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Graph-theoretic design and analysis of key predistribution schemes

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Key predistribution schemes for resource-constrained networks are methods for allocating symmetric keys to devices in such a way as to provide an efficient trade-off between key storage, connectivity and resilience. While there have been many suggested constructions for key predistribution schemes, a general understanding of the design principles on which to base such constructions is somewhat lacking. Indeed even the tools from which to develop such an understanding are currently limited, which results in many relatively ad hoc proposals in the research literature. It has been suggested that a large edge-expansion coefficient in the key graph is desirable for efficient key predistribution schemes. However, attempts to create key predistribution schemes from known expander graph constructions have only provided an extreme in the trade-off between connectivity and resilience: namely, they provide perfect resilience at the expense of substantially lower connectivity than can be achieved with the same key storage. Our contribution is twofold. First, we prove that many existing key predistribution schemes produce key graphs with good expansion. This provides further support and justification for their use, and confirms the validity of expansion as a sound design principle. Second, we propose the use of incidence graphs and concurrence graphs as tools to represent, design and analyse key predistribution schemes. We show that these tools can lead to helpful insights and new constructions.

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  1. Alon, Noga, Milman, Vitali D.: \(\lambda _i\), Isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B 38(1), 73–88 (1985)

  2. Alon N., Spencer J.H.: The Probabilistic Method. Wiley, Hoboken (2000).

  3. Bailey R.A., Cameron P.J.: Combinatorics of optimal designs. Surv. Comb. 365, 19–73 (2009).

  4. Bailey R.A., Cameron P.J.: Using graphs to find the best block designs. Preprint. (2011).

  5. Bapat R.B., Dey A.: Optimal block designs with minimal number of observations. Stat. Probab. Lett. 11(5), 399–402 (1991).

  6. Blackburn S.R., Gerke S.: Connectivity of the uniform random intersection graph. Discret. Math. 309(16), 5130–5140 (2009).

  7. Bollobás B.: Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability. Cambridge University Press, Cambridge (1986).

  8. Buser P.A.: A note on the isoperimetric constant. Ann. Sci. l’Éc. Norm. Supér. 15(2), 213–230 (1982).

  9. Cakiroglu S.A.: An upper bound on the algebraic connectivity of regular graphs. Preprint. (2014).

  10. Cameron P.J.: Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge (1994).

  11. Cameron P.J.: Permutation Groups. Cambridge University Press, Cambridge (1999).

  12. Cameron P.J.: Random strongly regular graphs. Electron. Notes Discret. Math. 10, 54–63 (2001).

  13. Cameron P.J.: Strongly regular graphs. Preprint. (2001).

  14. Cameron P.J., van Lint J.H.: Graph Theory, Coding Theory and Block Designs. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1975).

  15. Çamtepe S.A., Yener B.: Combinatorial design of key distribution mechanisms for wireless sensor networks. In: Computer Security—ESORICS 2004. Lecture Notes in Computer Science, vol. 3193, pp. 293–308. Springer, Heidelberg (2004).

  16. Çamtepe S.A., Yener B.: Key distribution mechanisms for wireless sensor networks: a survey. Rensselaer Polytechnic Institute, Computer Science Department, Technical Report TR-05-07 (2005).

  17. Çamtepe S.A., Yener B., Yung M.: Expander graph based key distribution mechanisms in wireless sensor networks. In: IEEE International Conference on Communications, ICC 06, pp. 2262–2267 (2006).

  18. Chan H., Perrig A., Song D.: Random key predistribution schemes for sensor networks. In: SP ’03, Proceedings of the 2003 IEEE Symposium on Security and Privacy. IEEE Computer Society, pp. 197–213. IEEE Computer Society Press, Los Alamitos (2003).

  19. Cheeger J.: A lower bound for the smallest eigen value of the Laplacian. Probl. Anal. 625, 195–199 (1970).

  20. Chen W.-K.: Applied Graph Theory: Graphs and Electrical Networks. North-Holland, Amsterdam (1976).

  21. Chen C.-Y., Chao H.-C.: A survey of key distribution in wireless sensor networks. Secur. Commun. Netw. 7(12), 2495–2508 (2011).

  22. Cheng C.-S., Bailey R.A.: Optimality of some two-associate-class partially balanced incomplete-block designs. Ann. Stat. 19(3), 1667–1671 (1991).

  23. Chung F.R.K.: Spectral graph theory. In: CBMS Conference on Recent Advances in Spectral Graph Theory. American Mathematical Society, Providence (1997).

  24. Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs. CRC Press, Boca Raton (2010).

  25. Davidoff G., Sarnak P., Valette A.: Elementary Number Theory, Group Theory and Ramanujan Graphs. London Mathematical Society Student Texts, vol. 55. Cambridge University Press, Cambridge (2003).

  26. Desmedt Y., Duif N., van Tilborg H., Wang H.: Bounds and constructions for key distribution schemes. Adv. Math. Commun. 3(3), 273–293 (2009).

  27. Di Pietro R., Mancini L.V., Mei A., Panconesi A., Radhakrishnan J.: Redoubtable sensor networks. ACM Trans. Inf. Syst. Secur. (TISSEC) 11(3), 1–22 (2008).

  28. Dodziuk J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984).

  29. Dodziuk J., Kendall W.S.: Combinatorial Laplacians and isoperimetric inequality: from local times to global geometry. Control Phys. 150, 68–74 (1986).

  30. Dolev D., Dwork C., Waarts O., Yung M.: Perfectly secure message transmission. J. ACM 40(1), 17–47 (1993).

  31. Erdös P., Rényi A.: On the evolution of random graphs. In: Publication of the Mathematical Institute of the Hungarian Academy of Sciences, pp. 17–61 (1960).

  32. Eschenauer L., Gligor V.D.: A key-management scheme for distributed sensor networks. In: Proceedings of the 9th ACM Conference on Computer and Communications Security—CCS ’02, pp. 41–47. ACM, New York (2002).

  33. Friedman J.: Some graphs with small second eigenvalue. Combinatorica 15(1), 31–42 (1995).

  34. Friedman J., Wigderson A.: On the second eigenvalue of hypergraphs. Combinatorica 15(1), 43–65 (1995).

  35. Harary F: Graph Theory. Addison-Wesley, Reading (1969).

  36. Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–562 (2006).

  37. Kendall M., Martin K.M.: On the role of expander graphs in key predistribution schemes for wireless sensor networks. In: Research in Cryptology. Lecture Notes in Computer Science, vol. 7242, pp. 62–82. Springer, Heidelberg (2012).

  38. Kendall M., Kendall E., Kendall W.S.A.: A generalised formula for calculating the resilience of random key predistribution schemes. ICAR Cryptol. 2012, 426 (2012).

  39. Kendall M., Martin K.M., Ng S.L., Paterson M.B., Stinson D.R.: Broadcast-enhanced key predistribution schemes. ACM Trans. Sens. Netw. 11(1), 6:1–6:33 (2014).

  40. Kleinberg J., Rubinfeld R.: Short paths in expander graphs. IEEE Annu. Symp. Found. Comput. Sci. 37, 86–95 (1996).

  41. Lanphier D., Miller C., Rosenhouse J., Russell A.: Expansion properties of Levi graphs. ARS Comb. 80(3), 1–7 (2006).

  42. Lee J., Stinson D.R.: A combinatorial approach to key predistribution for distributed sensor networks. In: IEEE Wireless Communications and Networking Conference, pp. 1200–1205. IEEE, New Orleans (2005).

  43. Lee J., Stinson D.R.: Deterministic key predistribution schemes for distributed sensor networks. In: Selected Areas in Cryptography. Lecture Notes in Computer Science, vol. 3357, pp. 294–307. Springer, Heidelberg (2005).

  44. Lee J., Stinson D.R.: Common intersection designs. J. Comb. Des. 14(4), 251–269 (2006).

  45. Levi F.W.: Finite Geometrical Systems: Six Public Lectures Delivered in February, 1940, at the University of Calcutta. The University of Calcutta (1942).

  46. Nilli A.: On the second eigenvalue of a graph. Discret. Math. 91(2), 207–210 (1991).

  47. Paterson M.B., Stinson D.R.: A unified approach to combinatorial key predistribution schemes for sensor networks. Des. Codes Cryptogr. 71, 433–457 (2014).

  48. Purdy E.: Locally expanding hypergraphs and the unique games conjecture. PhD Thesis, University of Chicago (2008).

  49. Shafiei H., Mehdizadeh A., Khonsari A., Ould-Khaoua M.: A combinatorial approach for key-distribution in wireless sensor networks. In: IEEE Global Telecommunications Conference, pp. 1–5. IEEE, Houston (2008).

  50. van Lint J.H., Schrijver A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1(1), 63–73 (1981).

  51. Wallis W.D.: Combinatorial Designs. Pure and Applied Mathematics. Marcel Dekker, New York (1988).

  52. Xiao Y., Rayi V.K., Sun B., Du X., Hu F., Galloway M.: A survey of key management schemes in wireless sensor networks. Comput. Commun. 30(11–12), 2314–2341 (2007).

  53. Yağan O., Makowski A.M.: On the existence of triangles in random key graphs with a note on their small-world property. Preprint. (2013).

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The authors are extremely grateful to Rosemary Bailey, Aylin Cakiroglu and Leonard Soicher for the many helpful conversations on designs and optimality. Research of Michelle Kendall conducted under EPSRC funding at Royal Holloway, University of London, UK.

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Correspondence to Michelle Kendall.

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Communicated by M. Paterson.

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Kendall, M., Martin, K.M. Graph-theoretic design and analysis of key predistribution schemes. Des. Codes Cryptogr. 81, 11–34 (2016).

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