetric. The orbit of P under G is not reduced to {P } since P is not symmetric. It is not either reduced to two elements, because the third element would be symmetric, so it is equal to the set {P, σ(P ), σ 2 (P )}. The stabilizer of P in G would then be reduced to a transposition. But the stabilizer of σ(P ) would contain a conjugate transposition, and this transposition would also fix P , as the action of G and G commute. So it is impossible, which proves that P must be symmetric. Therefore P is of the form

P (x, y, z) = c 1 (x 2 + y 2 + z 2 ) + c 4 (xy + xz + zy) + b 1 (x + y + z) + d 1 .
That is the condition (1) of theorem 3.1.

So the only case left is when at least one of the n i s is non-zero. In this case we have:

X 1 = A 0 + A 1 + A 2 + i n i C σ i ,
and

X 2 = A 0 + A 1 + A 2 + i n i C σ 2 i .
If i n i C i is not invariant under the action of the symmetry group, then there exist a divisor X 3 > A 0 + A 1 + A 2 different from X 0 , X 1 and X 2 . Then j X j > D, which is a contradiction and i n i C i is invariant under the action of the symmetry group. Moreover, if i n i C i lies over F q and corresponds to an absolutely irreducible factor of φ i (i.e. only one of the n i 's is equal to one and all the others are zero), there exists a divisor X 4 which is defined over F q and which contains C i , leading again to a contradiction. This corresponds to the condition (2) of theorem 3.1.

6. Some applications 6.1. Exceptional APN polynomials of degree 20.

In this section, we will use the theorem 3.1 to investigate the case where e = 5. The decomposition of φ 5 is given by proposition 1:

φ 5 = (x + αy + α 2 z)(x + α 2 y + αz),
where α is in F 4 -F 2 . Hence, the only symmetric factor of φ 5 is φ 5 itself and then the condition (3) of theorem 3.1 cannot hold. Also, the condition (1) is already treated in [START_REF] Caullery | The exceptional APN functions of degree 4e[END_REF] and the conclusion is that f is CCZ-equivalent to x 5 . So we only have to study the consequences of condition (2) on f . That is

φ = (φ 5 + L(x, y, z)) (Aφ 5 + R(x, y, z)) (Aφ 5 + σ (R(x, y, z))) Aφ 5 + σ 2 (R(x, y, z)) ,
where L is a symmetrical polynomial of F q of degree 1 and R is a symmetrical polynomial of F q 3 of degree 4.

The first thing we show is that L(x, y, z) = a(x + y + z) + b = 0. As φ does not have any absolutely irreducible component, (φ 5 + L) cannot be absolutely irreducible. Hence, there exist two polynomials G(x, y, z) and H(x, y, z) in F2 [x, y, z] such that G × H = φ 5 + as 1 + b. Writing G i and H i the homogeneous components of degree i of G and H respectively, we get:

φ 5 = G 1 × H 1 .
Without loss of generality we can assume that G 1 = x + αy + α 2 z and H 1 = x + α 2 y + αz. Also,

a(x + y + z) = G 0 (x + α 2 y + αz) + H 0 (x + αy + α 2 z),
and hence

G 0 + H 0 = a G 0 α + H 0 α 2 = a G 0 α 2 + H 0 α = a
Plugging G 0 = H 0 + a into the last two equations we get H 0 = aα and H 0 = a(α + 1), that is a = H 0 = G 0 = 0 and thus b = 0, so L(x, y, z) = 0. Now, as φ = 20 j=0 a j φ j , we have for every j = 0, . . . , 20, φ 5 divides a j φ j . Hence φ = a 20 φ 20 + a 10 φ 10 + a 5 φ 5 .

That is f is equal to a 20 x 20 +a 16 x 16 +a 10 x 10 +a 8 x 8 +a 5 x 5 +a 4 x 4 +a 2 x 2 +a 1 x+a 0 . As the class of APN polynomial is invariant under the addition of q-affine polynomial, we can restrict ourselves to f = a 20 x 20 + a 10 x 10 + a 5 x 5 . Clearly, f is of the form ϕ(x 5 ) where ϕ(x) is a q-affine polynomial of degree 4, hence f is EA (thus CCZ) equivalent to the polynomial x 5 .

To sum up, what we proved is that the exceptional APN function of degree 20 are CCZ-equivalent to the function x 5 . As this function is APN only on every extension of F 2 of odd degree we get that m is an odd number and this concludes the proof of theorem 3.3.

Other examples.

The case e = 9 can be solved in the same way than the precedent one. But the impossibility of showing that φ 9 + L(x, y, z) is not absolutely irreducible if and only if L is zero leads to a long calculation which is not of real interest here but one can prove that f is CCZ-equivalent to x 9 .

One can also ask if there exist e such that the condition (3) can happen. We provide an example here.

Take e = 2 6 + 1. Clearly, e is a Gold exponent so the decomposition of φ 65 is given by proposition 1. That is φ 65 = α∈F 2 6 -F2 (x + αy + (α + 1)z). Now, let β be a generator of F 2 6 , then the polynomial ψ = (x + βy + (β + 1)z)(x + β 7 y + (β 7 + 1)z)(x + β 8 y + (β 8 + 1)z) (x + β 56 y + (β 56 + 1)z)(x + β 55 y + (β 55 + 1)z)(x + β 62 y + (β 62 + 1)z) is symmetric, defined over F 2 3 (and then on F q 3 ) and ψ, σ(ψ) and σ 2 (ψ) are relatively prime if F q does not contain F 2 3 . That means that the polynomial ψ meets the condition (3) of theorem 3.1. Again, some long calculations would be necessary to investigate the consequences of this division.

In conclusion, I think that this method reaches its limit here and I would suggest to try a different approach to solve the remaining cases.