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

  2. 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

  3. 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.

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

  5. 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.

  6. 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 .

  7. 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) ∈ .

  8. 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 .

  9. 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

  10. Proof TA 1 2 3 Randomly chooses N

  11. 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

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

  13. Construct a KPS from KDP B1 TA B2 Randomly chooses Bl Bm

  14. Construct a KPS from KDP P

  15. Proof

  16. 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

  17. Blundo et al scheme

  18. 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

  19. 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

  20. 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 ).

  21. Notations • Notation 1: The vector space r -linear functions over Fq • Notation 2: The vector space r –linear symmetric functions over Fq

  22. 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

  23. 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.

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

  25. ∈U