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Convex Analysisの二,三の進展について

From the Publisher:
Among the many approaches to formal reasoning about programs, Dynamic Logic enjoys the singular advantage of being strongly related to classical logic. Its variants constitute natural generalizations and extensions of classical formalisms. For example, Propositional Dynamic Logic (PDL) can be described as a blend of three complementary classical ingredients: propositional calculus, modal logic, and the algebra of regular events. In First-Order Dynamic Logic (DL), the propositional calculus is replaced by classical first-order predicate calculus. Dynamic Logic is a system of remarkable unity that is theoretically rich as well as of practical value. It can be used for formalizing correctness specifications and proving rigorously that those specifications are met by a particular program. Other uses include determining the equivalence of programs, comparing the expressive power of various programming constructs, and synthesizing programs from specifications. 
This book provides the first comprehensive introduction to Dynamic Logic. It is divided into three parts. The first part reviews the appropriate fundamental concepts of logic and computability theory and can stand alone as an introduction to these topics. The second part discusses PDL and its variants, and the third part discusses DL and its variants. Examples are provided throughout, and exercises and a short historical section are included at the end of each chapter.

Dynamic Logic

A secret-sharing scheme is a method by which a dealer distributes shares to parties such that only authorized subsets of parties can reconstruct the secret. Secret-sharing schemes are an important tool in cryptography and they are used as a building box in many secure protocols, e.g., general protocol for multiparty computation, Byzantine agreement, threshold cryptography, access control, attribute-based encryption, and generalized oblivious transfer.

In this survey, we describe the most important constructions of secret-sharing schemes; in particular, we explain the connections between secret-sharing schemes and monotone formulae and monotone span programs. We then discuss the main problem with known secret-sharing schemes - the large share size, which is exponential in the number of parties. We conjecture that this is unavoidable. We present the known lower bounds on the share size. These lower bounds are fairly weak and there is a big gap between the lower and upper bounds. For linear secret-sharing schemes, which is a class of schemes based on linear algebra that contains most known schemes, super-polynomial lower bounds on the share size are known. We describe the proofs of these lower bounds. We also present two results connecting secret-sharing schemes for a Hamiltonian access structure to the NP vs. coNP problem and to a major open problem in cryptography - constructing oblivious-transfer protocols from one-way functions.

https://www.cs.bgu.ac.il/~beimel/Papers/Survey.pdf

Secret-sharing schemes: a survey

Publisher Summary This chapter presents an introduction to some of the basic issues in the study of program logics. The chapter describes various forms of first-order Dynamic Logic and discusses their syntax, semantics, proof theory, and expressiveness. The chapter discusses the power of auxiliary data structures such as arrays and stacks, and a powerful assignment statement called the nondeterministic assignment. Program logics differ from classical logics in that truth is dynamic rather than static. In classical predicate logic, the truth value of a formula is determined by a valuation of its free variables over some structure. The valuation and the truth value of the formula it induces are regarded as immutable. In program logics, there are explicit syntactic constructs called programs to change the values of variables, thereby changing the truth values of formulas. There are two main approaches to modal logics of programs: (1) the exogenous approach, exemplified by Dynamic Logic and its precursor, the Partial Correctness Assertions Method; and (2) the endogenous approach, exemplified by Temporal Logic and its precursor, the Inductive Assertions Method.

Logics of Programs.

The present invention provides a method and apparatus for the production and labeling of objects in a manner suitable for the prevention and detection of counterfeiting. Thus, the system incorporates a variety of features that make unauthorized reproduction difficult. In addition, the present invention provides an efficient means for the production of labels and verification of authenticity, whereby a recording apparatus which includes a recording medium, having anisotrophic optical domains, along with a means for transferring a portion of the recording medium to a carrier, wherein a bulk portion of the recording medium has macroscopically detectable anisotrophic optical properties and the detecting apparatus thereon.

Authentication method and system

A secret sharing scheme permits a secret to be shared among participants of an n-element group in such a way that only qualified subsets of participants can recover the secret. If any nonqualified subset has absolutely no information on the secret, then the scheme is called perfect. The share in a scheme is the information that a participant must remember.

In [3] it was proved that for a certain access structure any perfect secret sharing scheme must give some participant a share which is at least 50\percent larger than the secret size. We prove that for each n there exists an access structure on n participants so that any perfect sharing scheme must give some participant a share which is at least about $n/\log n$ times the secret size.^1 We also show that the best possible result achievable by the information-theoretic method used here is n times the secret size.

^1 All logarithms in this paper are of base 2.

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The Size of a Share Must Be Large

The present invention relates to a method for determining the authenticity of an object, the method comprising the steps of: providing an authentication code, determining the positions of particles being distributed in an object, encoding of the positions to provide a check-code, using the check-code and the authentication-code to determine the authenticity of the object.

A perfect secret sharing scheme based on a graph G is a randomized distribution of a secret among the vertices of the graph so that the secret can be recovered from the information assigned to vertices at the endpoints of any edge, while the total information assigned to an independent set of vertices is independent (in statistical sense) of the secret itself. The efficiency of a scheme is measured by the amount of information the most heavily loaded vertex receives divided by the amount of information in the secret itself. The (worst case) information ratio of G is the infimum of this number. We calculate the best lower bound on the information ratio for an infinite family of graphs the entropy method can give. We argue that almost all existing constructions for secret sharing schemes are special cases of the generalized vector space construction. We give direct constructions of this type for the first two members of the family, and show that for the other members no such construction exists which would match the bound yielded by the entropy method.

An impossibility result on graph secret sharing

Given a graph G, a perfect secret sharing scheme based on G is a method to distribute a secret data among the vertices of G, the participants, so that a subset of participants can recover the secret if they contain an edge of G, otherwise they can obtain no information regarding the key. The average information rate is the ratio of the size of the secret and the average size of the share a participant must remember. The information rate of G is the supremum of the information rates realizable by perfect secret sharing schemes. Based on the entropy-theoretical arguments due to Capocelli et al [3], and extending the results of M. van Dijk [6] we construct a graph Gn on n vertices with average information rate below < 4/ log n. We obtain this result by determining, up to a constant factor, the average information rate of the d-dimensional cube.

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Secret sharing schemes on graphs.

Given a graph G , a perfect secret sharing scheme based on G is a method to distribute a secret data among the vertices of G , the participants , so that a subset of participants can recover the secret if they contain an edge of G , otherwise they can obtain no information regarding the key. The average information rate is the ratio of the size of the secret and the average size of the share a participant must remember. The information rate of G is the supremum of the information rates realizable by perfect secret sharing schemes.Based on the entropy-theoretical arguments due to Capocelli et al [4], and extending the results of M. van Dijk [7] and Blundo et al [2], we construct a graph G n on n vertices with average information rate below < 4/log n . We obtain this result by determining, up to a constant factor, the average information rate of the d -dimensional cube.

László Csirmaz

Papers

The Size of a Share Must Be Large

Authentication method and system

An impossibility result on graph secret sharing

Secret sharing schemes on graphs.

Secret sharing schemes on graphs