Relation between H\'{e}non maps with biholomorphic escaping sets

We study H\'{e}non maps with biholomorphic escaping sets. We show that if $H$ and $F$ are two H\'{e}non maps of degree $d$ with biholomorphic escaping sets, then there exist complex numbers $\alpha,\beta$ and $\gamma$ with $\alpha^{d+1}=\beta$ and $\beta^{d-1}=\gamma^{d-1}=1$ such that with the following linear automorphisms $$B(x,y)=(\gamma \alpha \beta^{-1}x, \alpha^{-1}y) \quad\text{and}\quad L(x,y)=(\gamma^{-1}\beta x,\beta y )$$one has $$ F\equiv L \circ B \circ H \circ B. $

Dynamical properties of families of holomorphic mapping

In the first part of the thesis, we study some dynamical properties of skew products of H\'enon maps of \mbb C^2 that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of H\'enon mappings. In analogy with the dynamics of the iterates of a single H\'enon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. Further, it is shown that the successive pullbacks of a suitable current by the skew H\'enon maps converge to a multiple of the fibered stable current. Second part of the thesis generalizes most of the above-mentioned results for a completely random sequence of H\'enon maps. In addition, for this random system of H\'enon maps, we introduce the notion of average Green functions and average Green currents which carry many typical features of the classical Green functions and Green currents. Third part consists of some results about the global dynamics of a special class of skew maps. To prove these results, we use the knowledge of dynamical behavior of pseudo-random sequence of H\'enon maps widely. We show that the global skew map is strongly mixing for a class of invariant measures and also provide a lower bound on the topological entropy of the skew product. We conclude the thesis by studying another class of maps which are skew products of holomorphic endomorphisms of \mbb P^k fibered over a compact base. We define the fibered Fatou components and show that they are pseudoconvex and Kobayashi hyperbolic.Comment: 83 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.03431; text overlap with arXiv:0810.0811 by other author

Attribute-Based Access Control for Inner Product Functional Encryption from LWE

The notion of functional encryption (FE) was proposed as a generalization of plain public-key encryption to enable a much more fine-grained handling of encrypted data, with advanced applications such as cloud computing, multi-party computations, obfuscating circuits or Turing machines. While FE for general circuits or Turing machines gives a natural instantiation of the many cryptographic primitives, existing FE schemes are based on indistinguishability obfuscation or multilinear maps which either rely on new computational hardness assumptions or heuristically claimed to be secure. In this work, we present new techniques directly yielding FE for inner product functionality where secret-keys provide access control via polynomial-size bounded-depth circuits. More specifically, we encrypt messages with respect to attributes and embed policy circuits into secret-keys so that a restricted class of receivers would be able to learn certain property about the messages. Recently, many inner product FE schemes were proposed. However, none of them uses a general circuit as an access structure. Our main contribution is designing the first construction for an attribute-based FE scheme in key-policy setting for inner products from well-studied Learning With Errors (LWE) assumption. Our construction takes inspiration from the attribute-based encryption of Boneh et al. from Eurocrypt 2014 and the inner product functional encryption of Agrawal et al. from Crypto 2016. The scheme is proved in a stronger setting where the adversary is allowed to ask secret-keys that can decrypt the challenge ciphertext. Doing so requires a careful setting of parameters for handling the noise in ciphertexts to enable correct decryption. Another main advantage of our scheme is that the size of ciphertexts and secret-keys depends on the depth of the circuits rather than its size. Additionally, we extend our construction in a much desirable multi-input variant where secret-keys are associated with multiple policies subject to different encryption slots. This enhances the applicability of the scheme with finer access control

El proceso de formación ambiental no formal a través de un proyecto artístico: El carnaval

Let {H-lambda} be a continuous family of Henon maps parametrized by lambda is an element of M, where M subset of C-k is compact. The purpose of this paper is to understand some aspects of the random dynamical system obtained by iterating maps from this family. As an application, we study skew products of Henon maps and obtain lower bounds for their entropy