In 2010, the fourth decade of the HIV pandemic arrived during a time of unprecedented success in HIV prevention. Globally, UNAIDS estimated that new HIV infections fell by 33% between 2001 and 2011; new infections among adults and adolescents fell by 50% or more in 26 countries (more than half of these countries were in sub-Saharan Africa), and new infections among children worldwide dropped by 52% [1,2]. The declines in new HIV infections are particularly evident in countries with sustained and more strategic investments, which take into account the specifics of local epidemics, increased political leadership and community engagement in response to the HIV epidemic, and scale up of HIV prevention and treatment programmes [2]. The rapidly growing delivery of antiretrovirals to women and infant feeding–based prevention programmes has resulted in a sharp decline in new HIV diagnoses among children. The encouraging declines in HIV infections can also be attributed to the improved effectiveness of combination antiretroviral treatment (cART), an expanded range of improved medications, the declining prices that make cART more accessible to people in low-income countries, growing coverage with HIV testing, and improved access to prevention and treatment services (particularly for women and young people in low-income countries). Global investment in the AIDS response jumped from US$3.8 billion in 2002 to US$18.9 billion in 2012. The new decade also saw a revolution in HIV prevention with ground-breaking scientific advances in HIV biomedical prevention, and specifically, proof that microbicides containing an antiretroviral agent can reduce sexual transmission of HIV to women by 39% [3], that earlier start of treatment by HIV-positive people (treatment as prevention or TasP) can reduce the risk of onward transmission by as much as 96% [4], and that consistent, correct use of a daily antiretroviral tablet by men who have sex with men (MSM) can achieve substantial reductions in HIV infections (pre-exposure prophylaxis or PrEP) [5]. In response to the excitement and optimism surrounding the preventative effects of antiretroviral medications, the UN member states considered and unanimously approved the new Political Declaration on HIV/AIDS at a special session of the General Assembly in New York in 2011 [6]. At the core of the 2011 UN Political Declaration are ambitious new HIV prevention targets calling on governments to commit to reducing sexual transmission of HIV by 50%, reducing HIV transmission though injecting drug use by 50% and eliminating mother-to-child transmission of HIV by 2015. These targets are aimed at reinvigorating the commitment towards achieving the Millennium Development Goal #6 to combat HIV/AIDS [7]. In the past two years, research on HIV biomedical prevention has focused on adapting the new prevention strategies to the context of local HIV epidemics [8]. There have been equal measures of optimism and pessimism expressed about the ability of new prevention strategies to halt the HIV pandemic. Based on the evidence that starting treatment earlier can increase health benefits and extend life for people with HIV [9,10], many clinicians are already recommending early treatment for both medical and TasP purposes. Regarding PrEP, the widespread reaction is caution in recommending this strategy. Such reluctance is based on concerns about the common adherence issues in the studies of PrEP [11], the obvious relationship between level of adherence to daily medication schedule and its preventative effect [12] and side effects and drug-resistant HIV [13], among others. Despite little evidence that PrEP use can affect behaviour, many concerns have been voiced about the future of safe sex practices, particularly condom use among MSM, if new biomedical prevention strategies are introduced. More research is needed to investigate this issue using appropriate study designs. In its current form, daily PrEP may benefit only a small number of people with very high and ongoing risk for HIV infection, and other PrEP regimens must be explored. The overarching concern about new prevention strategies, particularly PrEP, is that the cost and burden of providing them are currently unacceptable for most, even high-income, countries. As a result, there have been calls for more evidence and a very slow progress in implementing these two new exciting HIV prevention developments. Regarding PrEP, only two countries to date have prescription guidelines for people at high risk of HIV infection [14,15]. We now have the knowledge and new tools to revolutionize HIV prevention, and we have the bold new Political Declaration with ambitious targets. It must be acknowledged that the task of bringing HIV infections down to zero seems daunting from where we stand now in late 2013. Despite the global success in lowering the HIV infection rates by 33% [1], sub-Saharan Africa has seen only a 25% decline [16]. Some regions have seen increases (8% in Eastern Europe and central Asia [17], 19% in East Asia [18] and 37% in the Middle East and North Africa [19]). This lack of progress has been associated with insufficient resources, inadequate coverage of women with antiretroviral treatment and HIV testing programmes not reaching the population groups at high risk for HIV infection. While sexual behaviour has changed to become safer in some countries and populations, sexual risk taking has increased in other settings. This is the case in most high-income countries in North America, western and central Europe and Australasia, where MSM are central to local HIV epidemics. In these countries, both high-risk sexual practices among MSM and HIV infections have been on the rise [20,21]. Condom use has increased in some countries, but declined in others. Proven effective interventions (e.g., prevention of mother-to-child transmission (PMTCT) and needle- and syringe-exchange programmes) have not achieved sufficiently high coverage in many countries [1]. Although trends in risky sexual practices have been linked to the trends in HIV incidence [22] and population-level behaviour change to the reduction in HIV prevalence, there are still challenges in linking behaviour-change programmes to specific HIV outcomes on the population level [2]. While new expectations have been raised about the role of antiretrovirals for HIV prevention, mixed progress was observed in access to cART, and only 61% of people eligible for treatment under the 2010 WHO guidelines received it (this is as little as 34% under the 2013 WHO guidelines). The Political Declaration has for the first time named and acknowledged the importance of such population groups as MSM, people who inject drugs and sex workers for HIV prevention, but in many settings, stigma and access to treatment and prevention services for these groups are still important challenges. Many low- and middle-income countries have stepped up their local investments in HIV prevention [2], but, regrettably, the lack of resources has remained a major issue: only US$18.9 billion was available from all sources for the AIDS response in 2012, and this was estimated to be 16–26% short of annual need [1]. It is at this time of some successes in HIV prevention and challenges in how to optimize the available resources and tools that the aspirational Political Declaration of commitment to fight the pandemic is necessary. This year's International AIDS Day marks the midpoint towards the deadline set by the Political Declaration in 2011. It is an opportunity for governments and each of us to revisit and reinvigorate the universal commitment to bring HIV infections to zero. Like never before, we have good cause to expect the next generation to be AIDS-free and new HIV infections to move towards zero. In the face of the 75 million people who have suffered from HIV/AIDS and the many more affected, the international community should keep the promise and bring this HIV pandemic to an end. None were declared. The Kirby Institute receives project funding from the Australian Government Department of Health and Ageing. The views expressed in this publication do not necessarily represent the position of the Australian Government. IZ has prepared the manuscript and approved its final version.
Cryptographic credentials allow user authorizations to be granted and verified. and have such applications as e-Passports, e-Commerce, and electronic cash. This thesis proposes a privacy protecting approach of binding biometrically derived keys to cryptographic credentials to prevent unauthorized lending. Our approach builds on the 2011 work of Adams, offering additional benefits of privacy protection of biometric information, generality on biometric modalities, and performance. Our protocol integrates into Brands’ Digital Credential scheme, and the Anonymous Credentials scheme of Camenisch and Lysyanskaya. We describe a detailed integration with the Digital Credential Scheme and sketch the integration into the Anonymous Credentials scheme. Security proofs for non-transferability, correctness of ownership, and unlinkability are provided for the protocol’s instantiation into Digital Credentials. Our approach uses specialized biometric devices in both the issue and show protocols. These devices are configured with our proposed primitive, the fuzzy ex-tractor indistinguishability adaptor which uses a traditional fuzzy extractor to create and regenerate cryptographic keys from biometric data and IND-CCA2 secure en-cryption protect the generated public data against multiplicity attacks. Pedersen commitments are used to hold the key at issue and show time, and A zero-knowledge proof of knowledge is used to ensure correspondence of key created at issue-time and regenerated at show-time. The above is done in a manner which preserves biometric privacy, as and delivers non-transferability of digital credentials. The biometric itself is not stored or divulged to any of the parties involved in the protocol. Privacy protection in multiple enrollments scenarios is achieved by the fuzzy extractor indistinguishability adapter. The zero knowledge proof of knowledge is used in the showing protocol to prove knowledge of values without divulging them.
For various services, OTP (One-Time Password) is increasingly employed to sign in services. The most popular OTP scheme is S/KEY. However, S/KEY scheme is vulnerable to hash collision. In order to solve this problem, Choi and Kim proposed an OTP scheme based on pairing operation. This scheme overcame hash collision problem. Unfortunately, the scheme is vulnerable against insider attack. In this paper, we propose insider attack-resistant OTP scheme based on pairing operation. Our scheme employed zero-knowledge proof to generate OTP values. In short, our scheme sends an OTP and two random numbers encoded with XOR; any adversary has no sense about the random numbers. Our scheme authenticates entities with an OTP and gain a hint on next OTP value from one of random numbers. Consequently, we present insider attack-resistant OTP scheme by eliminating shared parameter among insider (e.g. time stamp).
Abstract. Consider a joint distribution (X,A) on a set X ×{0, 1}ℓ. We show that for any family F of distinguishers f: X × {0, 1}ℓ → {0, 1}, there exists a simulator h: X → {0, 1}ℓ such that 1. no function in F can distinguish (X,A) from (X,h(X)) with advantage ǫ, 2. h is only O(23ℓǫ−2) times less efficient than the functions in F. For the most interesting settings of the parameters (in particular, the cryptographic case where X has superlogarithmic min-entropy, ǫ> 0 is negligible and F consists of circuits of polynomial size), we can make the simulator h deterministic. As an illustrative application of this theorem, we give a new security proof for the leakage-resilient stream-cipher from Eurocrypt’09. Our proof is simpler and quantitatively much better than the original proof using the dense model theorem, giving meaningful security guarantees if instantiated with a standard blockcipher like AES. Subsequent to this work, Chung, Lui and Pass gave an interactive variant of our main theorem, and used it to investigate weak notions of Zero-Knowledge. Vadhan and Zheng give a more constructive version of our theorem using their new uniform min-max theorem. 1
We consider a class of two-party function evaluation protocols in which the parties are allowed to<br/>use ideal functionalities as well as a set of powerful primitives, namely commitments, homomorphic<br/>encryption, and certain zero-knowledge proofs. We illustrate that with these it is possible to capture<br/>protocols for oblivious transfer, coin- ipping, and generation of multiplication-triple.<br/>We show how any protocol in our class can be compiled to a symbolic representation expressed<br/>as a process in an abstract process calculus, and prove a general computational soundness theorem<br/>implying that if the protocol realises a given ideal functionality in the symbolic setting, then the<br/>original version also realises the ideal functionality in the standard computational UC setting. In<br/>other words, the theorem allows us to transfer a proof in the abstract symbolic setting to a proof in<br/>the standard UC model.<br/>Finally, we show that the symbolic interpretation is simple enough in a number of cases for the<br/>symbolic proof to be partly automated using ProVerif
In the paper, we present a kind of quantum voting protocol, which is based on controlled quantum teleportation of local unitary operations in non-symmetric quantum channel. In this protocol, the umpire CA with zero knowledge proof quantum identity authentication ensures voter’s anonymous identity authentication. The counting institution Bob generates a high-dimensional Greenberger-Horne-Zeilinger entangled state to establish a high-dimensional quantum communication channel. Performing the local unitary operation on their low-dimensional quantum ballot, voter’s quantum vote is teleportated by asymmetric matrix measurement and scrutineer Charlie auxiliary measuring. With the scrutineer Charlie’s help, Bob achieves the voting result by the output of unitary operation. Compared with other general quantum operation teleportation quantum voting protocol, the protocol utilizes the quantum information and transmission of quantum channel, which have different dimensions, so single particle information cannot be stolen, and can prevent forgery. The electoral process is fair and undeniable, owing to Charlie’s supervision. Since the success probability of quantum teleportation of local unitary operations is 1, the quantum voting is reliable.
Henry Corrigan-Gibbs, Wendy Mu, Dan Boneh, Bryan Ford
The security of any cryptosystem relies on the secrecy of the system's secret keys. Yet, recent experimental work demonstrates that tens of thousands of devices on the Internet use RSA and DSA secrets drawn from a small pool of candidate values. As a result, an adversary can derive the device's secret keys without breaking the underlying cryptosystem. We introduce a new threat model, under which there is a systemic solution to such randomness flaws. In our model, when a device generates a cryptographic key, it incorporates some random values from an "entropy authority" into its cryptographic secrets and then proves to the authority, using zero-knowledge-proof techniques, that it performed this operation correctly. By presenting an entropy-authority-signed public key certificate to a third party (like a certificate authority or SSH client), the device can demonstrate that its public key incorporates randomness from the authority and is therefore drawn from a large pool of candidate values. Where possible, our protocol protects against eavesdroppers, entropy authority misbehavior, and devices attempting to discredit the entropy authority. To demonstrate the practicality of our protocol, we have implemented and evaluated its performance on a commodity wireless home router. When running on a home router, our protocol incurs a $1.7\times$ slowdown over conventional RSA key generation and it incurs a $3.6\times$ slowdown over conventional EC-DSA key generation.
The notion of zero-knowledge [GMR85] is formalized by requiring that for every malicious efficient verifier V ∗ , there exists an efficient simulator S that can reconstruct the view of V ∗ in a true interaction with the prover, in a way that is indistinguishable to every polynomialtime distinguisher. Weak zero-knowledge weakens this notions by switching the order of the quantifiers and only requires that for every distinguisher D, there exists a (potentially different) simulator SD. In this paper we consider various notions of zero-knowledge, and investigate whether their weak variants are equivalent to their strong variants. Although we show (under complexity assumption) that for the standard notion of zero-knowledge, its weak and strong counterparts are not equivalent, for meaningful variants of the standard notion, the weak and strong counterparts are indeed equivalent. Towards showing these equivalences, we introduce new non-black-box simulation techniques permitting us, for instance, to demonstrate that the classical 2-round graph non-isomorphism protocol of Goldreich-Micali-Wigderson [GMW91] satisfies a “distributional” variant of zero-knowledge. Our equivalence theorem has other applications beyond the notion of zero-knowledge. For instance, it directly implies the dense model theorem of Reingold et al (STOC ’08), and the leakage lemma of Gentry-Wichs (STOC ’11), and provides a modular and arguably simpler proof of these results (while at the same time recasting these result in the language of zeroknowledge). 0 1
Bryan Parno, Jon Howell, Craig Gentry, Mariana Raykova
To instill greater confidence in computations outsourced to the cloud, clients should be able to verify the correctness of the results returned. To this end, we introduce Pinocchio, a built system for efficiently verifying general computations while relying only on cryptographic assumptions. With Pinocchio, the client creates a public evaluation key to describe her computation; this setup is proportional to evaluating the computation once. The worker then evaluates the computation on a particular input and uses the evaluation key to produce a proof of correctness. The proof is only 288 bytes, regardless of the computation performed or the size of the inputs and outputs. Anyone can use a public verification key to check the proof. Crucially, our evaluation on seven applications demonstrates that Pinocchio is efficient in practice too. Pinocchio’s verification time is typically 10ms: 5-7 orders of magnitude less than previous work; indeed Pinocchio is the first general-purpose system to demonstrate verification cheaper than native execution (for some apps). Pinocchio also reduces the worker’s proof effort by an additional 19-60×. As an additional feature, Pinocchio generalizes to zero-knowledge proofs at a negligible cost over the base protocol. Finally, to aid development, Pinocchio provides an end-to-end toolchain that compiles a subset of C into programs that implement the verifiable computation protocol. 1
Authentication is a process by which an entity, which could be a person or intended computer, establishes its identity to another entity. In private and public computer networks including the Internet, authentication is commonly done through the use of logon passwords. Knowledge of the password is assumed to guarantee that the user is authentic. Internet business and many other transactions require a more stringent authentication process. The aim of this paper is to propose two authentication schemes based on general non-commutative rings. The key idea of the schemes is that for a given non-commutative ring; one can build polynomials on additive structure and takes them as underlying work structure. By doing so, one can implement authentication schemes, one of them being zero-knowledge interactive proofs of knowledge, on multiplicative structure of the ring. The security of the schemes is based on the intractability of the polynomial symmetrical decomposition problem over the given non-commutative ring.
Xiao Qiang Guo, Li Hong Li, Cui Ling Luo, Yi Shuo Shi
The Bit Commitment (BC) is an important basic agreement in cryptography . The concept was first proposed by the winner of the Turing Award in 1995 ManuelBlum. Bit commitment scheme can be used to build up zero knowledge proof, verified secret sharing, throwing coins etc agreement.Simultaneously and Oblivious Transfer together constitute the basis of secure multi-party computations. Both of them are hotspots in the field of information security. We investigated unconditional secure Quantum Bit Commitment (QBC) existence. And we constructed a new bit commitment model – double prover bit commitment. The Quantum Bit Commitment Protocol can be resistant to errors caused by noise.
Man Ho Au, Willy Susilo, Yi Mu, Sherman S. M. Chow
Dynamick-times anonymous authentication (k-TAA) schemes allow members of a group to be authenticated anonymously by application providers for a bounded number of times, where application providers can independently and dynamically grant or revoke access right to members in their own group. In this paper, we construct a dynamick-TAA scheme with space and time complexities ofO(log(k)) and a variant, in which the authentication protocol only requires constant time and space complexities at the cost ofO(k) -sized public key. We also describe some tradeoff issues between different system characteristics. We detail all the zero-knowledge proof-of-knowledge protocols involved and show that our construction is secure in the random oracle model under theq-strong Diffie-Hellman assumption andq-decisional Diffie-Hellman inversion assumption. We provide a proof-of-concept implementation, experiment on its performance, and show that our scheme is practical.
Auctions have a long history, having been recorded as early as 500 B.C. [Auction Theory, Academic Press, San Diego, USA, 2002]. Nowadays, electronic auctions have been a great success and are increasingly used in various applications, including high performance computing [Concurrency and Computatio n: Practice and Experience 14(13–15) (2002), 1507–1542]. Many cryptographic protocols have been proposed to address the various security requirements of these electronic transactions, in particular to ensure privacy. Brandt [International Journal of Information Security 5 (2006), 201–216] developed a protocol that computes the winner using homomorphic operations on a distributed ElGamal encryption of the bids. He claimed that it ensures full privacy of the bidders, i.e. no information apart from the winner and the winning price is leaked. We first show that this protocol – when using malleable interactive zero-knowledge proofs – is vulnerable to attacks by dishonest bidders. Such bidders can manipulate the publicly available data in a way that allows the seller to deduce all participants’ bids. We provide an efficient parallelized implementation of the protocol and the attack to show its practicality. Additionally we discuss some issues with verifiability as well as attacks on non-repudiation, fairness and the privacy of individual bidders exploiting authentication problems.
José Bacelar Almeida, Manuel Barbosa, Endre Bangerter, Gilles Barthe · 6 authors
Developers building cryptography into security-sensitive applications face a daunting task. Not only must they understand the security guarantees delivered by the constructions they choose, they must also implement and combine them correctly and efficiently. Cryptographic compilers free developers from this task by turning high-level specifications of security goals into efficient implementations. Yet, trusting such tools is hard as they rely on complex mathematical machinery and claim security properties that are subtle and difficult to verify. In this paper we present ZKCrypt, an optimizing cryptographic compiler achieving an unprecedented level of assurance without sacrificing practicality for a comprehensive class of cryptographic protocols, known as Zero-Knowledge Proofs of Knowledge. The pipeline of ZKCrypt integrates purpose-built verified compilers and verifying compilers producing formal proofs in the CertiCrypt framework. By combining the guarantees delivered by each stage, ZKCrypt provides assurance that the output implementation securely realizes the abstract proof goal given as input. We report on the main characteristics of ZKCrypt, highlight new definitions and concepts at its foundations, and illustrate its applicability through a representative example of an anonymous credential system
We present a constant-round concurrent zero-knowledge protocol for NP. Our protocol is sound against uniform polynomial-time attackers, and relies on the existence of families of collision-resistant hash functions, and a new (but in our eyes, natural) falsifiable intractability assumption: Roughly speaking, that Micali’s non-interactive CS-proofs are sound for languages in P.
A short, 30cm, test section was used to study the effect of electrohydrodynamic (EHD) forces on flow redistribution in a horizontal, shell and tube heat exchanger subject to both boiling and condensation. The use of a short test section allows for a consistent flow pattern across the test section length which provides further insight into the true effect of EHD. It was found that the voltage polarity of the applied voltages influences the flow distribution. For the current geometry studied, it was found that positive polarity voltages tend to pull liquid away from heat transfer surface and that negative voltages tended to repel more liquid toward the heat transfer surface. Using this knowledge we were able to show that positive voltages were more effective for convective condensation heat transfer enhancement, whereas negative voltages were more effective for convective boiling heat transfer enhancement. A twofold enhancement of convective boiling heat transfer was achieved for positive voltages and a 4fold enhancement was achieved for negative voltages. Similar pressure drop penalties were seen for both cases, approximately twice that of the no EHD case. Furthermore, the effect of DC level, peak to peak voltage, frequency and duty cycle waveform parameters on convective boiling enhancement were studied to explore the range of controllability for the current set of flow parameters. It was found that these various waveform parameters can induce different flow patterns and consequently different heat transfer and pressure drop configurations. In general the heat transfer is enhanced by EHD, but different pressure drop penalties can be achieved for a given enhancement ratio using different waveforms. High heat transfer for relatively low pressure drop was achieved using either negative DC signals or 50%duty cycle pulse waveforms. In some cases the enhancement is quite little compared to the pressure drop, for example the zero DC level, varying peak to peak voltage data. It is suggested that in a system where the heat exchanger pressure drop due to EHD is more dominant than the system pressure drop, it may be possible to use EHD as a method of retarding the system rather than enhancing it thereby broadening the scope of controllability. Finally we showed the proof of concept of using DC EHD as a rapid control mechanism for the load conditions. Using -8kVDC the water side heat flux could be varied by approximately ±3.2 kW/m<sup>2</sup> within 5 seconds. As a comparison, the same experiment was repeated using the refrigerant flow rate to control the load. Response times were similar for both experiments and although the power required for the flow rate control was less, the minimal variability in flow parameters for the EHD control make it a more attractive method of load control.
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Membrane-based Ion Separation Techniques
Currency Recognition and Detection
Innovative Microfluidic and Catalytic Techniques Innovation
Ever since the famous Eddington-Dyson solar eclipse expedition in 1919, it has been known that massive bodies cause space (or rather space-time) to curve. This happens not only locally, in the vicinity of celestial bodies, but also on the largest possible global scale. Einstein's first cosmological model of 1917 represented the finite universe by the kind of 3D spherical space that had been familiar to mathematicians for more than half a century. According to Einstein, the constant curvature K and radius of curvature R were given by the average density ρ of matter in the universe by where G is Newton's gravitational constant. Although Einstein's model only survived to about 1930, curved space remained an element in most later cosmological models. The question to be decided by a combination of theory and observation was the size of the cosmic curvature, as expressed by the curvature constant k = R2K. In the Einstein universe, k = +1. The present consensus view, in part based on the inflationary scenario, is that we live in a flat or Euclidean space, corresponding to k = 0, which implies that the universe is infinite in extent. However, this is a view that can never be proved observationally, not even in principle. Whereas the reality of curved space belongs to the 20th century, as a mathematical hypothesis it was discussed many decades before Einstein. The first scientist who not only realized the possibility of a closed universe, but advocated it as a model of the real universe, is little known today. Few cosmologists have ever heard about the German astrophysicist Karl Friedrich Zöllner, who as early as 1872 argued that the universe is finite, in the sense that cosmic space is positively curved (Jaki 1969, Kragh 2012). Zöllner's remarkable cosmology based on non-Euclidean geometry deserves more than just a footnote in the annals of cosmological thought. Naturally, questions about the curvature of space could only be asked after the recognition, in the first half of the 19th century, that geometries other than Euclid's are possible. As early as about 1815, Karl Friedrich Gauss in Göttingen came to the conclusion that Euclidean geometry is not true by necessity but can be justified only empirically. According to an often repeated myth - but it is a myth - he attempted to test the validity of Euclidean geometry by measuring geodetically the sum of angles in a triangle extending between three mountain peaks in the state of Hanover (Breitenberger 1984). While Gauss anticipated non-Euclidean geometry, it was left to the Hungarian mathematician János Bolyai and, independently, his Russian colleague Nikolai Ivanovich Lobachevsky to establish geometrical systems different from the venerable one of Euclid. Of the two pioneers, Lobachevsky was the more empirically oriented. As he said in a paper of 1835, the truth of geometry “can only be verified, like all other laws of Nature, by experiment, such as astronomical observations” (Lobachevsky 1898). K F Zöllner, steel engraving from 1882. What Lobachevsky called “imaginary geometry” soon became known as hyperbolic geometry, characterized by a curvature constant k = −1 (and therefore an imaginary radius of curvature). Not only did he prove that in this kind of space the angle sum in a triangle always exceeds 180°, he also suggested that the geometry of physical space might be tested by considering stellar parallaxes. For example, while in Euclidean space the parallax of a star tends toward zero as its distance increases toward infinity, Lobachevsky showed that in hyperbolic space there is a minimum parallax for all stars irrespective of how far they are from the Earth. In his first paper on the new geometry, dating from 1829, he used a value of 1″.24 for the parallax of Sirius - three times as great as the real one - to conclude that space was flat to an approximation much closer than the error of measurement. Nonetheless, rather than concluding that space was Euclidean, he considered his calculations to be inconclusive. Perhaps, he speculated, a deviation from flat space would turn up in future measurements of much larger heavenly triangles. In a famous lecture of 1854, the young Göttingen mathematician Bernard Riemann completed and generalized the earlier ideas of Gauss, Lobachevsky and Bolyai. Emphasizing that curvature is an intrinsic property of space, he argued that although there is any number of possible geometries, there are only three that can represent physical space. These spaces of constant curvature correspond to the three values of the curvature constant, k = 0, ±1. Riemann paid particular attention to the case of a closed spherical space, pointing out that in such a space “we must distinguish between unboundedness and infinite extent.” A space of constant positive curvature “must necessarily be finite provided this curvature has ever so small a positive value” (Riemann 1873). A physicist as well as a mathematician, he speculated that the metrical structure of space on a microscopic scale might be of importance for the physics of atoms and molecules. On the other hand, he did not take an interest in the space of the astronomers. Questions about the global properties of space he dismissed as “idle questions”. Non-Euclidean geometry circulated slowly in the mathematical community, and even more slowly among physicists and astronomers. Only in the 1870s, in large measure due to popular lectures by Hermann von Helmholtz and William Clifford, did Riemann's ideas become generally known and seen as a vision of a possible geometrization of physics. Johann Karl Friedrich Zöllner (1834–1882) is today recognized for his contributions to astrophysics and, in particular, his pioneering work in astrophotometry (Koerber 1899, Hermann 1982). A skilled experimentalist and designer of instruments, in 1858 Zöllner invented an astrophotometer to measure the feeble light from stars and planets. In 1862 he moved to Leipzig, where he was appointed professor and established an astrophysical research programme, the first of its kind. In addition to his experimental work, he also made important studies of theoretical problems in astronomy and physics. These included electrodynamics, solar theory, sunspots and the theory of comets. In his Natur der Cometen from 1872 (figure 2) he developed an electrical theory of comets that for a period was widely admired. Title page of Zöllner's 1872 book on the nature of comets, including his proposal of a closed-space universe. Zöllner was a tireless advocate of Heinrich Weber's theory of electrodynamics based on a fundamental force law acting between hypothetical charged particles. Not only did Zöllner accept Weber's force law and associated atomistic theory, he also argued that it was of universal significance and valid for all terrestrial and cosmic phenomena. He suggested that it could be translated into a law of gravitation superior to Newton's, in the sense that the latter was merely a special case of Weber's. In Zöllner's extended version of Weber's theory, the interaction between two charged particles of opposite sign differed slightly, by a factor of 1.7 × 10−40, from the interaction between two particles of the same sign. Thus, a very small residual force would remain between two bodies, and this residual electric force he identified with the gravitational attraction (Zöllner 1882). In effect, he recognized the later so famous (and still unexplained) ratio between the gravitational and the electromagnetic interaction, given by the pure number Fgrav/Fem ≅ 10−40. Among other things, he used his electro-gravitational theory in an attempt to explain the anomalous motion of Mercury's perihelion, one of the major problems in astronomy until it was finally solved by Einstein. Natur der Cometen (Zöllner 1872) was a remarkable work in more than one sense. The major part of the 600-page book was not about comets, but instead a strange mixture of philosophy of science and unconstrained, chauvinistic charges of plagiarism. Zöllner's main targets were British scientists, including luminaries such as William Thomson and Charles Darwin, but he also attacked Helmholtz, one of the most powerful men in German science. The book aroused a storm of controversy and had the effect that Zöllner became increasingly marginalized as a scientist. Although much of the last decade of Zöllner's troubled life was occupied with philosophical speculations, spiritualism and endless controversies, he continued doing scientific work. Thus, he developed a theory of the origin of the Earth's magnetism according to which the magnetism was due to electrical currents in the fluid core of the Earth. Natur der Cometen included a chapter on “The Finitude of Matter in Infinite Space” in which Zöllner offered an original solution to Olbers' paradox in terms of a universe of constant positive curvature (Jaki 1969). In his systematic discussion of the finite versus the infinite in the universe, he assumed, for the sake of discussion, that there is only a finite amount of matter in the world. He then argued that in an unbounded (and therefore infinite) Euclidean space any finite amount of matter would evaporate and dissolve to zero density in an infinity of time. Given the actual existence of matter of non-zero density, he concluded that either is space finite or the universe has only existed for a limited period of time. Unwilling to accept the latter hypothesis, he suggested that Riemann's geometry might provide the key that would unravel the secrets of the universe and dissolve the problems of a materially finite universe: “It seems to me that any contradictions will disappear … if we ascribe to the constant curvature of space not the value zero but a positive value, however small … The assumption of a positive value of the spatial curvature measure involves us in no way in contradictions with the phenomena of the experienced world if only its value is taken to be sufficiently small.” In this way he made Olbers' paradox disappear without having to assume a limitation of either cosmic time or space. While he noted with satisfaction that energy conservation would apply to his finite material universe, he did not address the problem caused by the increase of entropy in a spatially finite but temporally infinite universe. Clearly inspired by Riemann, and happy to admit the inspiration, Zöllner further speculated that curved space was dynamically active, in the sense of determining the laws of Nature. Not even the divine force law of Weber was true a priori but somehow of cosmological origin, a speculation that bears some similarity to the later Mach's principle. And Zöllner went further than Riemann: whereas the Göttingen mathematician had declared that physics represented the “domain of another science”, the Leipzig astrophysicist maintained that the science of the physical world belonged entirely to the field of Riemann's investigations. Later in the century a few mathematicians attacked the problem of Mercury's anomalous precession by assuming space to be non-Euclidean. In 1885–1886 Wilhelm Killing and Carl Neumann derived orbits for Mercury moving in spherical space, and in 1902 Otto Liebmann did the same in the case of hyperbolic space. Zöllner's innovative cosmological speculations attracted some attention in German philosophical circles, but were ignored by most physicists and astronomers. Not only was cosmology considered a somewhat disreputable field that scarcely belonged to science, the idea of a closed space was also widely associated with the (even more disreputable) notion of a fourth space dimension. To understand the lack of scientific response to Zöllner's universe, one must take into account his controversial ideas of a fourth dimension as the site of spiritual phenomena (Zöllner 1880). In 1877, after meeting the chemist William Crookes in London, Zöllner turned wholeheartedly to spiritualism (Treitel 2004). Convinced of the reality behind spiritualist manifestations, he investigated them in great detail, attempting to integrate the spirits with both Weberian physics and his own highly unorthodox version of Christian theology. The first major result of his efforts in this area of unconventional research was an elaborate Transcendental Physics published in 1878 and translated into English two years later (Zöllner 1880). As Zöllner saw it, the project of a transcendental physics including both material and spiritual phenomena was but a natural extension of the astrophysical project of accommodating terrestrial and celestial phenomena within the same theoretical framework. It was a strictly scientific project. Not satisfied with simply accepting the spirits of deceased persons, as they appeared in séances, Zöllner argued that they were visitors from a hidden fourth dimension of space. During the last decades of the 19th century, beliefs of this kind were widespread; Zöllner only took them more seriously than most. It was sometimes contended that if our space is curved, it must be contained in a flat space of a higher dimension, in the same way that a 2D space is embedded in our 3D space. Although 4D “hyperspace” was often mixed up with ideas of non-Euclidean geometry, in reality there is no connection between them. William Clifford dismissed the connection as groundless, as did other mathematicians. A curved space does not need to be curved “in” another space. Zöllner's belief in a spiritual fourth dimension received inspiration from his knowledge of non-Euclidean geometry, which he sometimes used for purposes of illustration, but it did not depend on it. Nor did his claim of a fourth dimension rely exclusively on his belief in a spiritual world, for he held the claim even before his conversion to spiritualism. In a book of 1876 he argued that a fourth dimension was needed for epistemological reasons, in order to understand the symmetry between 3D objects, such as left- and right-handed gloves. The phenomenal objects in our 3D world must be “projections of objects in a space of four dimensions” (Zöllner 1876). He considered the insight to be of revolutionary importance to science as it heralded a change in the world view on a scale comparable to the one Copernicus had initiated. His colleagues in physics and astronomy were not immune to the fascination of the fourth dimension, but they rejected his interpretation of it. Zöllner was the only scientist in the 19th century who found it probable, and not merely possible, that space is curved in accordance with Riemann's geometry. He was also the only one to use the hypothesis to solve a cosmological problem, namely Olbers' paradox of the dark night sky. From the late 1870s, non-Euclidean geometry attracted increasing interest among mathematicians and philosophers and a few astronomers followed suit. One of them was the Irishman Robert Stawell Ball, Royal Astronomer of Ireland and from 1892 professor of astronomy and geometry in Cambridge. Without committing himself, he suggested that parallax investigations might show space to be non-Euclidean. Characteristically, his guarded preference for a closed cosmic space turned up in his popular publications only. In The High Heavens of 1893, he expressed sympathy with the hypothesis, vaguely suggesting that a finite universe was more satisfactory than the consensus view of an infinite space filled with stars. Another astronomer of distinction, the American Simon Newcomb, also dealt with the possibility of a closed-space universe, if only cautiously and apparently without believing in it. In the first edition of his classical text Popular Astronomy, he discussed whether the heat radiated by the Sun and stars would be lost forever. Noting that this would not be the case in a spherical universe, he nonetheless denied taking a Riemannian cosmic space seriously. It was “too speculative to admit of discussion” he said (Newcomb 1878). He followed up on the subject in correspondence with the philosopher-scientist Charles S Peirce, who was much more sympathetic to curved space. Indeed, for a decade Peirce defended the idea enthusiastically, suggesting various astronomical methods by means of which the curvature might be measured. Newcomb advised him to calm down: “The task of getting the scientific world to accept any proof that space is not homoloidal [flat] is hopeless, and you could have no other satisfaction than that of doing a work for posterity” (Eisele 1957). The most elaborate pre-relativistic attempt to link astronomy with non-Euclidean geometry appeared in 1900, in a paper by the 26-year-old German astrophysicist Karl Schwarzschild (published in translation in 1998). I cannot go into the substance of this work, except noting its main results concerning the possible curvature of space. In the case of a hyperbolic space, Schwarzschild found R > 4×106 AU, and for the closed space he estimated a lower bound of R > 108 AU. Although he saw no way to go beyond this rather indefinite conclusion, from a philosophically point of view he preferred a closed universe, which he thought was more “satisfying to reason” (Schwarzschild 1998). So did Einstein, 17 years later. A knot experiment Zöllner made with the American medium Henry Slade. The ends of the cord were sealed together, yet Slade's “spirits” tied several knots in the cord. To Zöllner (1880), it proved the reality of a fourth space dimension. Following up on Schwarzschild's analysis, Paul Harzer at the University of Kiel argued that the universe might well consist of a finite stellar system located in a larger spherical space. He estimated the size of the entire universe by the time it would take a ray of light to circumnavigate it. For this journey round the world, Harzer (1908) gave the figure 8700 years. Neither Schwarzschild nor Harzer seems to have been aware of Zöllner's earlier work, at the time long forgotten. Ever since Lobachevsky, non-Euclidean geometry was associated with astronomy and yet it was a subject most astronomers were to were for one of them that space was not considered part of science. The motion of celestial bodies was the of not the space in which the motion took Newcomb for the of astronomers he among both and to of space as an in To interest in the astronomical community, of space would have to be or for problems of astronomical on both While astronomers realized that the curvature of space was they also realized that the kind of bound for the curvature that measurements was to distinguish curved from flat space. Given this no that they saw no to the Euclidean space that had them so well in the space be curved, the curvature radius would be so large that for all purposes it was infinite - that space could be considered So Among the few problems of cosmological that might have astronomers to curved space was the question of whether space is finite or infinite in extent. The question might be seen as merely as it often but it had such as Olbers' only in one Zöllner's discussion of was the problem by that the stellar universe might be closed in accordance with Riemann's His solution to the most of Olbers' in terms of and saw no between the dark night and an infinity of stars. The main for the to the of space non-Euclidean was just they had no need for the
In this paper, through introducing the Williams public-key cryptosystem in detail, the analysis of the characteristics of the system, and the combination with zero knowledge proof, we set up a zero-knowledge proof scheme based on Williams public-key cryptosystem. The scheme will enrich the theory of cryptography, and particularly zero-knowledge proof theory.