Eric Maestri
email: eric.maestri@conspaganini.it

Gyorgy Ligeti

L ' Atelier

COMPOSITION AS AN ACT AND EXISTENTIAL TRACE

In this paper, I use the words of philosophy to investigate my compositional practice. I turn specifically to Martin Heidegger's thought, as developed in the central part of Being and Time, in relation to the Being-in. From the reflections of the German philosopher, I develop an interpretive scheme to explain my compositional approaches, highlighting a specific aspect of my practice, which I define through the notions of compositional act, compositional instant, existential trace, and projection. In this paper, I discuss these notions and advance a hypothesis on musical language, presenting musical instruments as signs of the acoustic experience and the musical language as a reflection and presentation of this experience.

Introduction

At times, composers and philosophers view musicians who refer to disciplines such as philosophy to speak, present, or think about their own music with suspicion. In one of the composition exams that I took to complete my diploma at a French conservatory, I was asked to write a short dissertation in addition to the scores. In this dissertation, I attempted to interpret my compositional practice through reflections made by the composer György Ligeti 1 and the philosopher Roman Ingarden 2 . I concentrated especially on certain excerpts that Ligeti had dedicated to the syntactic aspects of the musical language. According to the Hungarian composer, musical syntax takes place in a virtual space, a space in which music is conceived through the notion of "object" 3 . I presented this Ligetian hypothesis in relation to Ingarden's conception of the musical work, as an intentional object whose forms inhabit a specific musical space. In my dissertation, I argued that my music was based on a similar idea; that the music I wrote tried to resonate with this virtual space, soliciting the psychological limit between music and the ordinary experience of sound. In a nutshell, I was interested in identifying and presenting the moment when sounds, placed closed to each other, become music. Through such an act I turned to the listener: composing meant playing on his or her ability to imagine links between sounds. Unfortunately, the composition diploma jury made only one comment about it. Presumably because the jury knew that I had studied philosophy, one of the members argued, provocatively, that I had simply gathered together some notes taken during my studies in order to make my dissertation longer. While this was not true, his saying as much felt as though he had degraded my reflections.

Many years on, I now realize that the jury had not understood that the ideas illustrated in my dissertation really shaped my inspiration and stimulated my compositional practice. Re-reading them today, ten years later, I realize this even more strongly.

In this paper, I propose to establish a connection between music and philosophy, and to use philosophical reflections to express my compositional approach. It is neither a question of doing philosophy through music, nor elaborating a philosophical discourse. It is about placing my compositional practice in dialogue with philosophical thought, and making it resonate with some relevant theoretical aspects. This endeavor has one main reason: I believe that music benefits from words. Words, in this case the words of philosophy, can help musicians -as well as the audiencein articulating the complexity of their own musical acts. The transposition of musical practice into linguistic form can spur musical reflection in itself and induce music too.

Reflection as an act

Musical practice is often described through metaphors, especially visual and linguistic ones. For example, music can be said to have a form, be it composed of objects, defined by sentences, or shaped by motifs. These metaphors define the composer's operating space and frame his or her compositional approach. Composers also create their own metaphors in order to think about their music 4 . The elements of a compositional approach (e.g. objects, things, events, episodes, etc.) are potentially infinite, as many as composers are. Thinking of music through conceptual metaphors (e.g. sentences, surfaces, masses or directionality, processes, imaginary bodies, mathematical formulas, vector spaces, themes, architectural spaces, etc.) guides the search for sounds and the sense of their interrelation. These metaphors guide and condition the composer's work, emerging from a reflection on their musical experience. Thinking music is in certain ways a creative act, oriented towards the compositional outcome. Therefore, composition can be regarded as a reflexive activity, as much as thinking about music through certain metaphors can be understood in itself as an act of composition -an act in that it finds its justification in the concrete realization of a piece of music. This circle is based on acts, and therefore on explicit choices. Each compositional reflection on music is thus a form of act that presents itself as an event. This act, experienced instantaneously, remains a potential tool for future acts; the focus of this event is produced during the composition itself. The act of composing then begins with the progressive, seemingly interminable focusing of a musical vision, which constitutes the heart of the act5 . Through these acts, music decants progressively and becomes defined in its totality. Composing is an act that summarizes many different acts, including noting, verbalizing, drawing, listening, and reflecting. It uses words and inserts them in a circle; in turn, the act itself becomes part of it. The compositional act uses words to think about itself, but expresses itself through the choices that become sound. Sound is therefore the basis of these same choices, and at the same time the consequence of the words expressed during its implementation. This implementation erases the words in their activity, leaving them as traces. In this sense, I suggest that reflecting on composition is in itself a compositional act.

The compositional act

In this section, I tackle the main question of this paper: I will discuss the theoretical framework of my compositional approach through a reflection relying on the language of philosophy. Firstly, I define the compositional act principally as an act of communication. By communication I mean the articulation of being together as human beings. Subsequently, I define my compositional practice as an act in a shared space characterized by a set of acts that follow one another and overlap, all oriented -implicitly or consciously -towards a final decision about the possibility of making public that set of choices. The compositional act then results from a series of decisions oriented towards this final decision, the culminating moment of a tension that grows through a series of previous acts that are subsequently erased, albeit leaving traces6 . This decision concludes an action, it summarizes it. Therefore, not all decisions are of equal importance: the decision that closes the act is then the most important. Once the act is finished, its outcome will be part of the series of ensuing actions that did will not necessarily concern the composer at the origin of that specific act7 .

To address the hypothesis of the compositional act, I propose a model drawing on the paragraphs that Martin Heidegger dedicates to the theory of the Being-in in his book Being and Time8 . In this theory, the philosopher discusses the structure of the human being (Dasein) in his being-in-theworld. Heidegger considers the fact that the human being is in the world as a constitutive aspect of his being, and hence conceives this relationship as fundamental. Following this line of thought, I suggest that composing music is inevitably an act constituted by this being-in-the-world. Music itself populates this world. All human beings can experience music, not just those who make it and realize it. Music is an activity that creates specific entities (i.e. musical works) and, more generally, everything that can be referred to as music. Musical activity shares essential aspects with most human activities. Thus, if we accept Heidegger's hypothesis, the structure of being a musician, of any kinds (as well as that of anyone who enjoys music), is then fundamentally characterized by the structure of Being-in.

The Heideggerian Being-in model

Martin Heidegger describes Being-in as the mode of being-in-the-world: human beings dwell somewhere, practice activities, travel in space, use things, or act according to objectives. What Heidegger calls Being-in defines a certain structure of the interaction with and through the world. The existential structure of Being-in is defined by four fundamental elements: attunement, understanding, interpretation, and discourse.

"Attunement" defines the mood range. Human beings are always fundamentally affected by their mood, and are always in constant relationship with their "thrownness", which refers to the facticity of the Dasein as delivered over to somewhere through its mood. It indicates a substantial foundness of existence. For Heidegger, the human being is "delivered over the fact that it must always already have found itself, found itself in a finding which comes not from a direct seeking, but from fleeing" 9 . This attunement that assaults us comes neither from the outside nor from the inside. It is simply a way of being-in-the-world. This type of situation puts the human being to face with himself, as a fact with which we cannot escape but only come to terms with. The attunement is a "disclosive submission to the world out of which things that matter to us can be encountered" 10 . The fact of being-in-the-world is then the basis of our behaviors. This means that this structure of Being-in is constitutive and possesses a certain existential necessity of its own. According to Heidegger, the human being tries to act according to his attunement. This action is constituted by a movement of evasion, a diversion, from such a situation, and constitutes one of the fundamental moments of Being-in. In the case of this article, the act of composing is conceived of precisely as a sort of evasion when confronted with the sonic experience.

Alongside attunement, the other co-original aspect of Being-in is "understanding". Understanding is the "mode of being of Da-sein as a potentiality"11 . It is about the relationship with the world as a possibility. Understanding has the existential structure of the project. The project is a way of being which plans -that is, anticipates -its own possibilities. This original way of understanding situates the individual as far away as possible from his own attunement. The project then presents the possibility of its own development. In this sense, "understanding harbors in itself the possibility of interpretation, that is, the appropriation of what is understood."12 This type of development is defined by Heidegger as "interpretation". Interpretation is "not the acknowledgment of what has been understood, but rather the development of possibilities projected in understanding."13 This interpretation is not abstract, but is based on the encounter and interaction with the reality in the everyday experience. Such a relationship defines what Heidegger calls Bewandtnis14 , i.e. a certain conformity, involvement, between the tension caused by the attunement of the individual and his or her project. In this sense, involvement plays a fundamental role in interpretation because it somehow defines its success. Heidegger clarifies this perspective by introducing the concept of meaning, or sense -i.e. that in respect of which the project becomes understandable, explainable as something. Meaning is that "upon which of the project in terms of which something becomes intelligible as something." 15 Involvement is important in the compositional act. This act can make sense when a certain piece of composed music satisfies the act of composing. This satisfaction concerns the composition and its public reception. In this sense, the potential public success of a composition in a given context comes into play at the very moment of the compositional act, as this is in part a constitutive aspect of it.

An "extreme derivative of interpretation" is the "statement"16 . For Heidegger, "communication" is one of the meanings of "statement". For the German philosopher, language "has its roots in the existential constitution of the disclosedness of Da-sein"17 ; that is, in the fact of living intimately in a shared space. This shared space constitutes the foundation of what he defines as "discourse", a further element of Being-in. Discourse is the "articulation of the intelligibility"18 , the set of elements that can be interpreted and possible, and it lies "at the basis of interpretation and statement." 19 According to this hypothesis, meaning and discourse are linked. If meaning is "what can be articulated in interpretation" 20 , then, discourse articulates the "totality of significations" as an "existential language" 21 . These meanings are always meaningful and have access to the word -that is, they are conveyed by language. Consequently, "the way in which discourse gets expressed is language" 22 . In this sense, discourse is constitutive of the essence of the individual: it articulates the comprehensibility of the human being as communication; it acquires its worldliness, that is, it becomes a usable entity, through language. Communication is an articulation of the human being as a whole, it is a sharing of the common attunement. Therefore, communication cannot be reduced to what we commonly call information, that is the transmission of a message from the interiority of one subject to another.

Communication is the expression of Being-with, the way of being among other existences. Therefore, discourse has the character of this type of expression. Individuals express themselves as being-in-the-world, therefore already outside themselves: "all discourse about … which communicates in what it says has at the same time the character of expressing itself. […] Da-sein expresses itself […] because as being-in-the-world it is already outside when it understands. What is expressed is precisely this being outside, that is the actual mode of attunement"23 .

Consequently, the tone of the voice, the cadence and the perceived time are indexes of the attunement or emotions of the speaker in, and conditioned by, a given context. Therefore, according to Heidegger, the poetic discourse communicates the existential possibilities of the attunement24 . Discourse can then be defined as the "articulation in accordance with significance of the attuned intelligibility of being-in-the-world"25 . Some languages are also non-verbal. However, they are all anchored to this basic existential structure. In this sense, the compositional act concerns individual choices, which are constitutively linked to the context the composer lives in: the compositional act is an act composed of acts that are in a context. It acts in a context and directs the activity of composing towards that type of communication, which in turn has to do with the attunement and its diversions, understandings, and interpretations of its being-in-the-world.

To pose and to compose

Attunement, understanding, interpretation, and discourse are the fundamental moments of Beingin. They are not consecutive, but rather co-exist. According to Heidegger, language, understood as an "expression of discourse" -that is, of that thing that summarizes all the possible senses and contexts to which an expression can refer -articulates the space of communication. Communication is the very interaction that takes place through language. Therefore, language and communication coincide. By "language" I mean everything that forms a part of communication: words, gestures, more or less intentional movements, sound, even silence. Sound is an available language tool. It communicates because it is an event caused by bodies. It communicates the matter of a body or other types of things. Music uses sound, or refers to sound, as it is an act of communication that benefits from the sound qualities of bodies. Music uses sound to communicate, and defines itself as such. Among the sounds that music uses to communicate are musical sounds, which are part of our acoustic experience just like all other sounds. Music inhabits our sound world, and is one sound among others.

These general considerations outline the framework in which I situate my compositional approach. My attunement is populated by sounds. The encounter with a given sound is the starting point of my composing, as evasion of my individual attunement. If a sound presents itself to me and I perceive it as consistent with my sonic attunement, I choose it. So, first of all, I pose that sound. This kind of act already accomplishes an objective. Such an act does not take place in an abstract way but, as such, expresses a musical possibility. The act of posing a sound is therefore an act of choosing a sound, already positioning it in a certain space. That sound belongs in the instant of the choice in a given moment.

To pose the sound means to choose it in a certain place. That place is defined by a surrounding space. The act of posing sound is analogous to what Heidegger calls "understanding". Once the sound is chosen and placed in a space, the positioning of the other sounds constitutes an interpretation. This interpretation is at the basis of the subsequent acts and choices: it is the interpretation of that given sound in that given place and moment through another sound. Progressively, this act of posing takes on a more complex form, which in turn references other sounds similarly posed. This compositional act is then cumulative, defined by the addition of binding sounds. Gradually, posing becomes composing as an interpretation of the meaning of the placed sound. In fact, the interpretation of the sound that is placed is accomplished through other sounds. Consequently, posing and composing enter into a circle as part of the same act. Understanding a sound means having interpreted it in a given position, and in the function of a sound that will be added to it later on. The act of posing the sounds side by side as a series of successive acts follows Heidegger's model of understanding and interpretation. These two types of actions make it possible for a single act to concretize the sense of the object through its connection.

This compositional act is projective. The sound connections made and their interpretation, which is effected though during the composition, are at the same time conceived as valid in performance. The compositional act is firstly the result of a project understood during the conception of the piece. Secondly, the choices are made according to the fact that these sounds will be instantiated in a possible future during a possible performance. The choice results from the projection that I, as a composer, make of the piece: imagining the moment of the concert, its future intra-world reification, as Heidegger would probably say, that is, the listening by possible listeners. This moment, which is the public presentation of the work, intimately constitutes the compositional act, which is then the place of the deferred imagination of the moments when written music becomes part of the world and, then, begins to be part of the experience of other individuals and becomes a sound among others. This is when language comes into play. The discourse that takes place through music, composed of sounds, comes into contact with the whole of the gamut of possibilities endowed with meaning. This discourse is inhabited by all the other discourses that inhabit the possible experience of the listeners. Music becomes part of their experience, nourishing the hermeneutic circle of each one.

The act of projecting the piece of music integrates the compositional choices as they are made anticipating the listening of others. The fact of designing the sound that is composed in a context in which my intention is not expressed through words but only through sounds requires specific choices that allow the sounds to express a certain content. In this sense, composition can be understood as an interpretation of sounds within the possibilities offered by the type of possible discourses. This passage concretizes the musical form of communication. If understanding identifies the possible, interpretation articulates its possibilities, conveying its meaning, oriented in a deferred way towards the moment of listening. The organization of sound is therefore inhabited by the space outside the sound object itself and by the future, towards which it heads as this potentially has meaning.

Composition as the projection of a projection

Posing and composing are not separated from discourse. Posing a sound object occurs both in a given space and in relation to a discourse, i.e. the context and the sense that orients itself in it. In Heidegger's view, meaning is understood as the "existential phenomenon in which the formal framework of what can be disclosed in understanding and articulated in interpretation become visible as such" 26 . The choice of the object then synthesizes the (musical) being-in-the-world. Such sound is positioned and chosen according to a pre-comprehension of the discourse towards which it is oriented, as sense and as involvement: the composer draws his or her satisfaction from the object he or she projects at the level of discourse, that is, the success, in a given context, of his or her musical choices. The choice of a sound is then the result of a decision that anticipates the effect of the sound that will be heard. If the project is what is anticipated, then the piece I am composing will be projected at the time of the performance. The moment of the concert is when the choices I have made by posing the sounds and correlating them are expressed. In that moment, the succession of organized sounds are perceived as an organic whole by the listeners; it becomes part of their intraworld experience. The fact of projecting a music composition towards the indefinite future of a performance is then a projection of a project. A music composition interpreted in concert is not performed as such. The interpreter adapts the music to the audience, repeating it but not replicating it. The same piece of music, performed by different performers in different places, is different every

The fact of composing is then, for me, the projection of a projection as the composer creates the material for a future performance 27 . The fact of choosing a sound is not a personal matter alone; it is the projection of a real event that will take place at a certain time and in a given place. Its meaning is projected in such a way that in most cases, the communication takes place as it is thought 28 . All the variables of the sound projection are then always open, and must be re-spaced each time.

The fact that music can be projected in the absence of its author is a constitutive element of the act of composition. The compositional choices include the fact that the music can be projected without the composer: in the composer's relative absence, due to an impossibility or unwillingness to attend the performance; or in the composer's absolute absence, due to the fact that the author is no longer there. The attunement that characterizes this situation is what Heidegger defines as the "possibility of the impossibility of existence in general" 29 , that is, anxiety. The conception of the composition is therefore linked to the fact that this project can continue to exist even in the absence of its author, that it makes sense and that it produces the desired effect even when the composer is absent. This project is then the projection of a project, because it becomes the basis of the project of a subsequent interpretation; it is the project of what will be executed, prepared, and interpreted, that is to say another project, a new realization of what was composed. In this sense, every moment in which you choose a sound, pose it, and then compose it, includes this "possibility of the impossibility". Writing, in this sense, carries this possibility within itself, reified through the written track, which projects the project to be realized beyond the individual existence.

The understanding and the interpretation that the project achieves through writing is done through music. My compositional act is then an understanding and an interpretation, achieved via the sound of my musical attunement. This attunement is realized in the project of the piece, which is projected towards a possible future in which it will in turn be projected into a space and into the existences of the beings who project it and listen to it. I may not be present in this possible future; indeed, sooner or later, I certainly will not be there. The performance of the piece is defined then as a subsequent understanding and interpretation that carries out a new project from a project already written. The composition is then, in my view, a sort of deferred, anticipated, or imagined performance, carried out with or without myself. Composing is the exercise of imagining the future as a deferred present, constituted of choices made now that will imply choices and impressions in a future instant. This future is present during the composition, during the performance, and continues to be present from time to time. Thus, composing means building the listening of the others.

The compositional instant

27 Christopher Small considers the composer as the person who provides the materials for a performance. The composer is not, then, the author of a musical work, but the one who imagines an event that will be constructed from the materials he or she created. This definition of composition perfectly corresponds to the type of compositional approach that I propose in this text. cf. Christopher Small, Musicking. The Meaning of Performing and Listening (Middletown, Connecticut: Wesleyan University Press, 1998). 28 I attempted to elaborate this idea in a conference act published in 2016: Eric Maestri, "Notation as Temporal Instrument," in Proceedings TENOR 2016. International Conference on Technologies for Music Notation and Representation, ed. Chris Nash Richard Hoadley Dominique Fober (Cambridge, United Kingdom, 2016). 29 Heidegger, Being and Time, 242.

If the compositional act is essentially projective, what specific type of projection characterizes it? For some authors, music is based on what is sometimes defined as a musical idea30 . Arnold Schoenberg thought this notion through the concept of Faßlichkeit, that is, of comprehensibility31 . In my case, this comprehensibility is not the results of the good combination of forms, but the output of choices and projections. Every note, sound, sentence, every element of the language in a given moment; all must be chosen and thought of as it will be realized, understood, and listened to. In this sense, my compositional act is oriented towards the conclusion of the act itself and its concrete realization. The act of composition considers the act of writing, incorporating the imagination of future acts that will concern the use of the materials created for this act itself. The compositional act consists of several acts and moments; in these moments, I imagine the possible acts of the musician and the listener. These moments are all oriented towards the conclusion of the act of projection of the piece through writing, recording or other means of memorization. However, this conclusion of the act of writing does not represent the final objective intended by the author. That moment is the concert, the moment in which the score, the files, the programs, indeed anything that can be performed, resonated, and interpreted, are (re)produced in front of an audience. The realization of the materials for the performance, which represents the end point of the composer's work, is thus the stage or other space for performance. This projection integrates the choice of sound. The instant in which the sound is chosen synthesizes the chosen sound and its projection in the possible future context. Understanding and interpretation allow for the indication of the production of sounds that then become entities towards which the projecting understanding is directed. The writing of the composition is therefore the progressive definition of this possible projection, comprised of all the instants and all the acts performed to accomplish it, and oriented towards making other acts possible, in order to finally perform it. This musical idea does not have the fixity of ideal objects, but it changes following the experience and potentially changes in an infinite way.

I define this series of successive and simultaneous choices, oriented to the projection of the piece as projected in a context outside the written realm, as the compositional instant. The compositional instant is the moment in which I choose a sound following an understanding of its meaning within the context for which, and in which, it is conceived. In this moment, I synthesize the comprehension, the subsequent interpretations, and the projection of the music project. The compositional instant projects this project into a future time, in which the project will be realized and then projected in turn. The compositional instant potentially anticipates and summarizes all the contexts and all the possible realizations. It is the moment in which I realize how the instant of that composed moment will be perceived by the listener.

Composition as existential trace

The image of the compositional instant, which underlines the notion of choice (from that of the individual sounds to that of their organization) indicates the crucial and fundamental core of my practice. I conceive of composition as an existential trace, in the sense that every instant I composed in an act leaves signs, each of which constitutes the starting point of subsequent ones. The compositional choices are then traces of the existential experience of composing, collected in an existential fragment of a series of these choices. This fragment is the result of choices whose temporality is exactly the same as that of individual existence. However, during the composition, I extract the existential experience of my composing and put it to use in writing. The process of understanding and interpretation is then extracted from the time of life and projected outwards. In fact, composing also means finishing the work and projecting it, in its entirety, towards the future. This projecting closes the hermeneutic circle, putting an end to the posing and the composing of the sounds. The compositional instant incorporates this aspect and delivers the work to the future. The finished piece remains the outcome of a succession of choices made, the sum of the compositional instants. These choices remain as traces, blocked and released by the author. The compositional act is thus an existential trace, because it is the result of the decisions taken in a certain time which leave a two-dimensional imprint, a trace, an accumulation of compositional instants. The composed music remains there, in a place and in the form of the choices made, which are there as a project to be designed.

In this sense, my music is characterized by two fundamental questions -one about the nature of the compositional choice, the other about the limit between sound and music. I constantly try to balance the choice of a sound I hear and that heard by others. The moment when that thought and positioned sound is perceived outside of myself leaves a trace, from which I then perceive the following sounds. I try to compose the listening, and this projection outside is realized through choices and their traces. I try to imagine how my choices will sound as a reflexive act. This autoanalytical observation stimulates and motivates me in my compositional act. However, the fact that one has to perform the acts, and therefore has to erase in an instant all the other acts that led to the final decision, makes this observation disappear as one of the many acts that precede the decision and the preparation of the next act. Here, the reflection is cancelled by the compositional instant.

Instants

A hypothesis on musical language

Both notions -of the compositional instant, and of music as a projected projection -are based on a hypothesis concerned with the nature of music. The composer composes at a time when the full realization of what he or she is composing is not yet achieved. The composer projects real sound events into the future. These events are made in such a way as to constitute sensitive units, such as can be understood by the anticipated listener. Indeed, composers imagine real sound events. I understand this as constituting the composer's fundamental act of trust.

I adopt a very empirical notion of language: language inhabits the world and is used as an entity that results from the interaction between individuals. It makes it possible to talk about bodies or experiences in their absence and to reflect on elements of mutual concern in order to know them and understand them. The composer uses sounds as the basis of his or her musical language. However, in what sense is music a language for a composer? What does the composer use to build this language? And, what are the fundamental units that the musician uses to compose, if indeed any do exist?

In order to think of music as a language, we need to ask ourselves what music is about. It is well known that the hypotheses about musical signification are manifold; and that, probably, many of them are at least partially true. Music is often understood by semiologists as a kind of symbolic form [START_REF] Nattiez | Musicologie Générale et Sémiologie[END_REF] , which potentially has an infinite number of possible interpretations. These interpretations are both personal -the signification -and contextual -the connotation. According to Susanne Langer, music is a particular type of symbolism suitable for the expression of "unspeakable things" [START_REF] Langer | Philosophy in a New Key. A Study in the Symbolism of Reason, Rite, and Art[END_REF] , for it articulates types of forms that spoken language cannot articulate 34 . According to Langer, music is the representation and formulation of emotions, moods, mental tensions, a "logical picture" of the personal feeling 35 . In this sense, the American philosopher seems to share Heidegger's view of discourse as "the articulation in accordance with significance of the attuned intelligibility of being-in-the-world" 36 . Music is an articulation of meanings, an "unconsummated symbol", whose essence is not assertion but "expressiveness" [START_REF] Langer | Philosophy in a New Key. A Study in the Symbolism of Reason[END_REF] . The musical significance is therefore implicit.

Thus, music can be seen as a kind of expression of the unspeakable. It is a language that is complementary to spoken language, dealing with a part of reality hardly identified by verbal language: the sound reality. I contend that music talks about this reality. However, if music does articulate the existential attunement, how does it do so? In my opinion, the articulation of the human invisible part, the composer's interiority -that is, of the experience that each being individually experiences -is made by activating the invisible part of the bodies, that is, by vibrating their matter. The act of making a body resonate reveals its invisible aspects. After all, when we want to be sure of the matter that makes up an object, we touch it and make it resound [START_REF] Casati | La Philosophie Du Son[END_REF] . Do we then use the sound coming from the bodies we experience as a metaphor, in order to articulate the internal experience coming from our own bodies? I believe yes. By resonating with the matter of bodies, human beings learn to resonate with their own matter, and thus express their non-visible sensations.

Composers use bodies to produce sound. Their knowledge consists precisely in knowing how and where to activate the sound sources that they wish to use in a specific way. To do this, composers must know the properties of the bodies they intend to use, and know how to activate them themselves or how to indicate to someone else -the performers -how to activate the resonance of that given body. This primary intention -the indication of the matter, the instrument, and its activation -is, for the composer, transmitted through the score, the use of which is taught orally in schools. Composers refer to that type of practice of activation of the sound source. In this sense, the musicians behave like most human beings and associate the sound they want to produce with sound sources. They rely on ordinary sound experience, which provides the context for understanding the music itself. Indeed, the ecological approach to auditory perception proposed by Gaver underlines this very effectively [START_REF] Gaver | What in the World Do We Hear? An Ecological Approach to Event Auditory Perception[END_REF] . When a score is performed, this something to which the composer refers (excluding interpretation) often conforms to what has been written. After all, the composer constructs future events knowing that there exists a strong probability that these compositional events may be interpreted (or reinterpreted) and reworked in the performance. On this, the composer and the performer negotiate, speak, and rehearse mobilizing their common sonic and musical experience. Composers use their writing as a set of signs whose references indicates at the same time the sound body, the way of activating it and the sound that such action is expected to produce. The combination of these three aspects, if respected, contribute to the realization of the musical idea. Therefore, the composer's act is grounded in a shared experience which has been mobilized. His or her notation refers to something that goes beyond the music as indicated by the sign. Writing denotes something and this something comes into play in defining the linguisticity of the music.

By "denotation", one usually meant the principal meaning of a word. For example, when we say or read the word "chair", we think of any chair. But, if we ask a musician what does a musical sound mean, how would he or she respond? They would answer that sound means the sound itself. However, does this sound means something else? We could say that the musical note means the frequency performed by a specific instrument; or that a certain musical succession of notes means that specific phrase performed by that instrument and musician; a specific emotion. But let's think of any sound indicated by a notation: what does that written or thought note, that chord, gesture mean then? What is the principal meaning of a musical note?

I advance the hypothesis that that note means that given sound made through an instrument, whether it is a body other than that of the musician or through the body itself of the musician. Then, when we answer that a sound means the sound itself, we consider the sound of a musical instrument. The sound of the instrument presents the instrument itself, that is, the sound considered is a musical sound. We listen to it "as if" it is musical [START_REF] Arbo | Wittgenstein et l'esthétique Musicale, Collection Du GREAM/Esthétique[END_REF] . Thus, to understand its signification, we have to think of the instrument as a sign. First of all, the musical sign means a sound produced by a sound body. Secondly, the denotation of music is the sound produced. Therefore, notation allows the use of the musical instrument to create sound events; it makes it possible for the given instrumentalist, or more instrumentalists, do things from certain sound bodies. Hence, musical language is based on the instruments as signs. Music does not associate sounds in the abstract, but rather writes sound configurations starting from the instrumental experience.

If we agree that music denotes instrumental sounds, do these sounds then mean anything at all? As I have argued so far, music denotes sound bodies, instruments and, therefore, sounds. It speaks of their matter through their timbre. But, in turn, does talking about the timbre of sound bodies, connecting them and creating new sounds from their connection, i.e. making music, mean anything? I think that music -talking about sound bodies -means the ordinary acoustic experience. This signification is given by signs that are not visual, or simply sonic. The sign of music is the musical instrument. The instrument is the sign that music uses, that is, a sort of arche-writing, as Bernard Sève claims. According to the French philosopher, the instrument inscribes the discretization of the flow of sound 41 in the very materiality of its body. It is therefore a kind of sign already of its own; a sign of the material that characterizes it, and therefore of all types of materials that the instrument plays. The sound reality then travels through the instrument: one experiences it through this same instrument. The instrument is the fundamental element of the language of music, and the score is mainly a deferred instrument. The instrument means that ordinary sound, as its sound, is listened to as such. It is therefore a reduction of the sound that is implicitly used by musicians to mean the sound experience through more abstract sound entities, detached from that experience and more specifically musical. According to this hypothesis, the instrument, through the types of sound it can produce, is an index of sounds.

Such a semantic relationship can allow "sound" to mean sound. In this sense, the musical instrument expresses the world of sound and allows for an implicit, musical, non-verbal understanding of it, which can be used as a language that articulates a certain type of human experience. It is not true, therefore, that music does not mean anything. If by "meaning", we mean an instance referred to as a signifier, then the "meaning" of music is the sound and its signifier, the instrument. So, if music is about the instrument making sound, and the instrument is about the ordinary acoustic experience, then music is about the acoustic experience through that instrument. In this sense, then, music could be considered as the "symbolic transformation of experience" that Susanne Langer talks about42 . In fact, by practicing music, one could gain a deep and conscious sound experience. However, this transformation of the experience would not pass through the word but would use the sound world to understand the sound world itself. The instrument, as a sign of sounds, allows for such an experiential transformation. The musical instrument then provides the music with its alphabet. The instrument itself is a sign: music speaks of sound through the instrument 43 .

Consequently, music denotes the musical instrument and, by referring to the instrument, it means the ordinary acoustic experience and, therefore, also the music itself. The composer's ability to create comprehensible sound experiences depends on this connection. Environmental and musical sounds constitute the totality of the sound experience of each one of us. According to this hypothesis, musical instruments can be seen as a first abstraction from the sound world (as the sound world is made of music as well), and thus allow for its more precise interpretation. Instruments are signs of the sound world. Music then has an educational and cognitive function. However, the type of knowledge which it gives access to is not scientific, but musical. Those who practice music by making it or listening to it can regularly and continuously have access to that kind of deepening of the sound experience. It is not, therefore, an intellectual knowledge that can be rationally demonstrated, but a practical knowledge, which can offer the basis for other types of knowledge without having to justify itself or be based on scientific knowledge.

The creation of music through writing, playing, improvising, being together singing and playing, all use the instrument to reflect on sound reality through sound. Music is a reflection on sound from instruments that allow us to produce these sounds. Music lets us know sound reality through the act of playing. Therefore, if music means the musical instrument which in turn means the sound experience, composing means creating a real sound experience which then becomes part of the ordinary sound experience as music. Composing music means adding sounds to the sound experience. These added sounds enrich this experience and its reflective potential, as they are human-made sounds and hence understandable and repeatable. The realization of new sound events that make the instruments play, makes our daily sound experience resound and allows for a new interpretation and awareness. This, in my opinion, is what music means and what its function is.

Conclusion

In this paper, I have tried to delineate the elements of my musical practice. To do this, I turned to the discipline of philosophy. This reference is a reflexive one, as I did not intend to indicate a certain philosophical truth, but rather to explain through philosophy what music cannot. I have identified the elements to conduct this reflection in the passages of Being and Time that Martin Heidegger dedicates to the examination of Being-in. The description of the structure of Being-in offers a model for identifying the elements of my musical practice and my inspirations. This structure is characterized by four components: attunement, understanding, interpretation, and discourse. I have argued that this model is similar to my compositional process. I pose a sound and this posing is already in itself an understanding whose interpretation takes place by association and contact with other sounds placed. This model has enabled me to explain my thought, supporting my idea of composition as a projection of a projection, and the notion -for me fundamental -of the compositional instant. If posing and composing are acts that constitute the compositional instant, these acts are oriented towards what Heidegger calls discourse, that is, towards an outside of the sound that projects it as a possible project constituted by the thought of the listening of others. In this sense, the project consists intimately in the construction of the listener's listening. This reflection led me to advance a hypothesis on musical language. I have argued that, in the observation of my compositional practice, musical language has a primary meaning. Music speaks of sound using instruments. I thus proposed to see the instruments as signs of sound. Musical writing can be then understood as a deferred musical instrument. This hypothesis thus aims to clarify how composing is an act that bears an effect on sound reality. The music then has a specific impact on experience. The musical language activates sound bodies, and through their activation can produce privileged access to the ordinary sound experience. The compositional choices made are then thought of as real tracks instantiated at the time of execution. The existential trace of the compositional choice then becomes a trace of information in the performance [START_REF] Windsor | Gestures in Music-Making: Action, Information and Perception[END_REF] . Those choices, which are realized in the sounds, are real events made for the experience of the listener: they are imagined acts projected with the hope of becoming real acts, trusting the future realizations of the project conceived.

Table 1 :

 1 A comparison of the Heideggerian and the compositional models.

		Heideggerian model	Compositional model
	α	Attunement	Sound attunement
	β	Understanding	Pose
	γ	Interpretation	Compose
	δ	Discourse	Projection of a projection

The Argentinean composer Horacio Vaggione rightly proposes that music should be thought of through the notion of composable space: Horacio Vaggione, "Composable Space: Concerning Some Operating Categories in Electroacoustic Music," in L'espace:Musique/Philosophie, 1998, 153-66.

Susanne Langer uses the notion of commanding form to describe this type of vision: Susanne Langer, Feeling and Form. A Theory of Art (New York: Routledge & Kegan Paul, 1953), 122.

Gaston Bachelard, L'intuition de l'instant (Paris: Gonthier, 1932), 22.

Gaston Bachelard, La Dialectique de La Durée (Paris: Presses Universitaires de France, 1950), 17.

Martin Heidegger, Being and Time, trans. Joan Stambaugh (Albany: State University of New York Press, 1996).

Heidegger, 128.

Heidegger, 130.

Heidegger, 134.

Heidegger, 150.

Heidegger, 139.

Involvement.

Heidegger, Being and Time, 142.

Heidegger, 150.

Heidegger, 150.

Heidegger, 150.

Heidegger, 150.

Heidegger, 150.

Heidegger,

Heidegger, 150.

Heidegger, 152.

Heidegger, 152.

Heidegger, 152.

Heidegger, 146.

Jean-Louis Leleu, La Construction de l'idée Musicale. Essais Sur Webern, Debussy et Boulez (Genève: Éditions Contrechamps, 2015), 39.

Arnold Schoenberg, Style and Idea (New York: Philosophical Library, 1950), 159.

Langer, Philosophy in a New Key. A Study in the Symbolism ofReason, Rite, and Art, 35.

Pierre Schaeffer observed that if we have conquered traditional music, it is thanks to the means of the musical instruments. It would be amazing if we could undertake an inventory of sound by ear. Pierre Schaeffer, Traité Des Objets Musicaux. Essais Interdisciplines (Paris: Editions du Seuil, 1966), 402.


Nesrine Kaâniche
email: nesrine.kaaniche@telecom-sudparis.eu

Samovar

Ashish Gehani
email: ashish.gehani@sri.com

Efficiently Validating Aggregated IoT Data Integrity

We address the problem of validating the integrity of data aggregated in applications that run on the Internet of Things (IoT). When data from multiple sources is sent to a single receiver, the receiver needs to efficiently verify the authenticity and integrity of data. For example, sensors from multiple IoT devices would send the measurements to a gateway for data aggregation and/or processing. The gateway needs to verify the authenticity and integrity of the readings from sensors. To address this need, we propose a lightweight homomorphic signature scheme that supports the execution of aggregation operations.

Our approach collects digital signatures from multiple sources and provides an efficient aggregated signature verification mechanism for the end receiver. The aggregation and verification mechanism preserves the privacy of intermediate nodes. We discuss how our homomorphic signature scheme resists forgery and replay attacks.

I. INTRODUCTION

During the last decade, the Internet of Things (IoT) penetrated our lives in many ways -most recently in smart monitoring and control applications at home and workplaces. IoT devices are smart, and often resource-constrained devices, and they interact in a collaborative way or a semi-centric fashion under different processing and communication architectures and technologies in order to fulfill a common goal [START_REF] Xia | Internet of things[END_REF]. For example, a set of resource-constrained IoT devices can collect environmental information regarding a targeted area and collaboratively transmit data back to a base station or a central aggregator to monitor air quality, earthquake activities, or traffic patterns.

In many cases, users of IoT applications are only interested in aggregated results after in-network processing, rather than detailed readings from individual nodes. To ensure the authenticity and integrity of the data in the aggregated results, we need to provide a mechanism for the aggregator nodes to create a digital signature that matches the aggregated data, and for the end receiver to verify this signature.

While there are known cryptographic mechanisms that can provide data confidentiality, data integrity, and data/device authentication in a traditional computing environment of personal computers and servers, these may require heavy computation and communication costs. Instead of relying on dedicated resourceful devices, we aim to take advantage of large and distributed computation and storage capacities of the different devices by requiring each device to provide signed information * while Dr. Kaâniche was visiting SRI International for collected data. Each device aggregates the data it has received so far and is responsible for creating matching signature for the aggregated data.

Aggregation has been often presented as a solution to take advantage of the local computing and storage capabilities in order to remove redundant information within network flows. Many aggregation mechanisms have been proposed in the literature [START_REF] Boneh | A survey of two signature aggregation techniques[END_REF]- [START_REF] Wang | Discrete logarithm based additively homomorphic encryption and secure data aggregation[END_REF]. The use of such approaches considerably distributes the heavy computation burden among different hosts, thereby reducing the resource consumption of the end receiver. However, aggregation affects the security properties of protection systems. For instance, if some information of a signed data flow is dropped to reduce redundancy, the end receiver may not be able to correctly verify the authenticity of the signed stream.

Contributions -Our aggregation signature scheme allows the resource saving of aggregation and also allows the nodes to create the corresponding signatures of any subset of aggregate data for authenticity without knowing the private key materials of the original senders of the aggregated data. We propose a new homomorphic signature scheme based on set operations. A set-homomorphic signature scheme allows signing sets of data such that anyone can derive both a signature on the union of two signed sets and a signature of any subsets of a signed set [START_REF] Johnson | Homomorphic signature schemes[END_REF]. We consider all data sent from each IoT device as an element of a set. This enables us to design an efficient signature scheme that allows the aggregating entity to do the following:

1) detect forged signatures before the data and signatures reach the end receiver. 2) generate a proper signature without private keys used to derive each single signature. 3) apply set operations on signed data flows, such as union, subset, and set difference.

Paper Organization -The remainder of this paper is organized as follows. First, we detail the requirements needed for our signature scheme and review the related approaches presented in the literature in Section II. Then, we introduce our homomorphic signature scheme in Section III and present a concrete construction in Section IV. Finally, we give a detailed analysis in Section V, before concluding in Section VI.

II. DIGITAL SIGNATURE SCHEMES FOR DISTRIBUTED ENVIRONMENTS

IoT devices involved in distributed environments provide ubiquitous and pervasive computing capabilities. However, these systems are often deployed in environments where collecting hosts are open to various attacks due to untrusted environments. We introduce our network model in Section II-A and present security and privacy requirements in Section II-B.

A. Network Model

Figure II-A shows an example system with IoT devices and a gateway. Each IoT device periodically broadcasts the data it received from other IoT nodes and its own data (e.g. readings from sensors.) d i stands for data from node i, and d ij stands for the aggregated data from nodes i and j. The aggregated data (d ij can be in any form, such as concatenation (d i ||d j ) average ((d i + d j )/2), or max (max(d i , d j )). Note that node 3 aggregates all it has received (d 1 ) and its own d 3 into d 13 , while node 4 aggregates d 123 and d 13 into d 1234 , without repeating d 1 and d 3 . In other words, the aggregation step removes the redundancy in data.

Our aggregated signature scheme enables node 3 to create a signature for d 13 using the signature for d 1 and its own private key without knowing node 1's private key. Similarly, node 4 can create a signature for d 1234 using the signatures of d 123 and d 13 and its own private key without knowing the private key of any other nodes.

B. Requirement Analysis

We suppose the aggregation process can be performed by any participating node, and it may result in the removal of some information to reduce the redundancy between the collected data stream transmitted from different nodes to minimize communication latency.

Considering that collected data can be output by different nodes, our signature scheme has to fulfill the following requirements:

• Unforgeability -even if an adversary can capture signatures of different contents, it should be unable to generate a new signature on new data contents. • Anti-replay -for two communication sessions, even if the content of the collected packets is the same, the generated signatures should be different. This prevents any adversary to re-inject responses previously signed by a specific host. • Privacy -an adversary must not be able to gain any knowledge about the origin and content of the removed parts of the signed data flow without having access to them or to link each part of an aggregate signature to its related generator. • Support of multiple signers -the computation of the aggregated signature should be possible even if multiple private keys are used. • Low processing and communication overhead -the proposed security mechanism should provide low processing and communication complexities due to resourceconstrained devices.

C. Literature Review

Various types of homomorphic signatures have been presented in the literature [START_REF] Boneh | A survey of two signature aggregation techniques[END_REF], such as sanitizable signatures [START_REF] Ateniese | Sanitizable signatures[END_REF], set-based signatures [START_REF] Johnson | Homomorphic signature schemes[END_REF], redactable signatures [START_REF] Slamanig | Generalizations and extensions of redactable signatures with applications to electronic healthcare[END_REF], linearly homomorphic signatures [START_REF] Boneh | Linearly homomorphic signatures over binary fields and new tools for lattice-based signatures[END_REF], and verifiably encrypted signatures [START_REF]Aggregate and verifiably encrypted signatures from bilinear maps[END_REF]. The constructions in [START_REF] Canard | Sanitizable signatures with several signers and sanitizers[END_REF], [START_REF] Samelin | Redactable signatures for independent removal of structure and content[END_REF] propose sanitizable and redactable signatures, respectively. They enable an authorized user to bring modifications to a signed message. That is, given a signature on a message, a signature on subsets of this message can be generated by the authorized entity. However, these techniques generate long signatures that require substantial computing resources and introduce communication overhead. The mechanisms in [START_REF]Aggregate and verifiably encrypted signatures from bilinear maps[END_REF], [START_REF] Neven | A simple transitive signature scheme for directed trees[END_REF], [START_REF] Bagherzandi | Identity-based aggregate and multisignature schemes based on rsa[END_REF] mechanisms are limited with regard to the supported homomorphic operations, allowing only polynomial, transitive, multiplication operations, respectively.

III. SYSTEM OVERVIEW

Our objective is to develop a new distributed aggregate signature scheme. We discuss the basic properties that should be fulfilled by such a scheme in section III-A. Then, we introduce our system model in Section III-B and our threat model in Section III-C.

A. Basic Properties Definition 3.1: Set-based Signature Scheme

Let M be a message space, Σ be a signature space, K pr a private key space, and K pub a public key space. A set-based signature scheme consists of the following three algorithms:

• gen : 1 ξ -→ K pr × K pub : this key generation algorithm takes as input the security parameter ξ and outputs the public and private keys.

• sig : K pr × 2 M -→ Σ × 2 M :
the signature generation algorithm takes as input the private key of the signing host and the set of messages. It outputs the related signature.

Note that 2 M is the power set of M. • vrf : K pub × 2 M × Σ -→ {0, 1}: the signature verification algorithm takes as input the signature and the public key related to the signing key, and it outputs 1; i.e accept; or 0; i.e reject.

The main difference between traditional signature schemes is that the sig algorithm operates on sets of messages in 2 M instead of operating on individual messages in M. Definition 3.2: Set-homomorphic Signature Scheme A set-based signature scheme {gen, sig, vrf} is called set homomorphic if these three operations exist • : K pr ×K pr -→ K pr , ♦ : K pub × K pub -→ K pub and ∇ : Σ × Σ -→ Σ that satisfy the following two properties for a set operation and for any sets S i and S j and for any sk i and sk j in K pr and any pk i and pk j in K pub :

• Homomorphism -

sig(sk i •sk j , S i S j ) = sig(sk i , S i )∇sig(sk j , S j ) (1)
• Correctness -

vrf(pk i ♦pk j , sig(sk i • sk j , S i S j )) = vrf(pk i , sig(sk i , S i )) ∧ vrf(pk j , sig(sk j , S j )) (2)
We note the ∧ symbol is the logical AND operator. Therefore, a Set-Homomorphic Signature Scheme (SHSS) consists of four algorithms and four functions such that:

SHSS = {{gen, sig, vrf, agg}, {•, ♦, ∇, }}, (3)
where the agg algorithm is defined as agg : Σ × Σ -→ Σ × A (cf; Equation 4). This algorithm returns an aggregate signature with a set of parameters in A. We have to note A is the set of parameters required to calculate the ♦ function.

agg(sig(sk i , S i ), sig(sk j , S j )) = sig(sk i , S i )♦sig(sk j , S j ) (4)

B. System Model

Our proposal is composed of six randomized algorithms defined as follows: stp : 1 ξ -→ P : this algorithm is run by a trusted entity. It takes as input the security parameter ξ, and it outputs a set of public parameters P. We suppose that P is an auxiliary input of all the following algorithms. gen u : P -→ {sk u , pk u } : this algorithm is executed by a trusted entity in order to generate a pair of keys for each requesting host H u . It takes as input the public parameters P, and it outputs the secret key sk u and the public key pk u of H u . sig : sk u × S -→ σ : this algorithm is performed by any involved host H u . It takes as input the secret key of the signing host sk u and the data set S. It outputs a signature σ. vrf : pk u × σ -→ {0, 1} : this algorithm is run by a verifying host. It takes as input the public key of the signing entity pk u and the signed set S. It outputs either 1; i.e accept; or 0; i.e. reject. agg :

{σ i } i∈[1,n] × A -→ Σ :

C. Security Model

To design the most suitable security solutions for distributed environments, we have to consider realistic threat models. For example, an external malicious adversary may intend to inject false information as a legitimate user, or to gain extra knowledge about collected signed data. Thus, this attacker is considered against the unforgeability, anti-replay, and the privacy-preserving properties, as defined in Section II-B.

IV. CONSTRUCTION

In this section, we first introduce some mathematical notations and present our cryptographic assumptions in Section IV-A. Then, we detail the different algorithms of our construction in Section IV-B.

A. Mathematical Background

In this section, we first define our collision-resistant hash function, in Subsection IV-A1. Then, we introduce bilinear maps in Section IV-A2 and detail our cryptographic assumptions in Section IV-A3.

1) Hash functions: h is a collision-resistant hash function. h : M → P, where M is the message space and P is a set of prime numbers such that |M| ≤ |P|. For all m i , m j ∈ M, if m i = m j , then h(m i ) = h(m j ) and gcd(h(m i ), h(m j )) = 1.

2) Bilinear maps: Let us consider three cyclic multiplicative groups G 1 , G 2 , and G T of prime order p. Let ê : G 1 × G 2 → G T be a bilinear map that has to fulfill the following properties:

• Bilinearity -for each g 1 ∈ G 1 , g 2 ∈ G 2 and a, b ∈ Z p ,
we have the equality ê(g 1 a , g 2 b ) = ê(g 1 , g 2 ) ab

• Computabilityfor each g 1 ∈ G 1 and g 2 ∈ G 2 , there exists an algorithm that efficiently calculates ê(g 1 , g 2 ) • Non-degeneracyfor each g 1 ∈ G 1 and g 2 ∈ G 2 , we have < ê(g 1 , g 2 ) >= G T

3) Complexity assumptions: For our construction, we consider the following complexity assumptions:

• Computational Diffie Hellman Assumption (CDH) -
Let G be a group of a prime order p, and g is a generator of G. The CDH problem is given the tuple of elements (g, g a , g b ) where {a, b} R ← -Z p , and there is no efficient probabilistic algorithm A CDH that computes g ab . • Computational co-Diffie Hellman Assumption (co-CDH) -Let G 1 , G 2 be two groups of a prime order p, g 1 , g 2 and their respective generators and ê an asymmetric pairing function. The co-CDH problem in the asymmetric bilinear group (G 1 , G 2 , g 1 , g 2 , ê) is, given a tuple of elements (g 1 , g 2 , g 1 a , g 2 b ) where {a, b} R ← -Z p there is no efficient probabilistic algorithm A co-CDH that computes g 1 ab .

B. Concrete Construction

Our SHSS construction is composed of three phases, namely the generation phase, the single signature phase, and the aggregation phase.

1) Generation phase: The generation phase is executed only one time, and it relies on the stp and gen u algorithms. The stp algorithm is given by Algorithm 1. We consider the prime order p, the groups G 1 , G 2 , and the pairing ê have already been chosen and are naturally included in the parameters P.

Algorithm 1 stp algorithm 1: select g 1 a generator of G 1 ; 2: select g 2 a generator of G 2 ; 3: select h a collision-resistant hash function defined in Section IV.A.1); 4:

return P = (G 1 , G 2 , ê, g 1 , g 2 , h)
Then, for each host H u , the trusted entity executes the gen u algorithm given by Algorithm 2.

Algorithm 2 gen u algorithm 1: sk u R ← -Z p ; 2: pk u ← g 2 sku ;
3: return (sk u , pk u )

2) Single signature phase: As introduced in Section III-B, the single signature phase is composed of two algorithms, namely sig and vrf, defined by Algorithm 3 and Algorithm 4, respectively.

H(S) ← H(S) * h(m i ); 7: end for 8: σ S ← [g 1 sku ] H(S) -1 ; 9: return σ S
The correctness of our proposed single signature, for any set S, is easily deduced using the bilinearity property of pairing functions (cf; Equation 5) such that:

ê(σ S , g 2 ) = ê([g 1 sku ] H(S) -1 , g 2 ) = ê([g 1 ] H(S) -1 , pk u ) (5)
This proves Equation 5is correct if and only if the public key pk u = g 2 sku used by the vrf algorithm corresponds to the private key used to generate the signature σ S .

H(S) ← H(S) * h(m i );

6: end for 7: X S ← [g 1 ] H(S) -1 ; 8: if ê(σ, g 2 ) = ê(X S , pk u ) then B ← 0 12: end if 13: return B

3) Aggregation phase: The aggregation phase includes the agg and vrfagg algorithms, as detailed in Section III. The aggregating node executes the agg algorithm and derives the resulting signature that relies on set operations, namely union, subset, and intersection operations. The resulting signature, detailed in definition 3.2, must ensure the homomorphism and the correctness properties (cf. Equation 1 and Equation 2).

Theorem 1 shows our proposed signature scheme is homomorphic with respect to the union operation. The sethomomorphism properties, with respect to two sets S i and S j , is generated by H i and H j devices, respectively. Our scheme can easily be extended to support multiple sets of data records.

Theorem 1: Union-homomorphic Signature Let stp, gen u , sig, and vrf be the algorithms introduced by Algorithm 1, 2, 3, and 4. The agg algorithm resulting in an aggregate signature sig(sk i • sk j , S i S j ) presented in Equation 4 is defined as follows, such that is the union operator, and • is the minus operator:

sig(sk i , S i )∇sig(sk j , S j ) = [sig(sk i , S i )] u [sig(sk j , S j )] v (6)
such that:

u = u + v H(S i ) H(S j ) , v = v + u H(S j ) H(S i )
and u and v are the two unique integers that satisfy:

gcd(H(S i ), H(S j )) = vH(S i ) + uH(S j )
Proof 1: To prove Theorem 1, we must first express H(S i ∪ S j ), using H(S i ) and H(S j ), such that

H(S k ) = l∈[1,n S k ] h(s l ),
where k ∈ {i, j} and n S k = |S k | (i.e., the number of elements of a set S k ).

Lemma 1: Given two sets S i and S j and given a hash function H as defined in Algorithm 3, there exist two unique integers u and v such that:

H(S i ∪ S j ) -1 = uH(S i ) -1 + vH(S j ) -1 (7)
Proof 2: Given the hash function H as defined in Algorithm 3 the H(S i ∪ S j ) corresponds to:

H(S i ∪ S j ) = lcm(H(S i ), H(S j )) (8)
In addition, knowing that lcm and gcd satisfy the following Equation:

lcm(H(S i ), H(S j ))gcd(H(S i ), H(S j )) = H(S i )H(S j )
And using the Euclidean algorithm, there exist two unique integers u and v, such that:

gcd(H(S i ), H(S j )) = vH(S i ) + uH(S j )
Consequently, we have:

lcm(H(S i ), H(S j )) -1 = uH(S i ) -1 + vH(S j ) -1 (9)
Based on Equation 8and Equation 9, we prove Lemma 1.

In the following, we demonstrate that the agg algorithm satisfies the homomorphism and correctness properties. In this paper, we only prove for the union operation, but our scheme is easily extensible to the subset and set difference operations.

• Proof of homomorphism -To prove that our agg ensures the homomorphism property, we have to express sig(sk i , S i ), referred to as σ i , and sig(sk j , S j ), referred to as σ j , with respect to both sk i and sk j .

= σ i u σ j v = [g 1 uski ] H(Si) -1 [g 1 vskj ] H(Sj ) -1 = [g 1 uski ] H(Si) -1 [g 1 vskj ] H(Sj ) -1 [g 1 vski ] H(Sj ) -1 [g 1 vski ] H(Sj ) -1 = g 1 ski[uH(Si) -1 +vH(Sj ) -1 ] g 1 v(skj -ski)H(Sj ) -1
Similarly, we can express

= σ i u σ j v = g 1 skj [uH(Si) -1 +vH(Sj ) -1 ] g 1 u(ski-skj )H(Si) -1
Consequently, by dividing by , we obtain the following result:

g 1 (ski-skj )[uH(Si) -1 +vH(Sj ) -1 ] = g 1 v(skj -ski)H(Sj ) -1 g 1 u(ski-skj )H(Si) -1 g 1 (ski-skj )[uH(Si∪Sj ) -1 ] = g 1 skiH(Si) -1 (u+v H(S i ) H(S j ) ) g 1 skj H(Sj ) -1 (v+u H(S j ) H(S i ) ) sig(sk i -sk j , S i ∪ S j ) = σ i u σ j v
where:

u = u + v H(S i ) H(S j ) , v = v + u H(S j ) H(S i )
• Correctness -This property shows how the verifier relies on the vrfagg in Algorithm 5 based on the aggregate signature [sig(ski,Si)] u [sig(skj ,Sj )] v , the public keys pk i and pk j of nodes H i and H j , respectively, and the aggregate set S i ∪ S j .

Based on Equation 2 and the properties of the bilinear function ê, we can write ê(g 1 (ski-skj )[H(Si∪Sj ) -1 ] , g 2 ) denoted by E, as follows:

E = ê(g 1 u skiH(Si) -1 -v skj H(Sj ) -1 , g 2 ) = ê(g 1 u skiH(Si) -1 , g 2 ) • ê(g 1 -v skj H(Sj ) -1 , g 2 ) = ê(σ i , g 2 ) u • ê(σ j , g 2 ) -v
As such, based on the vrfagg algorithm defined in Algorithm 5 and Equation 2, we deduce that vrfagg(pk i ♦pk j , sig(sk i -sk j , S i ∪ S j )) = 1, such that pk i ♦pk j = pki pkj , if and only if:

ê(σ i , g 2 ) = ê(g 1 H(Si) -1
, pk i ), and

ê(σ j , g 2 ) = ê(g 1 H(Sj ) -1
, pk j )

Algorithm 5 vrfagg algorithm 1: input: the public keys pk i and pk j of nodes H i and H j , the aggregate set S i ∪ S j and the aggregate signature Σ 2: output: boolean value B ∈ {0, 1} B ← 0 13: end if 14: return B This proves the correctness of our proposed aggregate signature, with respect to Definition 3.2.

3: H(S i ∪ S j ) = 1; 4: n ← |S i ∪ S j |; 5: for all k ∈ [1 . . . n] do 6: H(S i ∪ S j ) ← H(S i ∪ S j ) * h(s k ); 7: end for 8: X ← [g 1 ] H(Si∪Sj ) -1 ; 9: if ê(Σ, g 2 ) = ê(X,

V. SECURITY DISCUSSION

A. Unforgeability

As discussed in Section II-B, the unforgeability requirement prevents external adversaries from generating a new valid signature based on new generated data.

Obviously, A tries a forgery attack against the CDH and co-CDH assumptions considering the simple signature σ and the aggregate signature Σ are both products of an accumulator over the set elements and a group element associated to the secret key of the signer. Knowing these two group elements, namely the accumulator and the group element associated with the secret key of the signer, are both required for deriving the corresponding signature, A is led to break the co-CDH and CDH assumptions. The Exp unf is then considered with respect to the CDH-assumption, as it is considered stronger than the co-CDH assumption [START_REF] Boneh | Short signatures from the weil pairing[END_REF].

The complexity of the CDH assumption has been studied in [START_REF] Boneh | Short signatures from the weil pairing[END_REF] and was demonstrated to be hard to solve; i.e., a (t, ) CDH group is a group for which the Adv(A, t) ≤ for every PPT adversary running in a time t. Therefore, our signature scheme is unforgeable.

B. Privacy

The privacy-preserving property covers the privacy of contents, where the adversary should not be able to guess the origin and the content of the removed parts from signed data flow, and the privacy of users, where an intruder should not be able to link each signed part of an aggregate data stream to its respective generator.

Generally, the attacker A tries to distinguish between two honestly derived signatures by linking each simple signature of an aggregate signature Σ to its related generator, having only access to Σ.

The aggregate signature Σ results in merging two signatures σ b and σb, where b = (j, k), such that j refers the set index (i.e., j ∈ {0, 1}) and k to the index of the private key (i.e., k ∈ {1, 2}). Each single signature σ (j,k) , generated by C using the private key sk k , is based on a randomly chosen set S j . As such, both signatures σ b and σb are identically distributed in both cases. Thus, the attacker A is relying on a left-or-right oracle with respect to the CDH assumption. Hence, he cannot distinguish the oracle's outputs better than flipping a coin.

Similarly, the aggregate signature Σ generated based on two signatures associated with two different private keys, sk k (i.e., k ∈ {1, 2}), is uniformly distributed. Hence, two aggregate signatures derived based on two different combinations b and b are statistically indistinguishable, and the probability of predicting b is 1 2 with respect to the CDH assumption. As such, the A cannot distinguish two different aggregate signatures with a probability greater than 1 4 . Therefore, our scheme provides the privacy property.

VI. CONCLUSION

The need for secure mechanisms that enable the end receiver to efficiently verify the authenticity of aggregated data streams and the emergence of homomorphic signatures led us to present a cryptographic mechanism that enables efficient aggregation of signed data. In this paper, we proposed a new lightweight aggregate signature scheme based on set-homomorphic relations in distributed environments. Our construction permits distribution of the verification process among multiple IoT nodes. We proved its homomorphism and correctness. In addition, our construction preserves the privacy of non-end nodes such that an external adversary cannot distinguish between honestly derived signatures by linking each simple signature of an aggregate signature to its respective generator.

Fig. 1 .

 1 Fig. 1. An Example System

  this algorithm is performed by an aggregating node. It takes as input a set of n signatures and a set of parameters A required for the calculation of the resulting signature. It outputs an aggregated signature Σ. vrfagg : Σ × {pk i } i∈[1,n] × A -→ {0, 1} : this algorithm is executed by any verifying entity. It takes as input an aggregated signature Σ, the set of public keys of the involved hosts, and the set of aggregation parameters A. It outputs either 1 if the signature is accepted, or 0 if rejected.

Algorithm 3 sig algorithm 1 :

 1 input: host secret key sk u and the set S = {m 1 , m 2 , • • • , m n S } 2: output: the signature σ S 3: H(S) = 1; 4: n S ← |S|; 5: for i ∈ [1 . . . n S ] do 6:

Algorithm 4 vrf algorithm 1 :

 1 input: host u's public key pk u , the set S and the signature σ 2: output: boolean value B ∈ {0, 1} 3: H(S) = 1; n S ← |S|; 4: for i ∈ [1 . . . n S ] do 5:

ACKNOWLEDGMENT

This material is based upon work supported by the National Science Foundation under Grant ACI-1547467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.