Studying a Range Proof Technique — Exception and Optimisation

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables. Our proof technique uses the De Bruijn identity and properties of Pfaffians; it also allows to prove that the high powers of these eigenvalues are independent. These results extend to any potential beyond the Gaussian case, as long as radial symmetry holds; this includes for instance truncations of quaternionic unitary matrices, products of quaternionic Ginibre matrices, and the quaternionic spherical ensemble. We then study the eigenvectors of quaternionic Ginibre matrices. Angles between eigenvectors and the matrix of overlaps both exhibit some specific features that can be compared to the complex case. In particular, we compute the distribution and the limit of the diagonal overlap associated to an eigenvalue that is conditioned to be at the origin. This complements a recent study of overlaps in quaternionic ensembles by Akemann, Förster and Kieburg [1, 2].

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Predicate signatures from pair encodings via dual system proof technique

Journal of Mathematical Cryptology ◽

10.1515/jmc-2017-0007 ◽

2019 ◽

Vol 13 (3-4) ◽

pp. 197-228

Author(s):

Mridul Nandi ◽

Tapas Pandit

Keyword(s):

Dual System ◽

Regular Languages ◽

Adaptive Model ◽

Proof Technique ◽

Encoding Scheme ◽

Constant Size ◽

Generic Framework ◽

Generic Construction ◽

Key Policy ◽

Predicate Encryption

Abstract Recently, Attrapadung (Eurocrypt 2014) proposed a generic framework for fully (adaptively) secure predicate encryption (PE) based on a new primitive, called pair encodings. The author shows that if the underlying pair encoding scheme is either perfectly secure or computationally (doubly-selectively) secure, then the PE scheme will be fully secure. Although the pair encodings were solely introduced for PE, we show that these can also be used to construct predicate signatures, a signature analogue of PE. More precisely, we propose a generic construction of predicate signature (PS) from pair encoding schemes. Our construction provides unconditional signer privacy, and unforgeability in the adaptive model. Thereafter, we instantiate many PS schemes with new results, e.g., the first practical PS schemes for regular languages, the first attribute-based signature (ABS) scheme with constant-size signatures in the adaptive model, unbounded ABS with large universes in key-policy flavor, etc.

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Deep Belief Networks Are Compact Universal Approximators

Neural Computation ◽

10.1162/neco.2010.08-09-1081 ◽

2010 ◽

Vol 22 (8) ◽

pp. 2192-2207 ◽

Cited By ~ 75

Author(s):

Nicolas Le Roux ◽

Yoshua Bengio

Keyword(s):

Learning Algorithm ◽

Feedforward Neural Networks ◽

Generative Models ◽

Universal Approximation ◽

Belief Networks ◽

Proof Technique ◽

Deep Belief Networks ◽

Boolean Expression ◽

Universal Approximators ◽

Causal Variables

Deep belief networks (DBN) are generative models with many layers of hidden causal variables, recently introduced by Hinton, Osindero, and Teh ( 2006 ), along with a greedy layer-wise unsupervised learning algorithm. Building on Le Roux and Bengio ( 2008 ) and Sutskever and Hinton ( 2008 ), we show that deep but narrow generative networks do not require more parameters than shallow ones to achieve universal approximation. Exploiting the proof technique, we prove that deep but narrow feedforward neural networks with sigmoidal units can represent any Boolean expression.

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Densities in certain three-way prime number races

Canadian Journal of Mathematics ◽

10.4153/s0008414x20000747 ◽

2020 ◽

pp. 1-34

Author(s):

Jiawei Lin ◽

Greg Martin

Keyword(s):

Asymptotic Formula ◽

Prime Number ◽

Error Term ◽

Proof Technique ◽

Real Numbers ◽

Logarithmic Density ◽

Mutual Independence ◽

Error Terms ◽

The Relationship

Abstract Let $a_1$ , $a_2$ , and $a_3$ be distinct reduced residues modulo q satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which $\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$ . The relationship among the $a_i$ allows us to normalize the error terms for the $\pi (x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

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New Constructions of Identity-Based Dual Receiver Encryption from Lattices

Entropy ◽

10.3390/e22060599 ◽

2020 ◽

Vol 22 (6) ◽

pp. 599

Author(s):

Yuan Liu ◽

Licheng Wang ◽

Xiaoying Shen ◽

Lixiang Li

Keyword(s):

Main Part ◽

Bilinear Pairings ◽

Third Party ◽

Key Generation ◽

Proof Technique ◽

Efficient Manner ◽

The Public ◽

Computation Technique ◽

Identity Based ◽

Injective Map

Dual receiver encryption (DRE), being originally conceived at CCS 2004 as a proof technique, enables a ciphertext to be decrypted to the same plaintext by two different but dual receivers and becomes popular recently due to itself useful application potentials such secure outsourcing, trusted third party supervising, client puzzling, etc. Identity-based DRE (IB-DRE) further combines the bilateral advantages/facilities of DRE and identity-based encryption (IBE). Most previous constructions of IB-DRE are based on bilinear pairings, and thus suffers from known quantum algorithmic attacks. It is interesting to build IB-DRE schemes based on the well-known post quantum platforms, such as lattices. At ACISP 2018, Zhang et al. gave the first lattice-based construction of IB-DRE, and the main part of the public parameter in this scheme consists of 2 n + 2 matrices where n is the bit-length of arbitrary identity. In this paper, by introducing an injective map and a homomorphic computation technique due to Yamada at EUROCRYPT 2016, we propose another lattice-based construction of IB-DRE in an even efficient manner: The main part of the public parameters consists only of 2 p n 1 p + 2 matrices of the same dimensions, where p ( ≥ 2 ) is a flexible constant. The larger the p and n, the more observable of our proposal. Typically, when p = 2 and n = 284 according to the suggestion given by Peikert et al., the size of public parameters in our proposal is reduced to merely 12% of Zhang et al.’s method. In addition, to lighten the pressure of key generation center, we extend our lattice-based IB-DRE scheme to hierarchical scenario. Finally, both the IB-DRE scheme and the HIB-DRE scheme are proved to be indistinguishable against adaptively chosen identity and plaintext attacks (IND-ID-CPA).

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Concrete stream calculus: An extended study

Journal of Functional Programming ◽

10.1017/s0956796810000213 ◽

2010 ◽

Vol 20 (5-6) ◽

pp. 463-535 ◽

Cited By ~ 5

Author(s):

RALF HINZE

Keyword(s):

Generating Functions ◽

Proof Technique ◽

Extended Study ◽

Equational Reasoning ◽

The Rich ◽

Infinite Sequences

AbstractThis paper shows how to reason about streams concisely and precisely. Streams, infinite sequences of elements, live in a coworld: they are given by a coinductive datatype, operations on streams are implemented by corecursive programs, and proofs are typically concocted using coinduction. This paper offers an alternative to coinduction. Suitably restricted, stream equations possessunique solutions. This property gives rise to a simple and attractive proof technique, essentially bringing equational reasoning to the coworld. We redevelop the theory of recurrences, finite calculus and generating functions using streams and stream operators, building on the cornerstone of unique solutions. The paper contains a smörgåsbord of examples: we study recursion elimination, investigate the binary carry sequence, explore Sprague-Grundy numbers and present two proofs of Moessner's Theorem. The calculations benefit from the rich structure of streams. As the type of streams is an applicative functor we can effortlessly lift operations and their properties to streams. In combination with Haskell's facilities for overloading, this greatly contributes to conciseness of notation. The development is indeed constructive: streams and stream operators are implemented in Haskell, usually by one-liners. The resulting calculus or library, if you wish, is elegant and fun to use.

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