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##### Linear Key Predistribution Scheme

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**Linear Key Predistribution Scheme**ShahzadBasiri Imam Hossein university shahzad_basiri@yahoo.com Workshop on key distribution Tuesday, May 24, 2011**Outline**• Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality**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.**Previous Scheme**• Polynomial • Blom • Blundo • Based on Key Distribution Patterns • Mitchell and Piper • Trivial Scheme**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.**Assumption**• A subset P⊂Uis a privileged subset of the specification structure if there exists F⊂Usuch that (P, F) ∈ . • The family of the privileged subsets of is denoted by P( ). • For any P ∈ P( ),let us consider FP = {F⊂U : (P, F)∈ }. The elements of FP are called the P-forbidden subsets of .**Assumption**• For any P ∈ P( ), the family of P-forbidden subsets FP is monotone decreasing, that is, if F1 ∈ FP and F2 ⊂ F1, then F2 ∈ FP . • For any F⊂U, we consider the family PF of F-privileged subsets of , which consists of all subsets P⊂U such that (P, F) ∈ .**Assumption**• Let be a specification structure on a set of users U such that both FP and PFare monotone decreasing for any (P, F) ∈ . • The specification structure = {(P, F) ∈ : (F, P) ∈ } is called the dual specification structure of .**Outline**• Key Predistribution Schemes • Linear Key Predistribution Schemes • Definition • Previous methods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality**Proof**TA 1 2 3 Randomly chooses N**Outline**• Key Predistribution Schemes • Linear Key Predistribution Schemes • Definition • Previous methods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality**Previous Scheme**• Polynomial • Blom • Blundo • Based on Key Distribution Patterns • Mitchell and Piper • Trivial Scheme**Construct a KPS from KDP**B1 TA B2 Randomly chooses Bl Bm**Blundo et al scheme**1 TA 2 i ui = f (si , x2, . . . , xr ) N Randomly Choose f (x1, x2, . . . , xr ) Choose distinc public s1, s2, . . . , sl**Blundo et al scheme LKPS**• Let Er be the vector space of symmetric polynomials on r variables, with coefficients inFq and degree at most t on each variable**Outline**• Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality**Multilinear function**• Let V be a vector space over a finite field Fq . • will denote the vector spaceV×V ×・・ ・×V, where there are r factors in this product. • A mapping T : Fq is called a multilinear function if, for any i = 1, 2, . . . , r , • T (v1, . . . , vi +v’i, . . . , vr ) = T (v1, . . . , vi, . . . , vr )+T (v1, . . . , v’i, . . . , vr ) • and • T (v1, . . . , λvi, . . . , vr ) = λT (v1, . . . , vi, . . . , vr ).**Notations**• Notation 1: The vector space r -linear functions over Fq • Notation 2: The vector space r –linear symmetric functions over Fq**Outline**• Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality**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.**Proof**(P, F) ∈ (F, P) ∈ There exists a -LKPS with information rate