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

2. Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes C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.

7. 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.

8. 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) ∈.

9. 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.

10. Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes DDefinition PPrevious methods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

13. Proof Randomly chooses TA1 2 3 N

14. Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes DDefinition PPrevious methods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

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

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

18. Construct a KPS from KDP P

20. Proof Proof

21. 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

22. Blundo et al scheme

23. 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

24. Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

26. 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 ).

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

30. Outline KKey Predistribution Schemes LLinear Key Predistribution Schemes PPrevious Metods in KPS CConstructing Linear Key Predistribution Schemes DDulity in Linear Key Predistribution Schemes CConstructing Linear Key Predistribution Schemes by Duality

31. 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.

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

37. ∈ U∈ U