Bruno Lauga
email: b.lauga@arvalis.fr

Béatrice Balvay
email: 2beatrice.balvay@idele.fr

Laurent Topart
email: laurent.topart@orange.com

Juliette Leclaire
email: jleclaire@fiea.fr

Anthony Clenet
email: 5aclenet@smag-group.com

François Brun
email: francois.brun@acta.asso.fr

François Pinet
email: francois.pinet@irstea.fr

Catherine Roussey

Mehdi Sine

MULTIPASS : Managing the consents of access to farm data in a chain of trust to make new services emerge for farmers

Keywords: farm data, data management, consent, transparency, chain of trust

come   L'archive ouverte pluridisciplinaire

Farmers' expectations of such an ecosystem have been expressed during workshops. They want to better identify existing data flows, including actors, data processes, and data clusters. Based on the farmers' expectations, the MULTIPASS project stakeholders have proposed the architecture of an ecosystem integrating two consent management tools as "pilots". This ecosystem should take in charge the interoperability between each consent management tools or with future tools. This solution is based on a shared typology of data and data processes as well as on the specifications of the consent message content. All these elements should be easily accessible to meet the interoperability need of the ecosystem. It is also based on a router, which provides unified access to consent management tools (using API). In particular, it provides the farmer (beneficiary) with an exhaustive view of his/her consents (which can be distributed on several consent management systems), meeting farmers' expectations for transparency. It is also the point where a data provider can check whether the consent required to provide data exists, without needing to know which consent management system is concerned.

In this project, the stakeholders want to demonstrate to agricultural professional organizations the benefits and feasibility of a consent management ecosystem. By strengthening the confidence of farmers to share data, the project will allow the emergence of new knowledge and new services.

INTRODUCTION

Farmers are engaged in a progress for sustainable and productive agriculture. With the emergence of digital technologies, farms become a relevant source of data to meet the challenges of multiperformance agriculture. These data are the basis of the decision-making process. There is a datadriven agriculture based on the data transfer within the farm. These data also make it possible to create new knowledge or tools that improve the precision and relevance of agricultural operations in order to increase yields without negative impact on the environment.

Beyond the services provided, access to farmers' data depends on a clear understanding of their use, which must be done in a transparent way [START_REF] Brun | ACTA -Les Instituts Techniques Agricoles: L'accès aux données pour la Recherche et l'Innovation en Agriculture[END_REF]. This is a real concern for both farmers, who cannot control the uses of their data, and also data providers who have difficulties in determining the access and reuse permissions they can provide on the farmers' data they host. Access rights must be properly managed, as well as the farmer's consent for the uses of her/his data. The conditions related to this consent must be easily accessible and modifiable by the farmer. It is this chain of trust that the MULTIPASS project wants to implement.

Through the MULTIPASS project, the partners want to make available to farmers and data producers an interoperable farmers' consent management ecosystem, protecting data exchanges improving confidence to share their data with other organizations.

CHALLENGES FOR A CONSENT MANAGEMENT ECOSYSTEM

Towards a widespread use of consents

Consents are the adherence of one party to the request made by another. In the case of personal data, consent is one of the 6 legal bases provided by the EU General Data Protection Regulation (GDPR, 2016) which authorizes the implementation of data processing. The law does not require the systematic collection of consents before processing personal data, because other legal bases can be invoked to process these data such as a mission of general interest or a contractual commitment (CNIL, 2018). Nevertheless, consents will enable the management of agricultural data exchanges not specified in the contracts.

To authorize an agricultural data processing and to reinforce the transparency of these uses, the farmer must be able to express her/his consent as shown in Figure 1 :

Build a chain of trust

Our goal was to build an ecosystem of stakeholders to manage consents and to create the engagement rules of these actors. We defined the typology of stakeholders presented in Table 1 involved in any farm data exchanges and consents management.

Table 1. Typology of actors in a consent management ecosystem

Term Definition

Right holder

The person who has the rights on the data. The consent of this person is needed to exchange data. In the MULTIPASS project, she/he is a farmer or breeder.

Delegatee

The right holder has delegated to a person or an organization (i.e., a delegatee) the right to give consents on her/his behalf.

Consent manager The manager in charge of a consent management system.

Service provider The organization that sells service to farmers and that needs an access to data. It is the beneficiary of the consent.

Data provider

The manager of the service (database) in charge of providing the data to the service provider.

Consent recorder The organization that registers consents in the consent management system.

In this chain of trust, each actor has a responsibility and must satisfy good practices related to the use of agricultural data and consents.

Respecting good practices

Jurists seem to think that, in the absence of a specific legal regulation, the control of agricultural data is ensured only by contracts with the farmer [START_REF] Douville | Contrat et données agricoles[END_REF]. The control will not come from the law but from a voluntary commitment made by the parties to respect some good practices in data uses.

The French DataAgri code of conduct (FNSEA, 2018) leaded by the "Fédération Nationale des Syndicats d'Exploitants Agricoles" (FNSEA) and the "Jeunes Agriculteurs" (JA), and the European CODE OF CONDUCT (EU code of conduct, 2018) clearly goes in this direction.

In this context, farmers' expectations of such an ecosystem have been expressed in various workshops. Farmers regret that so far they had not been consulted much when the service providers processed their data. They expressed a need for transparency and want to better identify existing flows, including stakeholders, data uses and associated data categories. Based on farmers' expectations, the MULTIPASS project stakeholders have proposed one architecture of an ecosystem integrating two consent management tools as "pilots" and the conditions for their interoperability with each other or with future tools.

IMPLEMENTATION OF THE MULTIPASS ECOSYSTEM

Proposed architecture

There already exist consent management solutions dedicated to agriculture. These systems are often designed for particular needs. These different consent management systems can be freely chosen by the ecosystem stakeholders. Consents are stored in the consent management systems with the only constraint to register the information expected in the MULTIPASS ecosystem interfaces.

The main tool defined in the MULTIPASS ecosystem is a router that guarantees the interoperability of the different consent management systems. It allows a unified access to consents to provide a list of them (by right holders, service providers, etc.) or to verify the existence of consents before data exchanges. For this, it knows and can query the various consent management systems which will have interfaces (APIs) similar to those of the MULTIPASS router.

In particular, it provides the right holder with an exhaustive view of her/his consents (which can be distributed across several consent management systems), meeting farmers' transparency needs. The router also allows a data provider to check if the consent required for a data exchange exists, without needing to know in which consent management system it is managed. There is also a traceability of these controls. The use case diagram presented in Figure 2 shows the expected roles of each of the actors as well as the functional scope of the MULTIPASS router.

Figure 2. Use case diagram of MULTIPASS router

All actors other than right holders must register on the router before they can use it. Their registration is validated by a router's administrator. A right holder (or her/his delegatee) does not interact with the router. It is the role of the consent recorder to allow the input or the modification of consents. Only the right holder can see her/his consents once she/he is authenticated. The consent recorder cannot see them. This security is especially needed when the consent recorder is also a service provider (it must not see if the farmer works with its competitors). For this, either it will have made a contractual commitment in its contract with the farmer, or it will be committed by adhering to a charter or it will be obliged by GDPR in the case of personal data.

The management of the data repositories is the responsibility of a router's administrator.

Technical architecture

The router has a Java REST API that exposes business and administrative services. As consents are by nature sensitive data that must be secured, HTTPS is used for the exchanges. The OAuth protocol is used for authentication. A signature mechanism guarantees the API that the token issued during authentication process has been generated by the system. The passwords of the different users are stored in a SSHA hashed form in an LDAP server. A Java human-machine interface allows system administrators to manage the different users and data repositories (data categories and uses) of the router.

A reverse proxy "HA Proxy" is used to secure the application upstream. This system will also be used for load balancing between different downstream application servers. The PostgreSQL database that registers actors, data repositories and logs could eventually be transferred into an elastic stack.

Conditions for ecosystem interoperability

The router is an important part of the ecosystem interoperability. It is based on the main concept of consent. Consents are not managed in the database of the router, but only in the interfaces. The identification of the companies (farm, data or service provider) is done by the French SIRET identifier but the system allows the use of another identifier.

Table 2. Description of the concept of consent

Ontologies are one of the possible solutions for solving data interoperability issues. The word ontology covers a large number of different data sources ranging from thesauri to schemas shared on the Web through semantic Web technologies [START_REF] Roussey | Les ontologies en agriculture[END_REF]. In the MULTIPASS project, we studied different agricultural data exchange schemes, and in particular GIEA ("Gestion des Informations de l'Exploitation Agricole" -a model for Farm Information Management), a model created in France for data sharing [START_REF] Pinet | The use of UML as a tool for the formalisation of standards and the design of ontologies in agriculture[END_REF]. These schemes propose a vocabulary dedicated to agriculture, but too complex and not suitable for the uses in the context of consent management. The definition of consents will be associated with a typology of data and a typology of uses that remains to be defined. We recommend that these lists will be organized (hierarchies of category) and shared on the Web to meet the interoperability need of the ecosystem.

A Blockchain could constitute the ecosystem on its own, but the challenge at this stage is to explore its promises in terms of trust decentralization. For this, in the second phase of the project, two consent management tools will be compared within use cases. The first one is based on a trusted third party (France Génétique Elevage, 2016) and the second one will be based on Blockchain technology.

MULTIPASS does not have the ability to interfere with consent management systems. They have to verify that the person who registers a consent is the one for whom the consent is given. It is therefore recommended to clearly identify the users with the creation of identity providers for agriculture, as there are elsewhere (French administration, Google or Facebook). Finally, it is the responsibility of the consent manager to ensure the legal value of the consents collected. The participants of the MULTIPASS workshop held on Sept 27th, 2018 (bringing together socio-economic partners of the farmer) highlighted the overlap in the regulatory bases of contracts and consents. There may be a risk of contradiction between a consent and a pre-existing contract.

CONCLUSIONS

The project aims to demonstrate to agricultural professional organizations the benefits and feasibility of a consent management ecosystem through limited but concrete use cases in France. The Blockchain technology will be evaluated to explore its promises in terms of trust decentralization. The router designed by the partners will implement a proof of concept for interoperability between existing and future consent management systems. It provides a solution ("data passport") to farmers for the control on their data and on the transparency of the data uses.

By strengthening the confidence of farmers to share their data, the project will bring new knowledge and new services. It promotes open innovation, i.e. the emergence of agricultural applications coupled with farmers' data from any data source or connected object. In this context, the goals are (1) to avoid the risk of concentration of innovation, and (2) the creation of knowledge by the analysis of massive farm data, in a chain of trust. (2018)(2019)(2020) is funded by the French Ministry of Agriculture (in CASDAR program) and its partners.

MULTIPASS

Figure 1 .

 1 Figure 1. Example of a consent use for data exchanges

th EFITA International Conference | Rhodes Island, Greece | June 27-29, 2019

DISCUSSION 12 th EFITA International Conference | Rhodes Island, Greece | June 27-29, 2019


Popular song in Britain and in France in 1915 : processes and repertoires

Why 1915

It is important to have a tight focus. Books have been written on « British music hall from 1860 to 1920 », but for a deeper understanding, a closer focus is essential. Things in entertainment changed quickly, if less quickly than today .

ago. I'm interested to see which of the tools we have developed to study more recent pop music can be applied to the situation a century ago. So all the songs I will mention (and all the song titles on your handout) are from that one year.

What did the two industries have in common?

In both cases we are talking of music from the towns. Since 1850 the majority of the British population lives in towns, and in France the size and power of Paris Bordeaux or Marseille sis constantly growing.

In both cases, sheet music and live performance were at the centre of the industry. Gramophones are reserved to a small minority of the population : often middle class somewhat geekish amateurs rather than the ruling class .

In both cases it is the song, the 3 to 4 minutes song which dominates, and a singalong chorus is frequently used. The star system is well established and the private lives of the leading singers widely discussed in the popular press.

In both cases it is a mass industry. The new music halls being built in Britain or in France have 2 000 or 3 000 seats. The French café concerts from 200 to 1 000 seats.

The audience for the songs is socially mixed, mass and intergenerational. Twelve year olds listen to the same music in the streets and in the theatres as do adults and old people.

General levels of technology are similar. From the point of view of the song the key fact is that microphones will not be used on stage until the late twenties. This fact must have influenced not only the use of voice but also no doubt the relationship with the audience, and the important role given to singalong choruses.

The fact that the radio has not yet arrived has an important effect. Almost every night the singer is performing to people who have never heard the songs before. It is fair to say that this situation does not encourage musical innovation. The singalong sections in particular must be extremely easy to pick up.

What were the differences?

The British music industry is much bigger. Urbanization is far more advanced. Remember the French army of 1914 is fundamentally an army of peasants whereas the British army is an army of urban workers.

Concentration of capital in the entertainment industry is far more advanced.

In Britain, in 1915, music hall chains or individual theatres have shareholders, and are highly profitable enterprises In 1915, the London Coliseum paid out 25 per cent, the Leicester Palace 10 per cent and the Hackney Empire 6 per cent.

There was a far higher proportion of smaller venues (cafés concerts) in France than in Britain.

Theatre chains have far more power in GB . Theatres often had to join or fold In 1915, for example, the owners of the proudly independent Alhambra theatre in London was obliged for financial reasons to join one of the major chains, London Theatres of Variety.

This concentration has led in GB to the rise of revue, a new kind of evening entertainment which contrasts with the traditional music hall evening made up of individual turns.

Revue is a highly capitalized and artistically centralized show which could travel the country ( and even sometimes go on tours of America, Australia or Africa ) as a bloc. It was extremely popular. In Bristol, a city of around 300 000 people in 1915, the revue Passing Events sold 25,649 tickets the first week, a record for the town.

Revue benefitted from economies of scale. It involved large new groups of skilled workers, such as chorus girls, and introduced ever more lavish special effects to attract the audiences. Special effects for war scenes and wedding scenes came into their own. Special effects At the end of the revue Kultur, in 1915, one could see the actors playing German soldiers 'drowned' on stage in the enormous tank normally used for synchronized swimming displays.1

The British artistes far more likely to be unionized

The Variety Artistes Federation had between 3500 and 4000 members Members accepted union discipline and refused to sign contracts which paid less than the agreed union rate. In 1915, the union position was to refuse contracts which involved less than a week's work, and not to work on Sundays (even for charities, with the exception of war charities): several members were subject to disciplinary procedures for not having respected the majority decision.

Further research is required but it may be that the lyricists were not from the same social media in the two countries. In any case the list of known lyricists cited by Anne Simon includes « law graduates, doctors, classicists » And yet the songs tend to be in working class dialect, if it is a stage dialect. Perhaps in GB writing for the music hall was not respectable enough to attract classicists and doctors. Certainly the British sources suggest rather lyricists who hung around in down market pubs hoping to catch a singer who would by their song. This social difference, which needs to be checked up again, could also be linked to the stabilization of lyricist income in France ; in Britain, at this time, almost all lyricists sold for a single cash payment of a few pounds the rights to the song for performance or recording. They did not receive royalties.

France

The French music halls and cafés concerts, along with the theatres, were closed down on the outbreak of war for reasons of national emergency. Only towards the very end of 1914 were they opening again one by one, and this process continued into 1915. There were even songs written to protest about their reopening, singing « while our lads are risking their necks on the front lines » some dare re-open music and tango clubs ! » This serves to remind us also of the essential difference between the British and the French experience of the war : the French fought on their own soil.

What were the reasons for the main differences?

Clearly the main reason for the differences between the two industries was the level of urbanization and industrialization, far more pronounce din Britain.

Repertoires

What were the main characteristics of the two repertoires? What did the two repertoires have in common?

[Note that the corpuses were constituted in a different manner. The British Library stores thousands of songs in sheet music form, one enthusiast transfers old cylinder recordings onto CDs, and otherwise songs are cited in the trade press or elsewhere.

In France there is official centralized censorship, so the police headquarters kept a copy of all songs authorized and all songs refused authorization and all songs authorized on condition that certain words were changed.]

In both cases songs would often refer to everyday life of the working classes. unlike, for example, the traditional rural song very insistently championed by Cecil Sharp and his folk revival movement at the time.

In both countries the use of working class language was essential to the appeal. That is to say, in both cases the experience of the working classes was validated.

Construction of neighbourliness:2 P'r'aps you don't know Johnny Brown who sold the flowers and fruit? I stood 'im 'cos he used to make me laugh The other day he rolled up in Khaki, if you please I says, 'Lumme! don't it suit yer, Jack. Not 'arf!' Everyday life reflected by references to specific events which changed people's lives 1915 French song (Simon 44) about getting married by proxy in that year it was made legal for soldiers to marry even if they could not be present Or the 1915 GB song « the Ladies football club » which reflected the intrigued or worried reactions about the extremely rapid rise of women's football, since men's football had become politically impossible to stage In both cases, love was an important part of the repertoire, work far less so' : and in fact experience in the workplace was almost completely absent, as has indeed been the case with the vast majority of popular music repertoires today.

In both cases there was a kind of censorship and moral limits on what could be said.

In both cases the lyrics could be supplemented flexibly by the use of gesture and voice, in order to go further in certain theatres or for certain audiences or to avoid a censorship which is based in text alone.

Useful songs for the war drive

A number of songs work at building the sense of national unity of purpose, in a specific, intimate, neighbourly manner, which no doubt makes these songs more effective among the ordinary people than are speeches by bishops or ministers. about fraternity in the trenches, a touching tale of how a soldier always allows his friend who has no family to read his letters form home « a soldier is better than a brother, if we're sharing bread, we can share our family » he says.

British songs also work at the war drive, of course You can see on your handout a series of songs about the war. Never let the old flag fall/be a soldier, lad of mine/ pack up your trouble sin your old kit bag… Or the Gb song « All the boys in khaki get the nice girls »' Which is intended to encourage men to join the volunteer army (in France the army is a conscript army from day one.

There is plenty more analysis which can be done on exactly how the war is presented in these songs ( the glory of war is a very rare theme, and the word « home » will very quickly become the most common word in British song titles, as dreaming of the end of the war becomes the easiest theme to sell.

It is worth noting however that the many enthusiastic recruitment songs sung in Britain in August September and October of 1914 are no longer anywhere near as popular. In the Greatest Hits of the Year collection produced by Francis and Day one of the major music publishers, in December 1915, there is not a single recruitment song (although as you can see on your handout some were still being sung). What were the main differences? In Britain, the centrality of humour and jolliness is key. Shows have a circus flavour and the music hall evening will typically include animal imitators, jugglers, conjurors etc. . Performers generally saw themselves as showmen and showwomen rather than as artists.

Melodrama, which had been a feature of British music hall songs thirty years earlier, seems to have disappeared. Tragedy was almost never seen in the songs.

If hardships of everyday life in the war could be dealt with it was almost exclusively through humour.

In France there is a far higher level of realism, in particular about personal or even intimate matters which are banned de facto on the British stage.

So we find in the French repertoire [Anne Simon p 20] that (quoting from Anne Simon) « the songs express the longing of separated couples for affection and sex ; they express the acute torment of those mutilated by the war, they express the terrible sadness of mothers grieving for their lost sons ». This is absolutely untrue of the British repertoire.

One 1915 song which did not get through the censor, spoke of a female narrator who says how she loves to stare at the lovely bodies of men in scanty sportswear. These songs were censored… but they were presented to the censor so one might imagine the songwriter felt they would have been well received by audiences.

Other realistic of intimate songs were not censored. Here are three examples

One song (Simon 134)which spoke of the « terrible torture » of men being without women at the front.

Another 1915 song (Simon 267) speaks of the children born who will never know their fathers cos they died in the war But another (Simon 152) speaks of sexual frustration among the women who wait at home. Again this is quite explicitin one song the woman narrator explains to her husband that she has been completely faithful and has not even masturbated while he was away.

Another still (Simon 157) recounts how the fiancée who had been reticent about sex gives everything to her soldier boyfriend when he comes home on leave Botrel which was given away as a free supplement to the « petit journal », one of the most important Parisian daily newspapers, which sold 800 000 copies a day.

Elle adore entrer en danse Quand, pour donner la cadence -Verse à boire ! -A préludé le canon Buvons donc ! La polka dont elle se charge S'exécute au pas de charge

In Britain in 1915, and for decades before and after, respectability was a major ideological force. But in France although of course there were always attempts by elites to urge working class people to follow their tastes and manners, this was not configured as a major ideological force .

In December 1915, the manager of a Birmingham music hall was fined by magistrates after there were complaints about three songs sung by Harry Champion. Champion denied the charge of vulgarity, but admitted that he sometimes changed the words of his songs to make the audience laugh.10

This is why, when you search in Google for « respectable working class » you find 17 000 results, and for the French equivalent you only find 123

The power of natalisme was not the same. There is a 1915 French song making jokes about the importance of having contraception ready for when you are on leave : this is censored, because using contraception is refusing to make a baby for the fatherland Anti-German feeling was not the same There was a tradition of anti-German songs, demanding revenge for the annexation to Germany in 1870 of Alsace and Lorraine. But also the political response to the debacle of 1870 was the institution of compulsory military service.

By 1914 all French men did three years compulsory military service ( British men did none). Furthermore, in a largely rural country, as France was, such military service was an important point of socialbility, perhaps the only time a man expected to leave his home region. And it was also a source of many songs about drinking, friendship, and hating Germans

  So (Simon 48 ) 1915 French song where a soldier narrator says n so many words You see, my little wife, we do get in right well Us working men and the aristocratic lot When the fight against the savages is over We'll all be mates together, you will see Tu vois ma petite femme qu'on fait bon ménage Les ouvriers et les aristos Quand on se battra plus contre ces sauvages On s're de vrais poteaux Another A French 1915 song ( Simon 165)

  Other songs about the war also aim at raising morale and reassuring those whose lives are damaged. Song about the Gare de l'Est 1915 : the railway station where soldiers on leave arrived and departed They've all got mud right up to their balls and a lot of them are covered with lice But still they are joyful and ready to fight De la Boue ils en ont jusqu'au couils sont même des fois remplis de poux Ils sont joyeux et plein d'entrain (Simon 46) Other songs celebrate the values of the homeland and its people French 1915 song (Simon 189) in praise of nurses who bring the hearts and souls of our wounded soldiers back to life. It is important to handle the huge changes going on in society. Another song reassures that the nurses are not sexually available (Simon 192). She takes care of the wounded says the song « while thinking of her own fiancée » Very similar to a GB song from 1916, the Rose of No Man's land It's the one red rose the soldier knows It's the work of the Master's hand, It's the sweet word from the Red Cross nurse, She's the rose of no-man's land. In both countries (Simon 248) there are songs about being proud of young boys playing at soldiers Both countries have songs talking of mothers « French 1915 song « don't cry dear mother etc » (Simon 234)

  So although this verse was censored in 1915, it was presented « Pour moi, la chose émouvante/ C'est de lorgner presqu'à nu/ Sous l'maillot, l'torse d'un beau gas/ Ayant du poil sous les bras ! » 3 (in the song "ah les petits homes)" Another 1915 French song was censored since it spoke in a most suggestive manner of a woman on her wedding night and the « cigar » which her man was to provide for her as a gift. Another 1915 song refused by the censors spoke of a soldier coming back from the wars impotent, but his wife being very understanding. Also a song was censured in 1915 Toute aussi directe, cette femme qui n'hésite pas à dire : « J'en demande, il m'en faut ! ». Il s'agit des hommes, bien entendu. Elle n'a pas de préférence, « un jeune ou bien un vieux », ou encore « mariés ou célibataires », « qu'ils soient riches ou qu'ils soient gueux » 4 , elle ne peut pas se passer d'hommes et attend d'eux plaisir et frissons…La chanson est censurée, comme deux autres chansons sur le même thème :

  Comic stupid Irishmen appear in songs such as 'Paddy Maloney's Aeroplane' (1915), in which Maloney claims to have invented a plane which has remarkable technology: His corrugated iron-plated wonderful machine Scatters a thousand frizzly wigs on every submarine With telescope and microscope, Maloney says it's right That he can see the enemy when they are out of sight! Pride and identity It is worth noting that the revue Irish, and Proud of It Too! inspired several other revues of a similar type, singing the praises of other communities and identities. Lancashire, and Proud of It was a success in the North, 7 and in 1915 the revue Jewish and Glad of It! played in London; 8 Scotch and Cockney followed in 1917. 9 Each was dedicated then to significant sections of the working-class audience, and followed a similar formula of playing on some of the positive stereotypes of the groups. Anti-German songs 7 Burnley Express, 19 February 1916. 8 The Era, 19 May 1915. 9 The Era, 3 January 1917. GB In 1915, he sang a very successful war song, 'Me Old Iron Cross', mocking the Kaiser and the German bravery medal: I'm the bloke that broke the bank at Monte Carlo I'm the hero of a dozen dirty nights I went down in a submarine to give the Kaiser one It went off bang and up I went and landed in the Sun There I met the Kaiser and he said, 'I'm up the stick If you can get me out of here I'll treat you mighty quick. Compare France 1915 Melodramatic song to show that dogs have more of a heart than do Germans (le chien du boche) Songs about how wonderful war is and realism about the experience of war Songs about women's fear -fiancée's letter 1914 which says every time I hear some boy has died I'm afraid it will be you -song 1915 about mothers whose sons have been killed, who curse the Kaiser Sexy songs about Rosalie la baïonettesong by Theodor

  RacismAs well as these star acts, there were at least a dozen other blackface professionals in wartime music halls, as well as many amateurs. In 1915, one could see a novel act on tour of a young 'negro imitator' only eight years of age.

	Quand y a la fringale
	On s'en va aux halles
	Là dans les paniers
	Ya de quoi bouffer
	On chipe en passant
	Une pomme au marchand. » 5
	Black soldier:
	Germany has found that the colours won't run
	No matter how you shoot
	We always stand our ground
	Songs about street kids 1915 « On connaît tout, on sait y faire And John Bull's very proud of his little khaki coon
	Nous pratiquons tous les métiers
	Aux bourgeois on ouvre les portières
	Leur z'y donnant des noms princiers […]
	Le soir nous changeons de boulot
	Pour mettre quelques ronds dans la caisse […]

Several 1915 French song

(Simon 135, 138, 144)

promises plenty of sex once they get leave after 14 months at the front. It is explicit compared with Britsh songs, but it is not bawdy.

Another 1915 song (Simon 179) speaks of the flannel shirts sent by the penfriends (marraines), regretting that the flannel shirts didn't come with the women penfriends inside.

Songs may mix talk of sex with a moral project for the nation. A 1915 French song (Simon 119) has a soldier narrator who presents the war as a source of moral renewal. Before the war he was a pimp and his partner a prostitute, he says, but the military medal he has won means he must live up to it, and he writes to his partner that she must give up prostitution ; he in his turn promises to become an « honest working man ».

6

One 1915 song, 'John Bull's Little Khaki Coon', blends a celebration of the participation of Black African regiments alongside the British Army with condescending racist content. The narrator is a 5 M.P., 1915, paroles et musique de Vincent Scotto, éditions Delormel/Fortin.

6

The

Performer, 18 February 1915.

HANDOUT British pop songs in 1915, some examples of titles

  Whereas there were , in Britain in 1914 many thousands of Germans , often very well considered by the local population. IN particular the « German bands » were an institution in popular music (and indeed they appear in the pre-war popular hit song « Down at the Old Bull and Bush »

	Of 155 songs from this year easily classified into a category,
	37 were love songs; 51 songs were directly about the war
	(including some patriotic songs); 6 about singing or music or
	the profession; 3 about food and drink; 22 speaking of
	Ireland/ Scotland/ Lancashire/ Yorkshire etc; 6 reflecting on
	women's new roles; 30 in the category "general humorous".

Of 131 British songs of 1915 for which the singer can be identified, 85 were sung by men and 46 by women

Love songs

If you can't get a girl in the summertime; It's the girls that make the seaside; Kitty Murray, I'm in a hurry; My dapper little flapper Don't say I ever made you love me; A broken doll; Is it nothing to you?; Everybody loves me but the girl I love; I ain't got nobody; Hearts are sometimes more than playthings; Give me your answer, dear, before I go; Little Rosalie, My Pretty Refugee; Now you've got your khaki on; On our happy wedding day; My bonnie lassie ; My dusky princess; My girl from Slumber Town; Normandy girl. Portobello Lass; You're going back to your girl in London Meet me at the corner of the street; Meet me by Moonlight, Mary; You've got me and I've got you; Nobody else but you; Just we two and the moon; Just one kiss, just another one It's lovely to be in love; I want loving all the time; If I thought you loved me; Somebody knows, somebody cares.

Songs about the war

  Sons of the Empire ; Patty proudly packs for privates prepaid paper parcels When we've wound up the watch on the Rhine; Somewhere in France, dear Mother Tommy in the Trenches; Tommy's learning French; We'll never let the old flag fall; Cassidy-private Michael Cassidy ; Pack up your troubles ; Our Brave Navy ; the Kaiser's gramophone; ; Our British soldier; The man behind the rifle Be a soldier, lad of mine; Tell my daddy to come again; Be a man All together shout bow wow wow (the British bulldog's bark); Cheer up little soldier man ! ; While the British Bulldog's watching at the door God give us victory; We must all fall in; The traditions of the Navy; While dodging German bullets in the trenches; Who's going to volunteer; When the war is over, mother dear; In these hard times The Ladies Football Club; The Editress; Which Switch Is the Switch, Miss, for Ipswich?; If the Girlies Could Be Soldiers;Dance of the Fire Brigade Girls; Woman's opinion of man Twas an old-fashioned song he was singing; Any old song my mother sings I love a piano If you want to get on in revue; Ragtime Ragshop That sentimental ornamental oriental tune; Little bit of cucumber, Bread and Marmalade; A little of what you fancy does you good Dreaming of far away A prairie life for me; Alabama jubilee; Just try to picture me Back Home in Tennessee; Somewhere in Sunset Land; Road to the Isles; The Mississippi Barbecue; Anywhere on Louisiana Bay General humorous A wedding. on a gee-gee. Advertise! At the vicar's fancy dress ball Father's Got the Sack from the Waterworks; I sobbed and I cried like a child I'd like to shake Shakespeare I'm William the Conqueror If I were king for a day Keep your hands in your trousers pockets My new hat Tight skirts have got to go; I'm glad I took my mother's advice; Frightened to go to sleep again; Paddy Maloney's Aeroplane; The Yiddisher Irish You must change your name to O'Reilly

	Once the Kaiser's army; All the boys in khaki get the nice girls; Auld Mother Scotland. : (They lie far awa' frae their ain native land.); Cockney in khaki; Mister Sergeant Michael Donoghue ; Michael O'Leary V.C Laddie in Khaki; We're all North Country lads and lasses; Bravo territorials; Follow the Sergeant I'm giving up my job to the Kaiser I didn't raise my boy to be a soldier ; Kitchener's boys; Mother's sitting knitting little mittens for the navy; Our Whistling Singing, music or entertainment Tommies; Role of women Food, drink and fun

'

Bibliography Mullen John, The Show Must Go On: Popular Song in Britain during the First World War, London, Ashgate, August 2015 Simon-Carrère, Anne, Chanter la Grande Guerre: les Poilus et les femmes, Paris, Editions Champ Vallon, 2014

The Encore, 7 January 1915.

This song can be heard on line at http://www.firstworldwar.com/audio/1916.htm (accessed 16 July 2014).

APPP, série BA 699, non visée par la censure entre 1915 et 1918, « Ah ! Les petits hommes ! », paroles de Pinel et Chimenès, musique de Lust et Hamel.

APPP, série BA 712, refusée le 28 juin 1915, paroles de Lucien Bouvet, musique d'Edouard Jouve, éditions Jouve.

TheGuardian, 21 December 1915.

-Refrain -Quand elle chante à sa manière Taratata, taratata, taratatère Ah que son refrain m'enchante, C'est comme un zoiseau qui chante Je l'appell' "la Glorieuse" Ma p'tit' Mimi, ma p'tit' Mimi, ma mitrailleuse Rosalie m' fait les doux yeux Mais c'est ell' que j'aim' le mieux.

One song written in 1915, but not approved by the censor (Simon 95) in which the narrator declares that the French soldiers will rape the beautiful German women just as beautiful French women have been raped by German soldiers What were the reasons for the main differences? We would not want to find answers only in stereotypes of the sexy French and the inhibited English.

The French star if he was the key player at a particular café concert perhaps had more artistic control than a GB star touring the country with a fifteen minute turn The power of respectability was not the same


Eleonora Guerrini
email: guerrini@lirmm.fr

Romain Lebreton
email: lebreton@lirmm.fr

Ilaria Zappatore
email: zappatore@lirmm.fr

Enhancing simultaneous rational function recovery: adaptive error correction capability and new bounds for applications

In this work we present some results that allow to improve the decoding radius in solving polynomial linear systems with errors in the scenario where errors are additive and randomly distributed over a finite field. The decoding radius depends on some bounds on the solution that we want to recover, so their overestimation could significantly decrease our error correction capability. For this reason, we introduce an algorithm that can bridge this gap, introducing some ad hoc parameters that reduce the discrepancy between the estimate decoding radius and the effective error correction capability.

I. INTRODUCTION

The family of Reed Solomon codes (RS) is a large class of very well studied algebraic codes. They are MDS codes, they perform list-decoding and have efficient decoding algorithms that can be viewed in a computer algebra setting as rational reconstruction problems. More specifically, we are interested on Interleaving Reed Solomon (IRS) codes. IRS codes are well studied and can be decoded efficiently by a bounded distance (BD) decoder beyond the unique correction capability radius for almost all error patterns (cf. [START_REF] Bleichenbacher | Decoding of interleaved reed solomon codes over noisy data[END_REF], [START_REF] Schmidt | Collaborative decoding of interleaved reedsolomon codes and concatenated code designs[END_REF], [START_REF] Puchinger | Decoding of interleaved reed-solomon codes using improved power decoding[END_REF]).

In this work we focus on the problem of solving a polynomial linear system with errors (PLSwE), introduced in [START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF] and [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF]. Since the solution of PLSwE is a vector of rational function, the PLSwE is a special case of the problem of reconstructing a vector of rational function given its evaluations, some of which could be erroneous (the simultaneous rational function recovery, shortly SRFR). In [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF] we proposed an algorithm, based on IRS decoding, that allows to solve SRFR (and in particular PLSwE), correcting more than [START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF] and [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] in a probabilistic setting. In this paper, we improve the technique of [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF], increasing the error correction capability. Since we want to recover a vector of rational functions y, which is solution of a polynomial linear system over a finite field, A(x)y(x) = b(x), we introduce a new bound on the error correction capability which also depends on the bounds on the degrees of A and b. Moreover, the knowledge of the degrees of the solution would allows us to reach an ideal error correction capability, but we do not know these degrees and their overestimation could significantly decrease the amount of errors we could correct. For this reason, we introduce a parameter oblivious algorithm for the PLSwE that allows us to get closer to the ideal error correction capability, without knowing the real degrees.

The paper is structured as follows: in Section II we recall standard facts for IRS codes, in Section III we introduce our problem (PLSwE) and we set up the model. In Section IV we introduce the generalization of the PLSwE, i.e the simultaneous rational function recovery (SRFR). We present a technique, based on IRS decoding, that allows to achieve a bigger error correction capability. In Section V, we propose our algorithm and in Section VI, we present our main theorem, the cornerstone of all our technical results. Finally in Section VII, we expose our open problems and conclusions.

II. INTERLEAVED REED-SOLOMON CODES

A Reed-Solomon (RS) code of length n and dimension k over F q can be defined as the set

C RS (n, k) = {(f (α 1 ), . . . , f (α n )) | f ∈ F q [x], deg(f ) ≤ k -1}
where α := {α 1 , . . . , α n } is the set of distinct evaluation points over F q . For l ≥ 1, an l-Interleaved Reed-Solomon (IRS) code is defined by the direct sum of l RS codes C RS (n, k i ) sharing the same set of evaluation points, i.e.

C IRS (n, k) = (c i ) 1≤i≤l ∈ (F q ) l×n | c i ∈ C RS (n, k i ) If k = k 1 = . . . = k l ,
(x) = (f 1 , . . . , f l ) ∈ (F q [x]) l×1 with deg(f ) := max 1≤i≤l (deg(f i )) ≤ k -1.
We now consider the decoding instance Υ = C + Ξ ∈ (F q ) l×n where C ∈ C IRS (n, k; l) and Ξ is the error matrix. We can see Υ = (y j ) 1≤j≤n and Ξ = (e j ) 1≤j≤n both in (F l q ) n . As error model we consider burst errors, i.e. the error positions are the nonzero columns of the error matrix Ξ. In detail, for any 1 ≤ j ≤ n, the set of error positions is E := {1 ≤ j ≤ n | e j = 0}. The number of errors is then |E|.

Since C = (f (α 1 ), . . . , f (α n )) for f ∈ (F q [x]) l×1 , deg(f ) ≤ k -1, for any 1 ≤ j ≤ n, y j = f (α j ) + e j . (1)
In order to decode C, we need to recover the vector of polynomials f (x).

For

1 ≤ i ≤ l, let Y i ∈ F q [x
] be the Lagrangian polynomials such that Y i (α j ) = y ij , for 1 ≤ j ≤ n, and deg(Y i ) < n and let Λ = j∈E (x -α j ) be the error locator polynomial. We observe that for any 1 ≤ i ≤ l, ΛY i ≡ Λf i mod n j=1 (x -α j ). This is a nonlinear equation in the unknowns Λ(x) and f (x). A classic approach for decoding RS codes (cf. [START_REF] Gao | A new algorithm for decoding reed-solomon codes[END_REF], [START_REF] Berlekamp | Error correction of algebraic block codes[END_REF]), that can be extended to IRS codes, consists in the linearization of these equations, by replacing Λ(x) and Λ(x)f i (x) with the unknowns λ(x) and ϕ i (x). In this way we obtain the key equation

λY i ≡ ϕ i mod n j=1 (x -α j ). (2)
In order to decode, it suffices to study the set S of (λ, ϕ 1 , . . . , ϕ l ) ∈ F (l+1)×1 q which verify (2) and such that deg(λ) ≤ |E| and deg(ϕ i ) ≤ |E| + k -1. IRS codes, can be decoded by efficient BD decoders beyond the unique decoding radius. These decoders succeed for almost all error patterns [START_REF] Bleichenbacher | Decoding of interleaved reed solomon codes over noisy data[END_REF]. With "almost all" we mean that there exists a polynomial R such that the decoder succeeds for all instances Υ satisfying R(Υ) = 0. A quite tight estimation of the probability of failure can be founded in [START_REF] Schmidt | Collaborative decoding of interleaved reedsolomon codes and concatenated code designs[END_REF] and improvements on the decoding radius recently appeared in [START_REF] Puchinger | Decoding of interleaved reed-solomon codes using improved power decoding[END_REF].

In the next section we will remark the parallel between the problem of solving polynomial linear systems with errors and the IRS decoding. (cf. [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF]).

III. POLYNOMIAL LINEAR SYSTEM SOLVING WITH ERRORS

Given l ≥ 1, we study the problem of solving a consistent full rank polynomial linear system over a finite field

F q , A(x)y(x) = b(x) (3)
where

A(x) ∈ (F q [x]) l×l is full rank and b(x) ∈ (F q [x]) l×1 .
Any solution of this system is a vector of rational functions, i.e. y(x) = f1(x) g1(x) , . . . , fl (x)

g l (x)
∈ F q (x) l×1 . Let g(x) be the monic least common denominator, then there is a unique solution

y(x) = f 1 (x) g(x) , . . . , f l (x) g(x) ∈ F q (x) l×1 (4)
that is also reduced, i.e. gcd(f 1 (x), . . . , f l (x), g(x)) = 1. Our main aim is to reconstruct such a solution. Note that this common denominator representation can be more compact and it appears frequently for solutions of linear systems computed by the Cramer's rule.

As in [START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF], [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF], [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF], we will analyze a scenario where some errors occur. In detail, we fix n evaluation points α and we suppose that any evaluation point is not a root1 of the polynomial g(x). In our model, there is a black box which, for any evaluation point α j , gives a solution y j ∈ (F q ) l×1 of the evaluated system of linear equations2 A(α j )y j = b(α j ). However this black box could do some errors in the computations.

Definition 1. (Erroneous evaluation points [START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF])

An evaluation point α j is erroneous iff

y j = f (αj ) g(αj ) . We denote by E := 1 ≤ j ≤ n | y j = f (αj ) g(αj )
the set of positions of the erroneous evaluations.

We can now formalize our problem, Definition 2. (Polynomial linear system solving with errors)

The problem of solving a polynomial linear system with errors (denoted PLSwE) consists in recovering the vector of rational functions (4), i.e. the unique solution of a consistent, full rank polynomial linear system (3), given

• n evaluation points α, • d f ≥ deg(f ), d g ≥ deg(g), d A ≥ deg(A), d b ≥ deg(b), • the black box output (y ij ) 1≤i≤l 1≤j≤n ,
• a bound on the number of erroneous evaluation points ε ≥ |E|.

Remark 3. We observe that if j ∈ E, then there exists a nonzero e j ∈ (F q ) l×1 such that y j = f (α j )/g(α j ) + e j .

In general, for any 1 ≤ j ≤ n, y j = f (α j )/g(α j ) + e j where e j = 0 iff j ∈ E.

We can conclude that the PLSwE can be seen as the extension of the problem of decoding an IRS code (see ( 1)) to the rational function case.

IV. SIMULTANEOUS RATIONAL FUNCTION RECOVERY

Definition 4. (Simultaneous rational function recovery)

Fix some parameters n, q, d f , d g , ε, α such that 0 ≤ d f , d g , ε < n ≤ q. An instance of the simultaneous rational function recovery problem (shortly SRFR) is a matrix (y j ) 1≤j≤n = (y ij ) 1≤i≤l 1≤j≤n ∈ (F q ) l×n such that there exists

• a reduced vector of rational functions f (x) g(x) ∈ (F q (x)) l×1 , where deg(f ) ≤ d f , deg(g) ≤ d g and ∀j, g(α j ) = 0.

• a matrix (e j ) 1≤j≤n = (e ij ) 1≤i≤l 1≤j≤n such that its column support E := {1 ≤ j ≤ n | e j = 0} satisfies |E| ≤ ε; which satisfy y j = f (α j )/g(α j ) + e j . The solution of an SRFR instance is (f (x), g(x)).

This problem was introduced in [START_REF] Pernet | High Performance and Reliable Algebraic Computing[END_REF] and in [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF]. We now present a recovering algorithm in the model of IRS codes. In detail, let (y j ) 1≤j≤n be an instance of the SRFR problem with parameters n, d f , d g , ε (we will omit q and α for simplicity). As for IRS codes we introduce the error locator polynomial Λ = j∈E (x -α j ) and the Lagrangian polynomials (Y i (x)) 1≤i≤l . We observe that f , g, Λ still satisfy ΛgY i ≡ Λf i mod n j=1 (x -α j ) so, as for IRS codes, we study the key equations

ψY i ≡ ϕ i mod n j=1 (x -α j ) for 1 ≤ i ≤ l, (5)
whose unknowns are the polynomials ϕ i (x) and ψ(x) such that deg

(ϕ i ) ≤ d f + ε and deg(ψ) ≤ d g + ε.
We observe that this is equivalent to study the evaluated system

[ϕ i (α j ) = y ij ψ(α j )] 1≤i≤l 1≤j≤n . (6)
In this case the unknowns are the d f + ε + 1 coefficients of any ϕ i (x) and the d g + ε + 1 coefficients of ψ(x). Let S y,d f +ε,dg+ε be the F q -vector space of (ϕ, ψ) = (ϕ 1 , . . . , ϕ l , ψ) ∈ (F q [x]) (l+1)×1 which verify (5) and the degree constraints deg(ϕ) ≤ d f + ε and deg(ψ) ≤ d g + ε.

Remark 5. Since we can consider the key equations in the polynomial [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] or evaluated version [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF], studying the solution space S y,d f +ε,dg+ε is equivalent to study the right kernel of the coefficient matrix M y,d f +ε,dg+ε (see [START_REF] Bleichenbacher | Decoding of interleaved reed solomon codes over noisy data[END_REF], [START_REF] Brown | Probabilistic decoding of interleaved RS-codes on the q-ary symmetric channel[END_REF]) of the evaluated key Equation (6). In detail,

M y,d f +ε,dg +ε =    V d f +ε+1 -D1V dg +ε+1 . . . . . . V d f +ε+1 -D l V dg +ε+1    (7)
where

V t = (α i-1 j ) 1≤j≤n 1≤i≤t
is an n×t Vandermonde matrix and D i is the matrix with y i1 , . . . , y in on the diagonal.

Theorem 6. (cf. [4]) If ε ≤ (n-d f -dg-1) 2
=: ε BK , then all solutions (ϕ, ψ) ∈ S y,d f +ε,dg+ε lead to the same vector of rational functions f /g, i.e. ϕ(x)

ψ(x) = f (x) g(x)
. This means that below this error correction capability ε BK , all the elements (ϕ, ψ) ∈ S y,d f +ε,dg+ε are polynomial multiples of the unique solution (Λf , Λg). Besides, it is possible to prove that S y,d f +ε,dg+ε = x i Λf , x i Λg 0≤i≤d f gE where

d f gE = min(d f -deg(f ), d g -deg(g)) + ε -|E|. (8)
Hence we can uniquely reconstruct the vector of rational functions f /g. We observe that if

d f = deg(f ) (or d g = deg(g))
and |E| = ε = ε BK the solution space S y,d f +ε,dg+ε is a vector space of dimension 1, spanned by (Λf , Λg).

In [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF], motivated by the analogy of the SRFR problem and the decoding of IRS codes, we proved that, under some assumptions on the error distribution, we can correct more than ε BK errors with a certain probability.

We now set up our probabilistic model. We focus on (y j ) 1≤j≤n , an instance of the SRFR with parameters n, d f , d g , ε with random errors. In detail, we suppose that y j = f (α j )/g(α j ) + e j , where f (x)/g(x) is a reduced fraction with degrees bounded by d f , d g and such that g(α j ) = 0. Moreover e j is uniformly distributed in (F q ) l×1 if j ∈ E and e j = 0 if j / ∈ E, for a fixed error position set E with |E| ≤ ε. Under this assumption we have the following.

Theorem 7. (cf. [6]) Fix n, d f , d g and ε GLZ = l(n-d f -dg-1) l+1
. Let (y j ) 1≤j≤n be an instance of the SRFR with random errors and parameters n, d f , d g = deg(g), ε = |E| = ε GLZ . Then the corresponding solution space S y,d f +|E|,deg(g)+|E| is a vector space of dimension 1, spanned by the solution (Λf , Λg), with probability at least 1 -deg(g)+|E| q .

In this work we extend the previous result of [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF] to the general case where we only know a bound d g ≥ deg(g) and a bound ε on the number of errors |E|, with ε ≤ ε GLZ = l(n-d f -dg-1) l+1

.

Theorem 8. Fix n, d f , d g and ε ≤ ε GLZ and take d f gE as in [START_REF] Berlekamp | Error correction of algebraic block codes[END_REF]. Let (y j ) 1≤j≤n be an instance of SRFR with random errors and parameters n, d f , d g , ε. Then with probability ≥ 1 -dg+ε q

we get S y,d f +ε,dg+ε = x i Λf , x i Λg 0≤i≤d f gE .

A. Simultaneous rational function recovery of a solution of a polynomial linear system with errors. By Remark 3 we can deduce that the PLSwE coincides with SRFR with parameters n, d f , d g , ε. The matrix (y j ) 1≤j≤n , which is the black box output, is then an instance of this problem. Hence, all the results of the previous section hold. Furthermore, since we want to reconstruct a vector of rational functions that is a solution of a polynomial linear system, it is possible to introduce a bound on the error correction capability which depends on the bounds on the degree of the polynomial matrix A(x) and on b(x) (as shown in [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF]). Let (y j ) 1≤j≤n be an instance of PLSwE with parameters n, d f , d g , d A , d b . Recall from the previous section that S y,d f +ε,dg+ε is the set of (ϕ, ψ) which verify [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] and such that deg(ϕ)

≤ d f + ε and deg(ψ) ≤ d g + ε. Theorem 9. (see [5]) Let ε KP S := n-max(d A +d f ,d b +dg)-1 2 . If ε ≤ ε KP S , then S y,d f +ε,dg+ε = x i Λf , x i Λg 0≤i≤d f gE .
The same result holds if we consider ε ≤ max(ε BK , ε KP S ).

There are some cases in which this error correction capability is bigger than ε BK . In fact, as proved in [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF], when all the bounds are tight,

ε BK < ε KP S iff deg(g(x)) > deg(A(x)).
In this paper, we will introduce a new bound on the error correction capability based on d A and d b under probabilistic assumptions. In particular, given a polynomial linear system as in (3), we suppose that the black box returns (y j ) 1≤j≤n where y j is uniformly distributed in (F q ) l×1 if j ∈ E (instead of y j = f (α j )/g(α j )). By Remark 3, y j = f (αj ) g(αj ) + e j and so our probabilistic assumption on the black box output is indeed an assumption on the error distribution, i.e. e j is uniformly distributed in (F q ) l×1 (instead of e j = 0), when j ∈ E. We will call PLSwE with random errors, the PLSwE in this error model.

Theorem 10. Fix n, d f , d g , d A , d b , take d f gE as in [START_REF] Berlekamp | Error correction of algebraic block codes[END_REF] and,

ε ≤ ε GLZ2 := l(n -max(d A + d f , d b + d g ) -1) l + 1 .
Let (y j ) 1≤j≤n be an instance of PLSwE with random errors with parameters n, d f , d g , d A , d b . Then with probability at least 1 -dg+ε q we get S y,d f +ε,dg+ε = x i Λf , x i Λg 0≤i≤d f gE . Thus the same result holds when ε ≤ max(ε GLZ , ε GLZ2 ).

We will prove Theorems 8 and 10 in Section VI.

V. PARAMETER OBLIVIOUS DECODING ALGORITHM

The PLSwE problem takes as input some degree bounds d f and d g , hence all our error correction capabilities until now depend on these bounds, e.g. ε GLZ = l(n-d f -dg-1) l+1

. Most importantly, our technique for solving the PLSwE requires such degree bounds to decode up to this capability.

Ideally we could decode up to l(n-deg(f )-deg(g)-1) l+1 errors by taking the bounds tight, i.e. d f = deg(f ), d g = deg(g). Our lack of knowledge of the real degrees deg(f ), deg(g) limits us to correct this ideal amount of errors. Indeed, the bounds d f and d g could overestimate the degrees of f (x) and g(x), thus significantly decreasing all our error correction capability bounds.

In this work, we propose a parameter oblivious ( [START_REF] Khonji | Outputsensitive decoding for redundant residue systems[END_REF], [START_REF] Pernet | High Performance and Reliable Algebraic Computing[END_REF]) algorithm that allows to get closer to the ideal error correction capability even without the knowledge of the real degrees.

In [START_REF] Khonji | Outputsensitive decoding for redundant residue systems[END_REF] the authors already observed that even for classic RS codes (Section II), the knowledge of a bound instead of the real degree of f , could decrease the error correction capability. They proposed an algorithm for standard RS codes that allows to correct up to n-deg(f )-1 2 ≥ n-k 2 errors. On the other hand, in [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] it was introduced an algorithm for solving the PLSwE up to |E| ≤ max(ε BK , ε KP S ) where

ε BK := n-max(deg(f )+dg,deg(g)+d f )-1 2 ≥ ε BK , ε KP S := n-max(d A +deg(f )),d b +deg(g))-1 2 ≥ ε KP S
In this work, we propose an algorithm that succeeds for almost all instances (y j ) 1≤j≤n of a PLSwE with parameters n, d f , d g , d A , d b , ε whenever |E| ≤ max(ε GLZ , ε GLZ2 ) where

ε GLZ := n -max(deg(f ) + dg, d f + deg(g)) -ε l -1 ≥ εGLZ ε GLZ2 := n -max(dA + deg(f ), d b + deg(g)) -ε l -1 ≥ εGLZ2
We will explain later where these bounds come from. Our new capability ε GLZ can be greater than ε BK , especially when ε is a tight bound on the number of errors. In particular, if we assume that |E| ≤ ε GLZ , or equivalently if ε = ε GLZ , then ε GLZ becomes

ε GLZ = l(n -max(deg(f ) + d g , deg(g) + d f ) -1) l + 1 ≥ ε BK .
The same holds for ε GLZ2 w.r.t. ε KP S .

The main idea consists in the introduction of some others parameters δ f , δ g , ξ and on the study of the solution space of the key equation 5 with new degree constraints δ f + ξ and δ g + ξ. As in [START_REF] Berlekamp | Error correction of algebraic block codes[END_REF] we define, δ f gE := min(δ f -deg(f ), δ g -deg(g)) + ξ -|E|. [START_REF] Pernet | High Performance and Reliable Algebraic Computing[END_REF] Informally speaking, we will see in the following theorem that the introduction of these new degree constraints will allow us to increase the error correction capability.

Theorem 11. (Parameter oblivious algorithm)

If |E| ≤ max(ε GLZ , ε GLZ2 ) then Algorithm 1 outputs (ϕ, ψ) for all instances (y j ) 1≤j≤n of the PLSwE. Moreover (ϕ, ψ) = (Λf , Λg) with probability ≥ 1 -2(dg+ε) q . If |E| > max(ε GLZ , ε GLZ2 ), then Algorithm 1 returns "|E| > max(ε GLZ , ε GLZ2 )" with probability ≥ 1 -

2(dg+ε) q

.

The fact that the algorithm can (probabilistically) detect if |E| exceeds the error correction capability could be used inside another algorithm that would dynamically increase the redundancy n by requesting evaluation on new points, (cf. [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF], Algorithm 4.1).

Remark 12. In order to compute the nonzero minimal degree solution (e.g. in line 7 of Algorithm 1), we can use two different approaches: [START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] uses column echelon form of the basis of the ker(M y,δ f +ξ,δg+ξ ) whereas [START_REF] Rosenkilde | Algorithms for simultaneous padé approximations[END_REF] proposes F q [x]-module techniques. The latter approach yields the best complexity, i.e. or "|E| > max(ε GLZ , ε GLZ2 )"

Algorithm 1 :

 1 Parameter Oblivious Algorithm Data: (y ij ) 1≤i≤l 1≤j≤n , an instance of PLSwE with parameters n, d f , d g , d A , d b , ε Result: (ϕ, ψ) (equal to (Λf , Λg) with high probability)

In[START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF] and[START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF], the authors study a more general case. They fix n distinct evaluation points, without any assumptions about the roots of g(x). In our work, we need this assumption in order to prove our results.

We suppose that for any evaluation points α j , the rank of the evaluated matrix A(α j ) still remains full. In[START_REF] Boyer | Numerical linear system solving with parametric entries by error correction[END_REF] and[START_REF] Kaltofen | Early termination in parametric linear system solving and rational function vector recovery with error correction[END_REF] there was also studied the rank drop case.

δ f + ξ ← n -d g -ε l -1; δ g + ξ ← n -d f -ε l -1

Let S y,δ f +ξ,δg+ξ be the solution space of the key equation 5 with degree constraints δ f + ξ, δ g + ξ.

 [START_REF] Puchinger | Decoding of interleaved reed-solomon codes using improved power decoding[END_REF]if S y,δ f +ξ,δg+ξ = {(0, 0)} then 4 return (ϕ, ψ) the non zero element of S y,δ f +ξ,δg+ξ with minimal degrees

S y,δ f +ξ,δg+ξ = {(0, 0)} then 7 return (ϕ, ψ) the non zero element of S y,δ f +ξ,δg+ξ with minimal degrees 8 return "|E| > max(ε GLZ , ε GLZ2 )"; O ∼ (l ω-1 n) arithmetic operations in F q where ω < 2.38 is the linear algebra exponent.

VI. TECHNICAL RESULTS

Theorem 13. Fix δ f , δ g , ξ ≥ 0 and let δ f gE as in [START_REF] Pernet | High Performance and Reliable Algebraic Computing[END_REF].

Let (y j ) 1≤j≤n be an instance of the PLSwE with parameters n, deg(f ), deg(g), |E|, deg(A), deg (b), where n ≥ min(N 1 , N 2 ) and

Then, with probability at least 1 -δg+ξ q

we have that S y,δ f +ξ,δg+ξ = x i Λf , x i Λg 0≤i≤δ f gE [START_REF] Brown | Probabilistic decoding of interleaved RS-codes on the q-ary symmetric channel[END_REF] By convention, if δ f gE < 0, we set

Proof. We start by proving that there exists a PLSwE instance (y j ) 1≤j≤ with the same parameters such that Eq. [START_REF] Brown | Probabilistic decoding of interleaved RS-codes on the q-ary symmetric channel[END_REF]. We take a partition E = l i=1 I i with the constraint |I i | ≤ |E|/l (it exists since l |E|/l ≥ |E|). For j ∈ E, we define 1 ≤ i(j) ≤ l as the unique index such that j ∈ I i(j) .

We separate two cases to prove that ϕ(x)g

then for all j ∈ E we choose y j ∈ (F q ) l×1 such that f (α j ) -g(α j )y j = ν i(j) , where ν i(j) ∈ (F q ) l×1 is a vector whose i(j)-entry is 1 and all the others are zero. We multiply by ψ(α j ) and we get ψ(α j )f (α j ) -g(α j )ψ(α j )y j = ψ(α j )ν i(j) . By key Equation [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF] we can replace ψ(α j )y j by ϕ(α j ). Fix i, then ∀j / ∈ I i , ψ(α j )f i (α j ) -g(α j )ϕ i (α j ) = 0. The number of roots of the polynomial ψ(x)

Hence, since this polynomial has more roots than its degree it is the zero polynomial.

Second, if min(N 1 , N 2 ) = N 2 , then for all j ∈ E we choose y j such that f (α j )-g(α j )y j = -A(α j ) -1 g(α j )ν i(j) or equivalently A(α j )y j -A(α j )f (α j )/g(α j ) = ν i(j) . Since

, after multiplying by ψ(α j ) and using the key Equation ( 6) we get

is greater than its degree and so it is the zero polynomial. We have that A(x)f (x) -g(x)b(x) = 0 and A(x)ϕ(x) -ψ(x)b(x) = 0. So if we multiply the first equation by ψ(x), the second by g(x) and we subtract we get

Hence in both cases, since f /g is a reduced fraction, there exists R ∈ F q [x] s.t. ϕ = Rf and ψ = Rg. Going back to Eq. ( 6), for all j ∈ E and i = i(j), we get

and the same for g which implies exactly that i ≤ δ f gE .

Let's now prove that Eq. ( 10) holds with high probability. We always have x i Λf , x i Λg ⊆ ker(M y,δ f +ξ,δg+ξ ) = S y,δ f +ξ,δg+ξ and Eq. ( 10) is the equality case. By the ranknullity theorem, we always have rank(M y,δ f +ξ,δg+ξ ) ≤ ρ where ρ := n(δ f + ξ + 1) + δ g + ξ -δ f gE and Eq. ( 10) is equivalent to rank(M y,δ f +ξ,δg+ξ ) = ρ. In the first part of the proof, we have proved that there exists an instance (y j ) 1≤j≤n such that rank(M y,δ f +ξ,δg+ξ ) = ρ, which means that there exists a nonzero ρ-minor. If we consider this ρ-minor as a polynomial in the variables (y ij ), we have shown that it is non zero. Note that it has total degree at most δ g + ξ because only the last δ g +ξ columns of M y,δ f +ξ,δg+ξ contain variables (y ij ) (see Eq. ( 7)). Therefore the Schwartz-Zippel lemma implies that it cannot be zero in more than δg+ξ q fraction of its domain. For those instances (y j ) 1≤j≤n that don't cancel this ρ-minor, we get that rank(M y,δ f +ξ,δg+ξ ) is equal to ρ and Eq. ( 10) holds.

Remark 14. Let (y j ) 1≤j≤n be an instance of the PLSwE with parameters n, d

, by Theorem 13, the solution space S y,d f +ε,dg+ε = x i Λf , x i Λg for 0 ≤ i ≤ δ f gE with probability at least 1 -dg+ε q . Equivalently, we can fix the number of evaluation points and let the error correction capability vary. Thus, let (y j ) 1≤j≤n be an instance of a PLSwE with parameters n,

. Then, by Theorem 13, the solution space S y,d f +ε,dg+ε = x i Λf , x i Λg for 0 ≤ i ≤ δ f gE with probability at least 1 -dg+ε q . Hence we have proved the Theorem 8 and Theorem 10.

Proof of Theorem 11. First we prove that |E| ≤ max(ε GLZ , ε GLZ2 ) iff there exists a choice of parameters in lines 1 and 4 such that x i Λf , x i Λg 0≤i≤δ f gE = {(0, 0)}. We observe that x i Λf , x i Λg 0≤i≤δ f gE = {(0, 0)} is equivalent to δ f gE ≥ 0. We suppose that ε ≤ ε GLZ (we can do the same in the other case) and consider the first choice of δ f + ξ and δ g + ξ as in line 1. Hence,

), the latter claim implies {(0, 0)} = x i Λf , x i Λg 0≤i≤δ f gE ⊆ S y,δ f +ξ,δg+ξ and Algorithm 1 always outputs (ϕ, ψ).

Second, we claim that for both choices of parameters δ f + ξ, δ g + ξ (lines 1 and 4) we have n ≥ min(N 1 , N 2 ). In fact, if

The same holds for the other affectation. The probability that both solution spaces S y,δ f +ξ,δg+ξ of lines 3, 6 are equal to x i Λf , x i Λg 0≤i≤δ f gE is at least 1-

2(dg+ε)

q by applying Theorem 13 on two different affectations.

Therefore, we can conclude that if |E| ≤ max(ε GLZ , ε GLZ2 ), then with probability at least 1 -

2(δg+ξ) q

, S y,δ f +ξ,δg+ξ is equal to x i Λf , x i Λg 0≤i≤δ f gE and since δ f gE ≥ 0, the minimal non zero element is (ϕ, ψ) = (Λf , Λg).

On the other hand, if |E| > max(ε GLZ , ε GLZ2 ), then δ f gE < 0 for both affectations, and so x i Λf , x i Λg 0≤i≤δ f gE = {(0, 0)}. But with probability at least 1 -

2(δg+ξ) q

, both solution spaces S y,δ f +ξ,δg+ξ are equal to x i Λf , x i Λg 0≤i≤δ f gE = {(0, 0)} so the algorithm will output "|E| > max(ε GLZ , ε GLZ2 )"

VII. CONCLUSION AND FUTURE WORK

In this work, we improve the result of [START_REF] Guerrini | Polynomial linear system solving with errors by simultaneous polynomial reconstruction of interleaved reed-solomon codes[END_REF] considering new bounds on the parameters and taking into account the degrees of A and b (as in ( 3)). We also present a parameter oblivious algorithm that allows us to correct more errors. Our algorithm is probabilistic and the failure probability depends on the parameters δ g and ξ. Remark that our bound on the failure probability is similar to the original result of [START_REF] Bleichenbacher | Decoding of interleaved reed solomon codes over noisy data[END_REF] for IRS codes. Actually, this bound on the decoding failure of IRS codes was strongly improved in [START_REF] Schmidt | Collaborative decoding of interleaved reedsolomon codes and concatenated code designs[END_REF]. Since the SRFR coincides with the reconstruction of a vector of rational functions by its evaluations, some of which erroneous, we can see the SRFR as the decoding of an interleaved rational code [START_REF] Pernet | High Performance and Reliable Algebraic Computing[END_REF]. Despite the similarity of this problem with the decoding of IRS codes, here we deal with a code which is not linear. This prevent the adaptation of most recent techniques for bounding the probability failure of IRS decoding algorithms. A future work is to provide a better comprehension of the interleaved rational code in order to better bound the failure probability.