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Shahzad Basiri Imam Hossein university shahzad_basiri@yahoo.com Workshop on key distribution Tuesday, May 24, 2011 Linear Key Predistribution Scheme

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Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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KKey Distribution Scheme A key predistribution scheme (KPS) is a method by which A trusted authority TA distributes secret information among a set of users in such a way that every user in a group in some specified family of privileged subsets is able to compute a common key associated with that group. Besides, certain coalitions of users ( forbidden subsets) outside a privileged group must not be able to find out any information on the value of the key associated with that group.

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Previous Scheme Polynomial Blom Blundo Based on Key Distribution Patterns Mitchell and Piper Trivial Scheme

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Goals One of the goals of this scheme is the construction of key predistribution schemes with good information rate for other families of specification structures. A new general model for the design of key predistribution schemes, which is based mainly on linear algebraic techniques, the linear key predistribution schemes (LKPSs). This new model, based on linear algebraic techniques, unifies all previous proposals.

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Assumption A subset P ⊂ U is a privileged subset of the specification structure if there exists F ⊂ U such that (P, F) ∈. The family of the privileged subsets of is denoted by P ( ). For any P ∈ P ( ), let us consider F P = { F ⊂ U : (P, F) ∈ }. The elements of F P are called the P- forbidden subsets of.

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Assumption For any P ∈ P ( ), the family of P -forbidden subsets F P is monotone decreasing, that is, if F 1 ∈ F P and F 2 ⊂ F 1, then F 2 ∈ F P. For any F ⊂ U, we consider the family P F of F - privileged subsets of, which consists of all subsets P ⊂ U such that (P, F) ∈.

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Assumption Let be a specification structure on a set of users U such that both F P and P F are monotone decreasing for any (P, F) ∈. The specification structure = {(P, F) ∈ : (F, P) ∈ } is called the dual specification structure of.

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Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes DDefinition PPrevious methods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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Proof Randomly chooses TA1 2 3 N

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Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes DDefinition PPrevious methods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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Previous Scheme Polynomial Blom Blundo Based on Key Distribution Patterns Mitchell and Piper Trivial Scheme

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Construct a KPS from KDP TA B2B2 BlBl BmBm B1B1 Randomly chooses

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Construct a KPS from KDP P

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Proof Proof

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Blundo et al scheme Blundo et al scheme TA 2 i N 1 Randomly Choose f (x 1, x 2,..., x r ) u i = f ( s i, x 2,..., x r ) Choose distinc public s 1, s 2,..., s l

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Blundo et al scheme

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Blundo et al scheme LKPS Let E r be the vector space of symmetric polynomials on r variables, with coefficients in F q and degree at most t on each variable

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Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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Multilinear function Let V be a vector space over a finite field F q. will denote the vector space V × V × ・・ ・ × V, where there are r factors in this product. A mapping T : F q is called a multilinear function if, for any i = 1, 2,..., r, T ( v 1,..., v i + v ’i,..., v r ) = T ( v 1,..., v i,..., v r )+T ( v 1,..., v ’i,..., v r ) and T ( v 1,..., λ v i,..., v r ) = λ T ( v 1,..., v i,..., v r ).

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Notations Notation 1: The vector space r -linear functions over F q Notation 2: The vector space r –linear symmetric functions over F q

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Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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Duality in LKPSs Under certain conditions, any -LKPS provides a LKPS for the dual specification structure and we relate the information rates of the two schemes.

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Proof (P, F) ∈ There exists a -LKPS with information rate (F, P) ∈

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∈ U∈ U

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