The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from a simple but ingenious idea of Halasz the proof uses only classical knowledge about the zeta function, results known since at least hundred years.
(English) The main goal of this dissertation is to develop new lattice-based cryptographic schemes. Most of the cryptographic protocols that each and every one of us use on a daily basis are only secure under the assumption that two mathematical problems, namely the discrete logarithm on elliptic curves and the factorization of products of two primes, are computationally hard. That is believed to be true for classical computers, but quantum computers would be able to solve these problems much more efficiently, demolishing the foundations of plenty of cryptographic constructions. This reveals the importance of post-quantum alternatives, cryptographic schemes whose security relies on different problems intractable for both classical and quantum computers. The most promising family of problems widely believed to be hard for quantum computers are lattice-based problems. We increase the supply of lattice-based tools providing new Zero-Knowledge Proofs of Knowledge for the Ring Learning With Errors (RLWE) problem, perhaps the most popular lattice-based problem. Zero-knowledge proofs are protocols between a prover and a verifier where the prover convinces the verifier of the validity of certain statements without revealing any additional relevant information. Our proofs extend the literature of Stern-based proofs, following the techniques presented by Jacques Stern in 1994. His original idea involved a code-based problem, but it has been reiteratedly improved and generalized to be used with lattices. We illustrate our proposal defining a variant of the commitment scheme, a cryptographic primitive that allows us to ensure some message was already determined at some point without revealing it until a future time, defined by Benhamouda et al. in ESORICS 2015, and proving in zero-knowledge the knowledge of a valid opening. Most importantly we also show how to prove that the message committed in one commitment is a linear combination, with some public coefficients, of the committed messages from two other commitments, again without revealing any further information about the messages. Finally, we also present a zero-knowledge proof analogous to the previous one but for multiplicative relations, something much more involved that allows us to prove any arithmetic circuit. We give first an interactive version of these proofs and then show how to construct a non-interactive one. We diligently prove that both the commitment and the companion Zero-Knowledge Proofs of Knowledge are secure under the assumption of the hardness of the underlying lattice problems. Furthermore, we specifically develop such proofs so that the arising conditions can be directly used to compute parameters that satisfy them. This way we provide a general method to instantiate our commitment and proofs with any desired security level. Thanks to this practical approach we have been able to implement all the proposed schemes and benchmark the prototype im-plementation with actually secure parameters, which allows us to obtain meaningful results and compare its performance with the existing alternatives. Moreover, provided that multiplication of polynomials in the quotient ring ℤₚ[𝑥]/⟨𝑥ⁿ + 1⟩, with 𝑝 prime and 𝑛 a power of two, is the most basic operation when working with ideal lattices we comprehensively study what are the necessary and sufficient conditions needed for applying (a generalized version of) the Fast Fourier Transform (FFT) to obtain an efficient multiplication algorithm in quotient rings as ℤₘ[𝑥]/⟨𝑥ⁿ − 𝑎⟩ (where we consider any positive integer 𝑚 and generalize the quotient), as we think it is of independent interest. We believe such a theoretical analysis is fundamental to be able to determine when a given generalization can also be applied to design an efficient multiplication algorithm when the FFT is not defined for the ring we are considering. That is the case of the rings used for the commitment and proofs described before, where only a partial FFT is available. (Español) El objetivo principal de esta tesis es obtener nuevos esquemas criptográficos basados en retículos. La mayoría de los protocolos criptográficos que usamos a diario son únicamente seguros bajo la hipótesis de que el problema del logaritmo discreto en curvas elípticas y la factorización de productos de dos primos son computacionalmente difíciles. Se cree que esto es cierto para los ordenadores clásicos, pero los ordenadores cuánticos podrían resolver estos problemas de forma mucho más eficiente, acabando con las bases sobre las que se fundamenta una multitud de construcciones criptográficas. Esto evidencia la importancia de las alternativas poscuánticas, cuya seguridad se basa en problemas diferentes que sean inasumibles tanto para los ordenadores clásicos como los cuánticos. Los problemas de retículos son los candidatos más prometedores, puesto que se considera que son problemas difíciles para los ordenadores cuánticos. Presentamos nuevas herramientas basadas en retículos con unas Pruebas de Conocimiento Nulo para el problema Ring Learning With Errors (RLWE), seguramente el problema de retículos más popular. Las pruebas de Conocimiento Nulo son protocolos entre un probador y un verificador en los que el primero convence al segundo de la validez de una proposición, sin revelar ninguna información adicional relevante. Nuestras pruebas se basan en el protocolo de Stern, siguiendo sus técnicas presentadas en 1994. Su idea original involucraba un problema de códigos, pero se ha mejorado y generalizado reiteradamente para poder aplicarse a retículos. Ilustramos nuestra propuesta definiendo una variante del esquema de compromiso, una primitiva criptográfica que nos permite asegurar que un mensaje fue determinado en cierto momento sin revelarlo hasta pasado un tiempo, definido por Benhamouda et al. en ESORICS 2015, y probando que conocemos una apertura válida. Además mostramos cómo probar que el mensaje comprometido es una combinación lineal, con coeficientes públicos, de los mensajes comprometidos en otros dos compromisos. Finalmente también presentamos una prueba de Conocimiento Nulo análoga a la anterior pero para relaciones multiplicativas, algo mucho más laborioso que nos permite realizar circuitos aritméticos. Todo esto sin revelar ninguna información adicional sobre los mensajes. Mostramos tanto una versión interactiva como una no interactiva. Probamos que tanto el compromiso como las pruebas de Conocimiento Nulo que le acompañan son seguras bajo la hipótesis de que el problema de retículos subyacente sea difícil. Además planteamos estas pruebas específicamente con el objetivo de que las condiciones que surjan puedan ser utilizadas directamente para calcular los parámetros que las satisfagan. De esta forma proporcionamos un método genérico para instanciar nuestro compromiso y pruebas con cualquier nivel de seguridad. Gracias a este enfoque práctico hemos podido implementar todos los esquemas propuestos y evaluar el rendimiento con parámetros seguros, lo que nos permite obtener resultados relevantes que poder comparar con las alternativas existentes. Por otra parte, dado que la multiplicación de polinomios en el anillo cociente ℤₚ[𝑥]/⟨𝑥ⁿ + 1⟩, con 𝑝 primo y 𝑛 una potencia de 2, es la operación más utilizada al trabajar con retículos ideales, estudiamos de forma exhaustiva cuáles son las condiciones suficientes y necesarias para aplicar (una versión generalizada de) la Transformada Rápida de Fourier (FFT, por sus siglas en inglés) para obtener algoritmos de multiplicación eficientes en anillos cociente ℤₘ[𝑥]/⟨𝑥ⁿ − 𝑎⟩, (considerando cualquier 𝑚 positiva y generalizando el cociente), de interés por sí mismo. Creemos que este análisis teórico es fundamental para determinar cuándo puede diseñarse un algoritmo eficiente de multiplicación si la FFT no está definida para el anillo considerado. Es el caso de los anillos que utilizamos en el compromiso y las pruebas descritas anteriormente, donde solo es posible calcular una FFT parcial.
Deebakkarthi Chinnasame Rani, Sai Ganesh Janakiraman, K. S. Chandra, Elambharathi Padmavathi Thangavel · 7 authors
Presented herein is a novel algorithm for multi-round, zero-knowledge proof (ZKP), devised specifically for authenticating factorisation proofs within a variety of cryptographic applications. This advanced algorithm, while maintaining computational complexity within acceptable bounds, offers a secure and proficient solution. The functionality of the algorithm is marked by multiple rounds of interaction between the Prover and Verifier. Initially, the Prover generates a random value and calculates a commitment. Subsequently, the Verifier issues a random challenge, eliciting a computed response from the Prover. To validate the proof, the Verifier verifies the equality of the commitment and the computed response. Efficaciousness of the proposed multi-round ZKP algorithm is demonstrated across diverse input sizes and parameters. Results indicate a success rate exceeding 90% on average, showcasing the robustness of the method. The recurring interaction between the Verifier and Prover enhances the Prover's authentication, thereby improving the algorithm’s reliability. Implementation of the algorithm, achievable through standard cryptographic tools and protocols, can fortify the security of multiple cryptographic applications. A significant application can be found in Digital Identity Management Systems (DIMS). Currently, these systems are vulnerable to a myriad of threats, including identity spoofing, data breaches, and internal security risks. The application of the ZKP algorithm can simultaneously augment security and withhold sensitive information, potentially transforming the DIMS security landscape. Future research may focus on improving the efficiency and scalability of the multi-round ZKP algorithm. There also remains a vast potential for exploring additional applications of this technique within various cryptographic domains.
Bernhard Fisseni, Deniz Sarikaya, Bernhard Schröder
Abstract We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics.
The opening scene in Saša Stanišić's latest work, Where You Come From, places the reader back in Višegrad, a storied town in today's Bosnia and Herzegovina and the site where his first semi-autobiographical novel, How the Soldier Repairs the Gramophone, ended. We see the narrator's grandmother Kristina as an elderly lady, shouting from the balcony down to her younger self on the street: ‘I'll come get you. Don’t move!’1 Living up to her promise, Kristina drags herself down three flights of stairs, without shoes, her feet covered by thin black stockings only. Cars hit the brakes. Traffic comes to an abrupt halt. In her effort to rescue her adolescent self, Kristina succeeds at stopping the steady flow of the world for a moment. The evocative opening places Kristina front and centre in the text while introducing a split in time: we are simultaneously in the years 2018 and 1943, as well as in some Yugoslav moment in-between, when the street on which the impossible encounter was to play out still bore the name of Josip Broz Tito.2 The formidable storyteller Stanišić tempts his audience in this vignette-like opening with a reassurance of sorts: where you come from – Herkunft (‘origin’) – can, indeed, be found at some mythic confluence of time and place. Origins are not only localisable; they can also be projected back in time, with kinship serving as a well-established narrative and ideological ploy for staking out historical claims to belonging. The very first scene thus appears to preempt the text that follows by giving a rather firm answer to the question that the book's title vividly raises. The answer – the Romantic idea of origins – seems to be reinforced when Stanišić takes his readers from Višegrad to the village of Oskoruša, and with that, deeper into the past. At Oskoruša's cemetery, many of the tombstones bear the narrator's last name: Stanišić. What could possibly be more affirming in a tale of origins than a mountain village, where even the ground is inscribed with proof of one's ancestry?3 Stanišić proves to be a formidable storyteller because he involves his readership in something like a theatre of origins, presenting us with the essential trappings of a ‘filiative’ form of belonging – a term we borrow from Edward Said and Timothy Brennan to designate communities predicated on inheritance, descent, and circumstances of birth – only to then reject the premise and ontological primacy ascribed to these ‘natural’ social bonds.4 Expressed in the well-known terms of National Socialist ideology, Blut (‘blood’) and Boden (‘soil’) not only merge here but become inseparable, with their union counteracting – that is, stabilising – the purportedly modern volatility of origins.5 From the perspective that this special issue of Critical Quarterly adopts, a striking moment comes to the fore: Stanišić's spectacle of origins plays out in what is commonly considered part of Europe's periphery, a tiny village ‘in the Bosnian mountains, in the farthest eastern regions of this perpetually tragic land’ (Where 27; 31). The opening is, of course, an ideological trap expertly set to ensnare the reader committed to the Romantic idea of unmediated belonging and unable to recognise the idea's kinship with the age-old stereotype of bellicose Balkan tribes. For both the yearning for pristine origins and the attribution of irreconcilable feuds to peripheral spaces share the presumption of filiative bonds and also of backwardness – a mode of existence caught outside of historical time and thus seemingly beyond the reach of the historical subject. But who exactly is the audience for Stanišić's demonstration of origins, one that is deeply informed by and that appears to reinforce West European stereotypes of the Balkans as Europe's backward periphery? For whom does this spectacle, in fact, take place? And just as importantly, what happens to the notion of origins over the course of the narrative? We begin with these questions because in the works of German-Bosnian author Saša Stanišić we find a particularly dynamic expression of the core-periphery mapping. We take Stanišić's oeuvre as exemplary for the writer whose critical and political affinities generate an aesthetic intervention that goes beyond the work of demystification (of categories such as core and periphery), summoning at the same time an active historical subject capable of changing the laws of history that are often mistaken for second nature. More specifically, examining Stanišić's latest novel, Where You Come From, we argue that Stanišić's intervention lies in subverting centre and periphery as stable entities and revealing how these categories are produced, employed, and exploited – in writing. His earlier work How the Soldier Repairs the Gramophone extends this critical work to the seemingly unalterable laws of history, reintroducing a force that could halt the process of peripheralisation and the endless proliferation of crises. The first critical move points to an alternative in a world in which we are told there are none. The second shows what it takes to actually change the world. Thus, while the first intervention targets discursive conventions, the terrain on which the second intervention plays out is not discursive. Born in Višegrad, Stanišić fled to Germany as a teenager during the Bosnian war. His first novel, How the Soldier Repairs the Gramophone (2008) (Wie der Soldat das Grammofon repariert, 2006), turned him into one of Germany's most celebrated authors overnight. And while this semi-autobiographical tale of migration from war-stricken Yugoslavia did not make it past the short list for the 2006 German Book Prize, it paved the way for Stanišić's success in the German literary markets, leading Boris Previšić to wonder if a “Yugoslavian Turn” was on the horizon of German-language literature and its largely orientalising depiction of the Balkans.6 In his second novel, Before the Feast (2016) (Vor dem Fest, 2014), Stanišić turned to a peripheral place within Germany. His peculiar spin on the German genre of the Heimatroman – most frequently translated as ‘regional novel’ – is set in the imaginary village of Fürstenfelde, in the former East German province of the Uckermark.7 Stanišić tells the story of a single night in Fürstenfelde, but what comes to life is the village's history in an assemblage of legends, family stories, and tales. The book continued Stanišić's critical acclaim, winning both the Alfred Döblin Prize in 2013 and the 2014 Leipzig Book Fair Prize. In Where You Come From (2021) (Herkunft, 2019), Stanišić revisits the Balkans to scrutinise this mythical place of the European imagination through the themes of origins, memory, and their disappearance. Where You Come From draws on a long Western European tradition of ‘inventing’, ‘imagining’, or ‘constructing’ Eastern Europe and the Balkans.8 In her now-classic monograph, Imagining the Balkans (1997), Maria Todorova traced the emergence of ‘Balkan’ and ‘balkanization’ as derogatory terms around the turn of the twentieth century and then drew attention to their resurgence in this denigratory capacity at the time the former Yugoslav state fell apart. ‘The persistent use of “Balkan” for the Yugoslav war has by now rekindled old stereotypes and licensed indiscriminate generalizations about the region’, Todorova writes.9 Even a cursory glance at the clichés that Todorova foregrounds reveals intriguing parallels with the inaugural moments in Where You Come From. Mountainous regions, extended clans and tribes, age-old feuds over land – these are common ahistorical staples in explaining the fall of Yugoslavia.10 If the similarities are not immediately visible, it might be due to the fact that Stanišić modifies the war scenario and appears to offer at first a reversal of established valorisations. While the location remains unchanged, what plays out against the background of Bosnian mountains is not violence but the alleged recovery of origins. No syllable you can cling to, zero rhythm, a bizarre sequence of sounds. From the very beginning: Osko – what's that supposed to be? who talks like that? – and then the plummet to the hissed end, -rusha. Hard and Slavic, the way things end in the Balkans. I could leave that in, people would probably accept it from me, seeing as I'm from the Balkans myself. Hard Slavic endings? (26; 30; italics in original) But for whom does this masterful juggling of firmly established clichés take place? Often a question that allows only for speculative answers regarding intended readership finds an explicit one in Where You Come From: the title of its second section and the phrase repeated throughout make it clear that the text is addressed ‘To the [German] Alien Registration Office’ (2; 6). Stanišić builds the authority responsible for granting German citizenship into the narrative world of Where You Come From as the text's intended reader: ‘Thirty years later, in March 2008, I was applying for German citizenship and had to submit at the Alien Registration Office a handwritten chronology of my life [Lebenslauf], among other things’, the narrator declares (2–3; 6–7).11 The entire section reads like the narrator's demonstrative attempt to find a form for his story – or, even a story itself – that proves adequate for his readership and the stated purpose of obtaining German citizenship. The text thus stages itself not solely as a work of literature belonging to the realm of aesthetics, but rather as one whose particular purpose (also) lies in achieving naturalisation. While the humorous tone that frequently tips over into irony unmistakably pokes at the text's the narrative the question of belonging at the of Where You Come What does a text to in to become be What are the narrative of grandmother Kristina the on the former Josip Broz in Višegrad, the section of Stanišić's work in the mode of was on March in Višegrad, on the (Where The also the moment in which of begin to as the of the with the same a my first the I to down on was that I was on March of some of which are the the narrator's and Where You Come From even more For the narrator that he the Alien Registration such as a narrative without I in the of the in which Where You Come From not with an of but with the story of the and his in the story of out a story beginning: what to – – I the story of the of my first which is, indeed, the of a section that comes in Where You Come From and to which we italics in What we in to the narrator's with the text of Where You Come From is also an on the process of a or that the reader that the to of Stanišić's The on and to a of the narrator to – in a as demonstrative as it is – that his German text is not only to the intended reader but also in the of the of the narrator's with Where You Come From and the that are these become at that the Alien Registration Office as the of the former also the intended reader of the is to Stanišić's ploy of that Where You Come From, itself to the German is not to of course, that the book's readers are to or are by the narrative the intended reader as to the question of belonging not in but with an to the particular historical and political At the same time, the that the text that the narrator Saša from Where You Come From, of reveals that the narrator is in the very of his tale of origins. The on the process of not only that this story is and into but that its takes place in to its The Alien Registration Office is not the only authority that on the of origins in questions of belonging. The whom the narrator in Bosnia the same question – you come – and the same to the primacy of (Where 30; come from one at the cemetery, with his what remains of their in this is supposed to to the of origins among the tombstones and on one's Even if to a place of one is to in some by and discursive The to a the to recognise his place of in the the narrator is at first by the he is from the of the family if in a a is to his is then to a family well – ‘in fact, to the of – and a of that from his ground (Where If the is a place where into the then the that in the the same from this back into the the imaginary of comes here – on the And for a from his well and the of his the narrator seems to that he can what he could see his place of the Romantic of he that was and like the to origins in the at the And while he from the of as he declares not to down over a does Stanišić his readers to their out to be a that the of by with if Where You Come From is firmly against the idea of the Balkans as a place for origins in the form of filiative it in Germany. often to be an – a that the of what the in German – is as just as the to the text for Stanišić's readers Oskoruša's like in a is not some place that for the modern to but a The on in the name and comes to the in his the Bosnian the the German about the and not about what the or how italics in takes in to what Stanišić to In an he the as is a and because it is by an you from an as to be an expression for an and to to their and from belonging. a and thus the of against the of the of and Stanišić's text revisits of belonging that are commonly to the author whose to in form in the German literary such as often of their both on belonging itself and the to places that the also The author with as the German has frequently a that for the of by way of that, the of Germany's these in these terms allows for the of the periphery as a place of does he his readers with a migration tale that the centre as a place of we can that in in Stanišić's by the narrative of the narrator's story for the Alien Registration Stanišić targets the of origins. In fact, the is in the origins a story that is with a audience and in both of which how they are and how of belonging are and the that the of the periphery plays in their Stanišić also targets of origins, the of the periphery and its to If Where You Come From the discursive of origins and the of the periphery that goes in with How the Soldier Repairs the Gramophone the core-periphery dynamic from the perspective of the historical subject. In other Stanišić's to beyond the seemingly stable categories of and in to a question of even what – and by whom – might the historical of peripheralisation be At first this that the fall of Yugoslavia and the of and are through the seemingly perspective of its storyteller from by what Boris Previšić a for his readers into a world of and stories, in which one can a depiction of the periphery as a place where still not giving way to the of the war through can it even in the reader to a war But beyond the and it can the narrator – particularly one on the of – also to life a historical subject capable of in its How can in and an historical subject be The in with the the within Stanišić's The Gramophone in a seemingly the narrator's he a and to his The of the is told that with this he become most in the to of things just as long as they are in with and the of the of we argue the to the of the of the former historical subject who was to change the laws of history and who at the of the the that the he is has its In fact, the of a that of what is told in the and that play with Balkan while the and at the same and his to the first into the only to into a within the by And while the appears to of – and thus the of the – a narrator to to the of his in Bosnia and to up on the in the of The last of the shows the through the narrative of appears to be for the of his old for some of his younger self was to earlier in the or had to the and this of would a the and of the a for the narrative of the world to the war. it would also an to the Stanišić's narrator is an that does not the of The narrative and even the of this of course, from a long list of authors who in what has in The of the narrative and only the of The that was to of to the that it has in the at the site where the war for the The finds at the of – or story that is not by the – to that could the world that has in that one can in some of while the are – would be at And in of this of the in the of Stanišić is not to up on its political of in the that both its existence and to the perspective of its narrator short of explaining the aesthetic and political that are at play in The In fact, we argue that – or what appears as a of things – the of the aesthetic and political The aesthetic intervention lies in with the of and the narrator's perspective as to in the political and of the historical subject by the war. And as a political an of the of this historical subject to the seemingly course of history and to change its In other from the of to the war in fact the former historical subject of its it to a life of to the over which it has The Gramophone this process of by which history into a mythical state of nature. is the that and the world of not the other way the by the war does not the narrator to in the the he a of a in a story of rather peculiar turn of phrase that Stanišić in the of his a answer to this of and its in peripheral of the in the – an the of in the Bosnian – takes place at a location Stanišić as The expression as in the German text – – as it does in is because the in both is a of an expression in Stanišić's first the as is often in the to a location and that even an make it In one could such a place as ‘in the of or, and most for as a – a peripheral site if there was The common of the is one a location to on and to a place that is the same as a that also appears in Stanišić's is attention because they how The Gramophone the periphery and this peripheral with the of a of one that to a rather than its or The is both a for and an of to in the – in the in which the narrator is told about the – about Stanišić's of the expression into Stanišić what the of the expression in a place outside of The is to be to its end on the that lies the back of who is to the of the is thus in this German and in the that the phrase the and critical it had in the which the is – more the which Višegrad is and impossible to – are by the of is a world in which it is to the of or in a world in which one can of a force capable of Stanišić's is on the that in this peripheral is to his and that these the to For this is, in fact, the only of in the and its not from but from very a historical – the of Yugoslavia and its during is thus not only to the particular political of but the existence of this political is the of for it to it is that, in of the of Yugoslavia and the political to the alleged he from the of his in the I a who could make things – – one reads throughout the Stanišić's of the is as it to its not solely the but also the which thus into and of in the past the of for are is the moment of that as an for writing. While this Stanišić's with the of of the the of Stanišić's is exactly to be more than an expression of to use The Gramophone, in fact, the question of how to the of which the form is to be an Stanišić to through in seemingly and the lies in the of a political and historical that to the narrator's even capable of in a and in that this the same at Thus, in to the first moment of the – which appears to a particular to as a political intervention – the second moment is a with the very of to or the is in this that Stanišić's of the More specifically, he not only the work of but also the work that some or political within in The Gramophone, is to work like a in the that the of a a in which the of the political work is not to giving in to the laws of historical – in the this giving in to the as the stereotype has it – or for an outside it be of or aesthetic nature. where does one turn to these the back of a to the of the out to its in the Stanišić plays on well-known of and German on the Western in the years of to these from both would in land during an and and play an The of this and is the of when are to in turn out The that the Bosnian in the course of the with the that the to be are when the declares the and Bosnian from the – by Before the Bosnian is to play a one which they The appears the their and play the of the to a In the of this is a that to a more the of these is the Bosnian and who this story with the narrator and and a Bosnian his the name of the of the in of a in which they to play and on which the of their and Bosnian – feet – something rather the of the or in other they the who the of the to this the site that is peripheral in some place as the at the core in an alternative of – one on the of some state of rather than on some mythic to what we had in The to the only it a of not by inheritance, descent, or circumstances of birth but by and for the – an and an active historical subject to itself from at the appears as the laws of
Fog computing composes of neighboring devices, which are connected as cluster to collect and compute data via given algorithms. Compared with cloud computing, it collects and processes data at the edge of device layer, instead of remote data transmission and task scheduling. Fog computing reduces the communication and computation overhead in networking and data aggregating, while preserves data privacy against the cloud center. However, due to the different devices capacities, trust relations, security settings, the concerns are that, if the given data is real and sufficient for analyzing program as input, whether the data processing is done as appointed, and how to prevent secret leakage during the computation. This paper proposes a trustless layer for secure data computing in fog network, where the function accepts “hidden” data as input and generates zero-knowledge computing proof which can be easily verified by public. The combination of secure computing functions and blockchain systems escort trustful business in fog environment.
With the rapid development of information technologies, industrial Internet has become more open, and security issues have become more challenging. The endogenous security mechanism can achieve the autonomous immune mechanism without prior knowledge. However, endogenous security lacks a scientific and formal definition in industrial Internet. Therefore, firstly we give a formal definition of endogenous security in industrial Internet and propose a new industrial Internet endogenous security architecture with cost analysis. Secondly, the endogenous security innovation mechanism is clearly defined. Thirdly, an improved clone selection algorithm based on federated learning is proposed. Then, we analyze the threat model of the industrial Internet identity authentication scenario, and propose cross-domain authentication mechanism based on endogenous key and zero-knowledge proof. We conduct identity authentication experiments based on two types of blockchains and compare their experimental results. Based on the experimental analysis, Ethereum alliance blockchain can be used to provide the identity resolution services on the industrial Internet. Internet of Things Application (IOTA) public blockchain can be used for data aggregation analysis of Internet of Things (IoT) edge nodes. Finally, we propose three core challenges and solutions of endogenous security in industrial Internet and give future development directions.
Mohammed Balfaqih, Zain Balfagih, Miltiadis D. Lytras, Khaled Mofawiz Alfawaz · 6 authors
The concept of a smart city is aimed at enhancing the quality of life for urban residents, and logistic services are a crucial component of this effort. Despite this, the logistics industry has encountered issues due to the exponential growth of logistics volumes, as well as the complexity of processes and lack of transparency. Consequently, it is necessary to develop an efficient management system that offers traceability and condition monitoring capabilities to ensure the safe and high-quality delivery of goods. Moreover, it is crucial to guarantee the accuracy and dependability of distribution data. In this context, this paper proposes a blockchain-enabled IoT logistics system for the efficient tracking and management of high-price shipments. A smart contract based on blockchain technology has been designed for automatic approval and payment, with the aim of distributing shipping information exclusively among legitimate logistics parties. To ensure authentication, a zero-knowledge proof is used to conceal the blockchain address. Moreover, an intelligent parcel (iParcel) containing piezoresistive sensors is developed to pack delivered goods during the shipping process for violation detection such as severe falls or theft. The iParcels are automatically tracked and traced, and if a violation occurs, the contract is cancelled, and payment is refunded. The transaction fee per party is reasonable, particularly for high-price products that guarantee successful shipment.
Procuratorates, as the prosecutor in public interest litigation (PIL), need to obtain evidence from other PIL stakeholders including citizens, companies, governmental agencies, IoT monitoring devices and so on. However, the evidence sharing is not smooth due to the lack of secure data sharing and privacy protection during case investigation and evidence collection. Therefore, the authors propose a consortium blockchain-based secure data sharing and privacy protection scheme named PILChain. The involved organizations are connected as peers in PILChain. The safety of uploaded evidence and user privacy can be guaranteed with a fine-grained access control and zero-knowledge identity proof. InterPlanetary File System is introduced to store large evidence files off-chain, further enhancing the data security and system scalability. The security of PILChain is analyzed in terms of access control, evidence confidentiality, evidence integrity, traceability, privacy, and scalability. Last, the authors evaluate the performance of the developed prototype system by implementing PILChain on Hyperledger Fabric.
Neural networks are widely used learning models to achieve successful results in many application areas today. However, proving and sharing the accuracy and reliability of these networks is often limited due to privacy and security challenges. In this study, a method of cryptographic proving the accuracy of neural networks without revealing their intrinsic components is presented. The method is presented by using the Circom programming language to create a circuit containing these elements by making use of the final weights, bias values, and inputs of the neural networks. The use of the Circom programming language makes it possible to convert neural network elements into electronic circuits. The resulting circuit contains the representation of the neural network model and mimics the transformation from inputs to outputs. It is also used with Groth16 which is a Zero Knowledge Proof system to prove the accuracy of the neural network without leaking private information. As in this study, the newly produced circuit can be used with the help of zkREPL or terminal. As a result, an experimental method is presented to prove the real-world performance of the neural network model and increase the reliability of the model, and using the knowledge found in the literature, an approach has been explored to be implemented to solve current security problems. In this way, the correctness of the model can be proven without directly telling the hidden inputs to the other party.
Open access
Adversarial Robustness in Machine Learning
Machine Learning and Data Classification
Physical Unclonable Functions (PUFs) and Hardware Security
Tariq Bontekoe, Dimka Karastoyanova, Fatih Türkmen
Privacy-preserving computation (PPC) methods, such as secure multiparty computation (MPC) and homomorphic encryption (HE), are deployed increasingly often to guarantee data confidentiality in computations over private, distributed data. Similarly, we observe a steep increase in the adoption of zero-knowledge proofs (ZKPs) to guarantee (public) verifiability of locally executed computations. We project that applications that are data intensive and require strong privacy guarantees, are also likely to require verifiable correctness guarantees, especially when outsourced. While the combination of methods for verifiability and privacy protection has clear benefits, certain challenges stand before their widespread practical adoption. In this work, we analyze existing solutions that combine verifiability with privacy-preserving computations over distributed data, in order to preserve confidentiality and guarantee correctness at the same time. We classify and compare 37 different schemes, regarding solution approach, security, efficiency, and practicality. Lastly, we discuss some of the most promising solutions in this regard, and present various open challenges and directions for future research.
Zero-knowledge Range Proof(ZKRP) has gained increasing interest due to its applications in blockchain and cryptocurrencies in particular.ZKRP provides a mechanism to prove that a hidden integer belongs to a given interval without revealing any information about hidden integers.Some ZKRP depends on an honest and reliable third party.Others avoid from the third party.Currently, one of the famous ZKRP is the so-called Bulletproofs proposed by Bunz et al.Applying well-known homomorphic encryption methods in realizing ZKRP is a prospective direction.This paper proposes a new ZKRP based on one of the well-known homomorphic encryption methods-the order-preserving encryption method.
It is crucial to ensure the privacy and authenticity of the owner’s information in car insurance claims. However, the current traditional car insurance claims scenario suffers from inefficiency, complex service, unreliable data, and data leakage. Therefore, considering the privacy and sensitivity of insurance information and car owner data, we can use blockchain, smart contracts, and zero-knowledge proof technology to improve the current problems. This paper proposes a novel car insurance claim scheme based on smart contracts, blockchain, and zero-knowledge proof. Our scheme focuses on preserving privacy in the car insurance authorization and claim process. We design a private smart contract for the creation and revocation of car insurance and public smart contract for the authorization and validation of car insurance. By using ZoKrates, generating zero-knowledge proofs off chain and verifying the proofs on chain reduces the amount of data storage and computation on chain and provides privacy protection for sensitive information. Experimental results confirm the efficacy of our scheme in terms of security and performance.
Fair machine learning is a thriving and vibrant research topic. In this paper, we propose Fairness as a Service (FaaS), a secure, verifiable and privacy-preserving protocol to computes and verify the fairness of any machine learning (ML) model. In the deisgn of FaaS, the data and outcomes are represented through cryptograms to ensure privacy. Also, zero knowledge proofs guarantee the well-formedness of the cryptograms and underlying data. FaaS is model--agnostic and can support various fairness metrics; hence, it can be used as a service to audit the fairness of any ML model. Our solution requires no trusted third party or private channels for the computation of the fairness metric. The security guarantees and commitments are implemented in a way that every step is securely transparent and verifiable from the start to the end of the process. The cryptograms of all input data are publicly available for everyone, e.g., auditors, social activists and experts, to verify the correctness of the process. We implemented FaaS to investigate performance and demonstrate the successful use of FaaS for a publicly available data set with thousands of entries.
Attribute-based anonymous credentials offer users fine-grained access control in a privacy-preserving manner. However, in such schemes obtaining a user's credentials requires knowledge of the issuer's public key, which obviously reveals the issuer's identity that must be hidden from users in certain scenarios. Moreover, verifying a user's credentials also requires the knowledge of issuer's public key, which may infer the user's private information from their choice of issuer. In this paper, we introduce the notion of double issuer-hiding attribute-based credentials (${\sf DIHAC}$) to tackle these two problems. In our model, a central authority can issue public-key credentials for a group of issuers, and users can obtain attribute-based credentials from one of the issuers without knowing which one it is. Then, a user can prove that their credential was issued by one of the authenticated issuers without revealing which one to a verifier. We provide a generic construction, as well as a concrete instantiation for${\sf DIHAC}$based on structure-preserving signatures on equivalence classes (JOC's 19) and a novel primitive which we calltag-based aggregatable mercurial signatures. Our construction is efficient without relying on zero-knowledge proofs. We provide rigorous evaluations on personal laptop and smartphone platforms, respectively, to demonstrate its practicability.
We present an implementation of a Web3 platform that leverages the Groth16 Zero-Knowledge Proof schema to verify the validity of questionnaire results within Smart Contracts. Our approach ensures that the answer key of the questionnaire remains undisclosed throughout the verification process, while ensuring that the evaluation is done fairly. To accomplish this, users respond to a series of questions, and their answers are encoded and securely transmitted to a hidden backend. The backend then performs an evaluation of the user's answers, generating the overall result of the questionnaire. Additionally, it generates a Zero-Knowledge Proof, attesting that the answers were appropriately evaluated against a valid set of constraints. Next, the user submits their result along with the proof to a Smart Contract, which verifies their validity and issues a non-fungible token (NFT) as an attestation of the user's test result. In this research, we implemented the Zero-Knowledge functionality using Circom 2 and deployed the Smart Contract using Solidity, thereby showcasing a practical and secure solution for questionnaire validity verification in the context of Smart Contracts.
The communities of blockchains and distributed ledgers have been stirred up by the introduction of zero-knowledge proofs (ZKPs). Originally designed to solve privacy issues, ZKPs have now evolved into an effective remedy for scalability concerns and are applied in Zcash (internet money like Bitcoin). To enable ZKPs, Rank-1 Constraint Systems (R1CS) offer a verifier for bi-linear equations. To accurately and efficiently represent R1CS, several language tools like Circom, Noir, and Snarky have been proposed to automate the compilation of advanced programs into R1CS. However, due to the flexible nature of R1CS representation, there can be significant differences in the compiled R1CS forms generated from circuit language programs with the same underlying semantics. To address this issue, this paper uses a data-flow-based R1CS paradigm algorithm, which produces a standardized format for different R1CS instances with identical semantics. By using the normalized R1CS format circuits, the complexity of circuits' verification can be reduced. In addition, this paper presents an R1CS normalization algorithm benchmark, and our experimental evaluation demonstrates the effectiveness and correctness of our methods.
We develop a general multiple scattering expansion (MSE) for computing Casimir forces between magneto-dielectric bodies and Casimir-Polder forces between polarizable particles and magneto-dielectric bodies. The approach is based on fluctuating electric and magnetic surface currents and charges. The surface integral equations for these surface fields can be formulated in terms of surface scattering operators (SSO). We show that there exists an entire family of such operators. One particular member of this family is only weakly divergent and allows for a MSE that appears to be convergent for general magneto-dielectric bodies. We proof a number of properties of this operator, and demonstrate explicitly convergence for sufficiently low and high frequencies, and for perfect conductors. General expressions are derived for the Casimir interaction between macroscopic bodies and for the Casimir-Polder interaction between particles and macroscopic bodies in terms of the SSO, both at zero and finite temperatures. An advantage of our approach above previous scattering methods is that it does not require the knowledge of the scattering amplitude (T-operator) of the bodies. A number of simple examples are provided to demonstrate the use of the method. Some applications of our approach have appeared previously [T. Emig, G. Bimonte, Phys. Rev. Lett. 130, 200401 (2023)]. Here we provide additional technical aspects and details of our approach.
Abstract Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are the most efficient proof systems in terms of proof size and verification. Currently, Groth’s scheme from EUROCRYPT 2016, $$\textsf{Groth16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Groth</mml:mi> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> , is the state-of-the-art and is widely deployed in practice. $$\textsf{Groth16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Groth</mml:mi> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> is originally proven to achieve knowledge soundness, which does not guarantee the non-malleability of proofs. There has been considerable progress in presenting new zk-SNARKs or modifying $$\textsf{Groth16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Groth</mml:mi> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> to efficiently achieve strong Simulation extractability, which is shown to be a necessary requirement in some applications. In this paper, we revise the Random oracle based variant of $$\textsf{Groth16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Groth</mml:mi> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> proposed by Bowe and Gabizon, BG18, the most efficient one in terms of prover efficiency and CRS size among the candidates, and present a more efficient variant that saves 2 pairings in the verification and 1 group element in the proof. This supersedes our preliminary construction, presented in CANS 2020 (Baghery et al. in CANS 20, volume 12579 of LNCS, Springer, Heidelberg. pp 453-461, 2020), which saved 1 pairing in the verification, and was proven in the generic group model. Our new construction also improves on BG18 in that our proofs are in the algebraic group model with Random Oracles and reduces security to standard computational assumptions in bilinear groups (as opposed to using the full power of the generic group model (GGM)). We implement our proposed simulation extractable zk-SNARK (SE zk-SNARK) along with BG18 in the library, and compare the efficiency of our scheme with some related works. Our empirical experiences confirm that our SE zk-SNARK is more efficient than all previous simulation extractable (SE) schemes in most dimensions and it has very close efficiency to the original $$\textsf{Groth16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Groth</mml:mi> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> .
Traditional blockchains cannot achieve the same transaction throughput as Web2, so their use cases are limited. Therefore, state sharding has been proposed to improve transaction throughput by dividing the blockchain network and managing states and transactions in parallel. However, Nightshade in the NEAR Protocol, a type of state sharding, provides a rollback protocol to cancel the generation of blocks containing inconsistent transaction results because processing cross-shard transactions (CSTXs) in a 2-phase commit may cause state inconsistency. We present a new attack that interferes with the generation of new blocks by repeatedly executing CSTXs that certainly causes state inconsistency, causing continuous rollback. We also propose a block-generation protocol called Synchro to incorporate all the state changes of each CSTX into the same block by coordinating the block prior to approving transactions in each shard. Synchro eliminates the occurrence of the state inconsistency caused by the CSTXs and the necessity of the rollback protocol. We use zero-knowledge proof to make Synchro scalable in the global validation phase. Although the actual overhead of the zero-knowledge proof has not yet been evaluated, we show that Synchro could achieve the same transaction throughput as Nightshade theoretically, depending on the future innovations in zero-knowledge proof techniques.
Biegon Kipkoech Collins, Alex Kibet, Andrew M. Kahonge
Amidst rapid technological advancement and digital transformation, ensuring privacy and data security is paramount. Governments and digital service providers face the challenge of establishing secure identification systems that protect individuals' personal information while enabling reliable authentication and seamless user experiences. Traditional identification methods often require individuals to disclose sensitive personal information, leading to privacy risks and potential data breaches. Zero-knowledge proofs (ZKPs) have emerged as a promising solution to address these concerns. By leveraging ZKPs, individuals can authenticate their identities or assert specific attributes without revealing sensitive data. This approach holds great potential for preserving privacy while enabling efficient and trustworthy verification processes. This paper explored ZKPs and how governments and digital service providers can utilize this technology to achieve secure identification while upholding privacy. A key focus was prototyping a secure identification protocol using ZKPs. Through practical implementation, this research aimed to demonstrate the reliability and effectiveness of ZKPs in real-world scenarios. Keywords : zero-knowledge proofs, privacy, digital identity, governments, digital services. DOI: 10.7176/ISDE/13-2-06 Publication date: September 30 th 2023
In order to solve the problems of imperfect transaction legitimacy verification strategies for protecting account balances and transaction amounts in privacy protection of consortium blockchain, and the low efficiency of the basic encryption algorithm Paillier, a <sup>+</sup>HomElG zero-knowledge proof protocol for consortium blockchain transfer privacy protection was proposed. A consortium blockchain transfer privacy protection application was constructed based on PBFT, which expounded the consensus interaction scenario of zero-knowledge proof of homomorphic encryption. The transaction amount and balance of account were encrypted by the <sup>+</sup>HomElG algorithm, and the zero-knowledge proof of the ciphertext was designed with the Σ protocol. The non-interactive zero-knowledge was designed through the idea of the Fiat-Shamir algorithm processes such as the proof of equality, the amount of the transaction greater than zero and the balance of the transfer party not less than zero in the proof of range. The protocol was proved to be correct, complete and zero-knowledge under the DDH. A consortium blockchain transfer privacy protection prototype system based on Hyperledger Fabric was constructed. The results verified that the protocol can realize ciphertext transactions to protect balance of account and transaction amount under the condition of non-interactive zero-knowledge proof. When the key length is 3072 bit and the data length is a 12-bit decimal integer, the efficiency of the <sup>+</sup>HomElG algorithm is 150.3 ms, and the efficiency of the proof of equality, the amount of the transaction greater than zero and the balance of the transfer party not less than zero in the proof of range are 482.3 ms, 209.3 ms and 261.3 ms respectively. Compared with the existing protocols, the proposed <sup>+</sup>HomElG algorithm is more efficient, and its transaction legitimacy verification strategies such as equality proof and range proof are more perfect and efficient. The proposed protocol can meet the privacy protection requirements of consortium blockchain transfer transactions.
The effective management of electronic health records (EHRs) is vital in healthcare. However, traditional systems often need help handling data inconsistently, providing limited access, and coordinating poorly across facilities. This study aims to tackle these issues using blockchain technology to improve EHR systems' data security, privacy, and interoperability. By thoroughly analyzing blockchain's applications in healthcare, we propose an innovative solution that leverages blockchain's decentralized and immutable nature, combined with advanced encryption techniques such as the Advanced Encryption Standard and Zero Knowledge Proof Protocol, to fortify EHR systems. Our research demonstrates that blockchain can effectively overcome significant EHR challenges, including fragmented data and interoperability problems, by facilitating secure and transparent data exchange, leading to enhanced coordination, care quality, and cost-efficiency across healthcare facilities. This study offers practical guidelines for implementing blockchain technology in healthcare, emphasizing a balanced approach to interoperability, privacy, and security. It represents a significant advancement over traditional EHR systems, boosting security and affording patients greater control over their health records. Doi: 10.28991/HIJ-2023-04-03-012 Full Text: PDF