Attacker who has access to all secrets of n or less

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attacker (who has access to all secrets of n or less nodes) cannot determine any illegitimate shared secret. However, with access to secrets of more than n nodes, the attacker can discover all pairwise secrets. For an ( n, p )-secure probabilistic KPS, an attacker who has access to all secrets of n randomly chosen nodes can compute only a fraction p of all illegitimate pairwise secrets. Alternately, an attacker with access to secrets of n nodes can discover any illegitimate pairwise secret with probability p . Note that as long as p is low (say 2 64 ), it is computationally infeasible for the attacker to even determine which illegitimate pairwise secret can be exposed using the secrets pooled from n nodes. For such probabilistic KPS 0 p ( n ) 1 is a monotonic (and increasing) function of n . Deterministic n -secure KPSs can actually be considered as special cases of ( n, p )-secure KPSs where p takes only binary values (0 or 1). Mathematically p ( x ) = 0 x n = 1 x > n Deterministic KPSs 0 x = 0 p ( x ) x < x 1 x → ∞ Probabilistic KPSs (4) Copyright © 2010. World Scientific Publishing Company. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:46 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 340572 ; Beyah, Raheem, Corbett, Cherita, McNair, Janise.; Security in Ad Hoc and Sensor Networks Account: ns224671
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42 M. Ramkumar Thus, while deterministic KPSs fail in a catastrophic manner, probabilistic KPSs fail gracefully. In this chapter we restrict ourselves to one representative scheme from each class - Blom’s deterministic KPS 10 and the ( n, p )-secure key subset and symmetric certificates (KSSC) 12 scheme. 3.2.2. Blom’s Deterministic Scheme The symmetric key generation scheme (SKGS) proposed by Blom 10 is based on maximum distance separation (MDS) codes over a finite-field - for ex- ample, the finite-field formed by the set of integers Z q = { 0 , 1 , . . ., q 1 } (where q is a prime). The KDC chooses (1) a public primitive element α Z q ; (2) ( n + 1) × ( n + 1) symmetric matrix D with ( n +1 2 ) independent values (secrets) chosen randomly from Z q . Corresponding to a node with ID A Z q the KDC computes g A = [ g A (0) , g A (1) , . . . , g A ( n )] T , g A ( i ) = α i × A . S A = Dg A = d A = [ s A 0 , s A 1 , . . . s A n ] T (5) The k = n + 1 secrets S A are then provided to A over a secure channel. Nodes A and B (with secrets d A = [ s A 0 · · · s A n ] T and d B = [ s B 0 · · · s B n ] T respectively) can compute K AB = ( d A ) T g B = ( d B ) T g A (as D is a sym- metric matrix). In other words, A and B compute K AB as K AB = n i =0 s A i α iB mod q by A n i =0 s B i α iA mod q by B (6) An n -secure SKGS is unconditionally secure as long as n or less nodes pool their secrets together. However, an attacker who has access to secrets assigned to more than n nodes can solve for all P = ( n +1 2 ) secrets chosen by the KDC, and thus compute any pairwise secret.
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  • Spring '12
  • Kushal Kanwar
  • Public key infrastructure, ........., Public-key cryptography, Pretty Good Privacy

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