Properties of distinct-difference configurations and lightweight key predistribution schemes for grid-based networks

Transcription

1 Properties of distinct-difference configurations and lightweight key predistribution schemes for grid-based networks Simon R. lackburn 1 Keith M. Martin 1 Tuvi Etzion 2 Maura. Paterson 1 1 Information Security Group Royal Holloway, University of London 2 Technion -Israel Institute of Technology epartment of omputer Science 19 May 2009 Maura Paterson s and KPSs for Grid-ased WSNs 1/ 17

2 Outline Key Predistribution for Grid-ased Networks Maura Paterson s and KPSs for Grid-ased WSNs 2/ 17

3 Precision griculture Maura Paterson s and KPSs for Grid-ased WSNs 3/ 17

4 Grid-ased Wireless Sensor Networks r restricted memory restricted battery power restricted computational ability vulnerable to compromise Maura Paterson s and KPSs for Grid-ased WSNs 4/ 17

5 Key Predistribution efinition (key predistribution scheme (KPS)) nodes are assigned keys before deployment nodes that share keys can communicate securely {k 1, k 5, k 7 } {k 3, k 5, k 12 } k 5 e.g. Eschenauer and Gligor: Each node randomly draws m keys uniformly without replacement from a keypool K Maura Paterson s and KPSs for Grid-ased WSNs 5/ 17

6 Goals for a KPS in a Grid-ased Network enable as many pairs of neighbouring nodes as possible to communicate securely minimise storage be resilient against node compromise Observation: it is not necessary for two nodes to share more than one key Maura Paterson s and KPSs for Grid-ased WSNs 6/ 17

7 ostas rrays one dot per row/column vector differences between dots are distinct applications to sonar, radar known constructions are based on finite fields Maura Paterson s and KPSs for Grid-ased WSNs 7/ 17

8 Translated ostas rrays Overlap in at Most One Point Maura Paterson s and KPSs for Grid-ased WSNs 8/ 17

9 Translated ostas rrays Overlap in at Most One Point Maura Paterson s and KPSs for Grid-ased WSNs 8/ 17

10 Translated ostas rrays Overlap in at Most One Point Maura Paterson s and KPSs for Grid-ased WSNs 8/ 17

11 Translated ostas rrays Overlap in at Most One Point Maura Paterson s and KPSs for Grid-ased WSNs 8/ 17

12 Translated ostas rrays Overlap in at Most One Point Maura Paterson s and KPSs for Grid-ased WSNs 8/ 17

13 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

14 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

15 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

16 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

17 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

18 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys E E E each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

19 Key Predistribution Using ostas rrays uses an n n ostas array each sensor stores n keys E E F F E F each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

20 Key Predistribution Using ostas rrays uses an n n ostas array G G E E G F F E F each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

21 Key Predistribution Using ostas rrays uses an n n ostas array G G H E H E G F F H E F each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

22 Key Predistribution Using ostas rrays uses an n n ostas array G G H E H E G I F I F H E I F each sensor stores n keys each key is assigned to n sensors two sensors share at most one key the distance between two sensors that share a key is at most 2(n 1) Maura Paterson s and KPSs for Grid-ased WSNs 9/ 17

23 efinition (istinct-ifference onfiguration (m, r)) m dots are placed in a square grid the distance between any two dots is at most r vector differences between dots are all distinct (5, 8) can be used for key predistribution in the same way as a ostas array more general than a ostas array more flexible choice of parameters Maura Paterson s and KPSs for Grid-ased WSNs 10/ 17

24 Upper ounds on m Theorem If a (m, r) exists, then m π 2 r + 3π1/3 2 5/3 r 2/3 + O(r 1/3 ) r + O(r 2/3 ). a (m, r) is contained in an anticode of diameter at most r and area at most (π/4)r 2 cover in circles of radius l count pairs (, d) where d is a pair of dots in (m, r) Maura Paterson s and KPSs for Grid-ased WSNs 11/ 17

25 Lower ounds on m Theorem There exists a (m, r) with m r o(r). Maura Paterson s and KPSs for Grid-ased WSNs 12/ 17

26 Sequences with istinct ifferences efinition Let be an abelian group. sequence {a 1, a 2,..., a m } is a 2 -sequence if all the sums a i1 + a i2 with 1 i 1 i 2 m are distinct. examples: Singer difference set Golomb ruler ose: 2 -sequence of size q in Z q 2 1 Maura Paterson s and KPSs for Grid-ased WSNs 13/ 17

27 Folding a 2 -Sequence {3, 13, 24, 29, 37, 41, 43, 44} (mod 63) Maura Paterson s and KPSs for Grid-ased WSNs 14/ 17

28 Results for the Manhattan Metric Theorem If a (m, r) exists then m 1 2 r + (3/2 4/3 )r 2/3 + O(r 1/3 ). There exists a (m, r) with m = 1 2 r o(r) Maura Paterson s and KPSs for Grid-ased WSNs 15/ 17

29 Efficient Key Predistribution for Grid-ased Wireless Sensor Networks, Information Theoretic Security, LNS 5155, 54-69, istinct ifference onfigurations: Multihop Paths and Key Predistribution in Sensor Networks. Two-imensional Patterns with istinct ifferences onstructions, ounds, and Maximal nticodes. Key Predistribution Techniques for Grid-ased Wireless Sensor Networks. Maura Paterson s and KPSs for Grid-ased WSNs 16/ 17

30 thank you! Maura Paterson s and KPSs for Grid-ased WSNs 17/ 17