60 research outputs found

    Security of Polynomial Transformations of the Diffie--Hellman Key

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    D. Boneh and R. Venkatesan have recently proposed an approachto proving that a reasonably small portions of most significant bits of the Diffie-Hellman key modulo a prime are as secure the the whole key. Some further improvements and generalizations have been obtained by I. M. Gonzales Vasco and I. E. Shparlinski. E. R. Verheul has obtained certain analogies of these results in the case of Diffie--Hellman keys in extensions of finite fields, when an oracle is given to compute a certain polynomial function of the key, for example, the trace in the background field. Here we obtain some new results in this direction concerning the case of so-called unreliable oracles

    Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations

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    Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any finite abelian group. Their algorithm stimulated a lot of interest in the cryptography community, especially in the context of `bit security'. This manuscript attempts to be a friendly and comprehensive guide to the tools and results in this field. The intended readership is cryptographers who have heard about these tools and seek an understanding of their mechanics and their usefulness and limitations. A compact overview of the algorithm is presented with emphasis on the ideas behind it. We show how these ideas can be extended to a `modulus-switching' variant of the algorithm. We survey some applications of this algorithm, and explain that several results should be taken in the right context. In particular, we point out that some of the most important bit security problems are still open. Our original contributions include: a discussion of the limitations on the usefulness of these tools; an answer to an open question about the modular inversion hidden number problem

    Bit Security of the Hyperelliptic Curves Diffie-Hellman Problem

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    The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent cryptanalytic results that the best known algorithms to attack hyperelliptic curve cryptosystems of genus g<3g<3 are the generic methods and the recent implementation results that hyperelliptic curve cryptography in genus 2 has the potential to be competitive with its elliptic curve cryptography counterpart. In this paper, we generalize Boneh and Shparlinksi\u27s method and result about elliptic curve to the case of Jacobians of hyperelliptic curves. We prove that the least significant bit of each coordinate of hyperelliptic curves Diffie-Hellman secret value in genus 2 is hard as the entire Diffie-Hellman value, and then we also show that any bit is hard as the entire Diffie-Hellman value. Finally, we extend our techniques and results to hyperelliptic curves of any genus

    On the hardness of approximating the permanent of structured matrices

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    We show that for several natural classes of "structured" matrices, including symmetric, circulant, Hankel and Toeplitz matrices, approximating the permanent modulo a prime p is as hard as computing its exact value. Results of this kind are well known for arbitrary matrices. However the techniques used do not seem to apply to "structured" matrices. Our approach is based on recent advances in the hidden number problem introduced by Boneh and Venkatesan in 1996 combined with some bounds of exponential sums motivated by the Waring problem in finite fields

    Hardness of Computing Individual Bits for One-way Functions on Elliptic Curves

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    We prove that if one can predict any of the bits of the input to an elliptic curve based one-way function over a finite field, then we can invert the function. In particular, our result implies that if one can predict any of the bits of the input to a classical pairing-based one-way function with non-negligible advantage over a random guess then one can efficiently invert this function and thus, solve the Fixed Argument Pairing Inversion problem (FAPI-1/FAPI-2). The latter has implications on the security of various pairing-based schemes such as the identity-based encryption scheme of Boneh–Franklin, Hess’ identity-based signature scheme, as well as Joux’s three-party one-round key agreement protocol. Moreover, if one can solve FAPI-1 and FAPI-2 in polynomial time then one can solve the Computational Diffie--Hellman problem (CDH) in polynomial time. Our result implies that all the bits of the functions defined above are hard-to-compute assuming these functions are one-way. The argument is based on a list-decoding technique via discrete Fourier transforms due to Akavia--Goldwasser–Safra as well as an idea due to Boneh–Shparlinski

    A contextuality witness inspired by optimal state discrimination

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    Many protocols and tasks in quantum information science rely inherently on the fundamental notion of contextuality to provide advantages over their classical counterparts, and contextuality represents one of the main differences between quantum and classical physics. In this work we present a witness for preparation contextuality inspired by optimal two-state discrimination. The main idea is based on finding the accessible averaged success and error probabilities in both classical and quantum models. We can then construct a noncontextuality inequality and associated witness which we find to be robust against depolarising noise and loss in the form of inconclusive events.Comment: 5 pages main text, 3 figures, 3 pages supplemental materia

    Semidefinite programming relaxations for quantum correlations

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    Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite relaxations can be adapted for a variety of research topics in quantum correlations, including nonlocality, quantum communication, quantum networks, entanglement, and quantum cryptography.Comment: To be submitted to Reviews of Modern Physic

    Cryptographic Pairings: Efficiency and DLP security

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    This thesis studies two important aspects of the use of pairings in cryptography, efficient algorithms and security. Pairings are very useful tools in cryptography, originally used for the cryptanalysis of elliptic curve cryptography, they are now used in key exchange protocols, signature schemes and Identity-based cryptography. This thesis comprises of two parts: Security and Efficient Algorithms. In Part I: Security, the security of pairing-based protocols is considered, with a thorough examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the relationship between the two instances of the DLP will be presented along with a discussion about the appropriate selection of parameters to ensure particular security level. In Part II: Efficient Algorithms, some of the computational issues which arise when using pairings in cryptography are addressed. Pairings can be computationally expensive, so the Pairing-Based Cryptography (PBC) research community is constantly striving to find computational improvements for all aspects of protocols using pairings. The improvements given in this section contribute towards more efficient methods for the computation of pairings, and increase the efficiency of operations necessary in some pairing-based protocol

    Semantic Spaces for Video Analysis of Behaviour

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    PhDThere are ever growing interests from the computer vision community into human behaviour analysis based on visual sensors. These interests generally include: (1) behaviour recognition - given a video clip or specific spatio-temporal volume of interest discriminate it into one or more of a set of pre-defined categories; (2) behaviour retrieval - given a video or textual description as query, search for video clips with related behaviour; (3) behaviour summarisation - given a number of video clips, summarise out representative and distinct behaviours. Although countless efforts have been dedicated into problems mentioned above, few works have attempted to analyse human behaviours in a semantic space. In this thesis, we define semantic spaces as a collection of high-dimensional Euclidean space in which semantic meaningful events, e.g. individual word, phrase and visual event, can be represented as vectors or distributions which are referred to as semantic representations. With the semantic space, semantic texts, visual events can be quantitatively compared by inner product, distance and divergence. The introduction of semantic spaces can bring lots of benefits for visual analysis. For example, discovering semantic representations for visual data can facilitate semantic meaningful video summarisation, retrieval and anomaly detection. Semantic space can also seamlessly bridge categories and datasets which are conventionally treated independent. This has encouraged the sharing of data and knowledge across categories and even datasets to improve recognition performance and reduce labelling effort. Moreover, semantic space has the ability to generalise learned model beyond known classes which is usually referred to as zero-shot learning. Nevertheless, discovering such a semantic space is non-trivial due to (1) semantic space is hard to define manually. Humans always have a good sense of specifying the semantic relatedness between visual and textual instances. But a measurable and finite semantic space can be difficult to construct with limited manual supervision. As a result, constructing semantic space from data is adopted to learn in an unsupervised manner; (2) It is hard to build a universal semantic space, i.e. this space is always contextual dependent. So it is important to build semantic space upon selected data such that it is always meaningful within the context. Even with a well constructed semantic space, challenges are still present including; (3) how to represent visual instances in the semantic space; and (4) how to mitigate the misalignment of visual feature and semantic spaces across categories and even datasets when knowledge/data are generalised. This thesis tackles the above challenges by exploiting data from different sources and building contextual semantic space with which data and knowledge can be transferred and shared to facilitate the general video behaviour analysis. To demonstrate the efficacy of semantic space for behaviour analysis, we focus on studying real world problems including surveillance behaviour analysis, zero-shot human action recognition and zero-shot crowd behaviour recognition with techniques specifically tailored for the nature of each problem. Firstly, for video surveillances scenes, we propose to discover semantic representations from the visual data in an unsupervised manner. This is due to the largely availability of unlabelled visual data in surveillance systems. By representing visual instances in the semantic space, data and annotations can be generalised to new events and even new surveillance scenes. Specifically, to detect abnormal events this thesis studies a geometrical alignment between semantic representation of events across scenes. Semantic actions can be thus transferred to new scenes and abnormal events can be detected in an unsupervised way. To model multiple surveillance scenes simultaneously, we show how to learn a shared semantic representation across a group of semantic related scenes through a multi-layer clustering of scenes. With multi-scene modelling we show how to improve surveillance tasks including scene activity profiling/understanding, crossscene query-by-example, behaviour classification, and video summarisation. Secondly, to avoid extremely costly and ambiguous video annotating, we investigate how to generalise recognition models learned from known categories to novel ones, which is often termed as zero-shot learning. To exploit the limited human supervision, e.g. category names, we construct the semantic space via a word-vector representation trained on large textual corpus in an unsupervised manner. Representation of visual instance in semantic space is obtained by learning a visual-to-semantic mapping. We notice that blindly applying the mapping learned from known categories to novel categories can cause bias and deteriorating the performance which is termed as domain shift. To solve this problem we employed techniques including semisupervised learning, self-training, hubness correction, multi-task learning and domain adaptation. All these methods in combine achieve state-of-the-art performance in zero-shot human action task. In the last, we study the possibility to re-use known and manually labelled semantic crowd attributes to recognise rare and unknown crowd behaviours. This task is termed as zero-shot crowd behaviours recognition. Crucially we point out that given the multi-labelled nature of semantic crowd attributes, zero-shot recognition can be improved by exploiting the co-occurrence between attributes. To summarise, this thesis studies methods for analysing video behaviours and demonstrates that exploring semantic spaces for video analysis is advantageous and more importantly enables multi-scene analysis and zero-shot learning beyond conventional learning strategies
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