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In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If A and B are sets and every element of A is also an element of B, then: :A is a subset of B, denoted by A \subseteq B, or equivalently, : B is a superset of A, denoted by B \supseteq A. If A is a subset of B, but

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Unique decomposition of homogeneous languages and application to isothetic regions

Nicolas René Jean Ninin

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

Efficient multistage secret sharing scheme using bilinear map

Mitra Fatemi, Reza Ghasemi

In a multistage secret sharing (MSSS) scheme, the authorised subsets of participants could recover a number of secrets in different stages. A one-stage multisecret sharing (OSMSS) scheme is a special case of MSSS schemes in which all the secrets are recovered simultaneously. In these schemes, in addition to the individual shares, the dealer should provide the participants with a number of public values associated with the secrets. The less the number of public values, the more efficient is the scheme. It is desired that the MSSS and OSMSS schemes provide computational security. In this study, the authors show that in the OSMSS schemes any unauthorised coalition of the participants can reduce the uncertainty of the secrets. In addition, in MSSS schemes recovering a secret causes reducing uncertainty of the unrecovered secrets. Furthermore, by introducing a new multi-use MSSS scheme based on weil pairing, they reduce the number of public values comparing with the previous schemes.

Inst Engineering Technology-Iet2014

Broadcast Encryption and Traitor Tracing for Conditional Access Systems

Alexandre Karlov

In the context of the thesis we are studying the notions of broadcast encryption and traitor tracing in an industrial framework of conditional access systems related to Pay-TV. Broadcast encryption represents a cryptographic primitive which allows confidential transmission of content on a broadcast channel in such a way that only an authorized subset of users defined by the broadcaster are able to access it. Traitor tracing is a mechanism for identifying rogue sources who shared their decryption keys. Combined together, these two methods can be used to identify compromised terminals and instantaneously revoke them. Intuitively these two techniques are of high interest for Pay-TV systems with respect to the threats of today. Today, the majority of industrial security system are based on the tamper-resistance against attacks, such as those exploiting, for instance, different side channel information such as time, power consumption, electromagnetic emanations to name a few. However during recent years these attacks became more and more sophisticated. Therefore the aim of this thesis is to propose new schemes and mechanisms to improve the state-of-the-art for the conditional access systems and more specifically Pay-TV systems so that they rely even more on a solid cryptographic basis. As a matter of fact, even though many cryptographic schemes closely related to the subject of this thesis existed since many years in academia, none of them could have been deployed directly and efficiently in the context of large scale Pay-TV system which puts a number of heavy constraints on the bandwidth as well as on the computational complexity of the receivers. Consequently we will also describe the system and its constraints in order to simplify the task of development and integration of future schemes in these two academic domains in the context of practical environment.

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