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The k-BDH Assumption Family: Bilinear Map Cryptography from Progressively Weaker Assumptions

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Topics in Cryptology – CT-RSA 2013 (CT-RSA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7779))

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Abstract

Over the past decade bilinear maps have been used to build a large variety of cryptosystems. In addition to new functionality, we have concurrently seen the emergence of many strong assumptions. In this work, we explore how to build bilinear map cryptosystems under progressively weaker assumptions.

We propose k-BDH, a new family of progressively weaker assumptions that generalizes the decisional bilinear Diffie-Hellman (DBDH) assumption. We give evidence in the generic group model that each assumption in our family is strictly weaker than the assumptions before it. DBDH has been used for proving many schemes secure, notably identity-based and functional encryption schemes; we expect that our k-BDH will lead to generalizations of many such schemes.

To illustrate the usefulness of our k-BDH family, we construct a family of selectively secure Identity-Based Encryption (IBE) systems based on it. Our system can be viewed as a generalization of the Boneh-Boyen IBE, however, the construction and proof require new ideas to fit the family. Our methods can be extended to produce hierarchical IBEs and CCA security; and give a fully secure variant. In addition, we discuss the opportunities and challenges of building new systems under our weaker assumption family.

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References

  1. Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM J. Computing 32(3), 586–615 (2003); Extended abstract in Proceedings of Crypto 2001

    Article  MathSciNet  MATH  Google Scholar 

  2. Gentry, C.: Practical Identity-Based Encryption Without Random Oracles. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 445–464. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  3. Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF Formulas on Ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  4. Abdalla, M., Pointcheval, D.: Interactive Diffie-Hellman Assumptions with Applications to Password-Based Authentication. In: S. Patrick, A., Yung, M. (eds.) FC 2005. LNCS, vol. 3570, pp. 341–356. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Boneh, D., Boyen, X.: Short Signatures Without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  6. Boyen, X.: The Uber-Assumption Family – A Unified Complexity Framework for Bilinear Groups. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 39–56. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Gentry, C., Halevi, S.: Hierarchical Identity Based Encryption with Polynomially Many Levels. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 437–456. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Waters, B.: Dual System Encryption: Realizing Fully Secure IBE and HIBE under Simple Assumptions. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 619–636. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  9. Hofheinz, D., Kiltz, E.: Secure Hybrid Encryption from Weakened Key Encapsulation. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 553–571. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  10. Shacham, H.: A Cramer-Shoup encryption scheme from the linear assumption and from progressively weaker linear variants. Cryptology ePrint Archive, Report 2007/074 (2007), http://eprint.iacr.org/

  11. Boneh, D., Boyen, X., Shacham, H.: Short Group Signatures. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 41–55. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Lewko, A.B., Waters, B.: Efficient pseudorandom functions from the decisional linear assumption and weaker variants. In: ACM Conference on Computer and Communications Security, pp. 112–120. ACM (November 2009)

    Google Scholar 

  13. Boneh, D., Montgomery, H., Raghunathan, A.: Algebraic pseudorandom functions with improved efficiency from the augmented cascade. In: Keromytis, A., Shmatikov, V. (eds.) Proceedings of CCS 2010, pp. 131–140. ACM Press (October 2010)

    Google Scholar 

  14. Boneh, D., Halevi, S., Hamburg, M., Ostrovsky, R.: Circular-Secure Encryption from Decision Diffie-Hellman. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 108–125. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  15. Camenisch, J., Chandran, N., Shoup, V.: A Public Key Encryption Scheme Secure against Key Dependent Chosen Plaintext and Adaptive Chosen Ciphertext Attacks. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 351–368. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  16. Naor, M., Segev, G.: Public-Key Cryptosystems Resilient to Key Leakage. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 18–35. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  17. Dodis, Y., Haralambiev, K., López-Alt, A., Wichs, D.: Cryptography against continuous memory attacks. In: Trevisan, L. (ed.) Proceedings of FOCS 2010, pp. 511–520. IEEE Computer Society (October 2010)

    Google Scholar 

  18. Freeman, D.M., Goldreich, O., Kiltz, E., Rosen, A., Segev, G.: More Constructions of Lossy and Correlation-Secure Trapdoor Functions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 279–295. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  19. Boyle, E., Segev, G., Wichs, D.: Fully Leakage-Resilient Signatures. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 89–108. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  20. Nechaev, V.I.: Complexity of a determinate algorithm for the discrete logarithm. Mathematical Notes 55(2), 165–172 (1994)

    Article  MathSciNet  Google Scholar 

  21. Shoup, V.: Lower Bounds for Discrete Logarithms and Related Problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  22. Boneh, D., Boyen, X.: Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  23. Benson, K., Shacham, H., Waters, B.: The k-bdh assumption family: Bilinear map cryptography from progressively weaker assumptions. Cryptology ePrint Archive, Report 2012 (2012), http://eprint.iacr.org/

  24. Boneh, D., Canetti, R., Halevi, S., Katz, J.: Chosen-ciphertext security from identity-based encryption. SIAM J. Comput. 36(5), 1301–1328 (2007)

    Article  MathSciNet  Google Scholar 

  25. Boyen, X., Mei, Q., Waters, B.: Direct chosen ciphertext security from identity-based techniques. In: Atluri, V., Meadows, C., Juels, A. (eds.) Proceedings of CCS 2005, pp. 320–329. ACM Press (November 2005)

    Google Scholar 

  26. Waters, B.: Efficient Identity-Based Encryption Without Random Oracles. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  27. Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Proceedings of ACM Conference on Computer and Communications Security 2006, pp. 89–98. ACM (November 2006)

    Google Scholar 

  28. Sahai, A., Waters, B.: Fuzzy Identity-Based Encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  29. Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: Dwork, C. (ed.) Proceedings of STOC 2008, pp. 187–196. ACM (May 2008)

    Google Scholar 

  30. Groth, J., Sahai, A.: Efficient Non-interactive Proof Systems for Bilinear Groups. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 415–432. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  31. Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Proceedings of STOC 1989, pp. 25–32. ACM (1989)

    Google Scholar 

  32. Verheul, E.R.: Evidence that XTR Is More Secure than Supersingular Elliptic Curve Cryptosystems. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 195–210. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  33. Moody, D.: The diffie hellman problem and generalization of verheuls theorem. Designs, Codes and Cryptography 52, 381–390 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Canetti, R., Halevi, S., Katz, J.: A Forward-secure Public-key Encryption Scheme. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 255–271. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  35. Babai, L., Szemerédi, E.: On the complexity of matrix group problems I. In: Proceedings of FOCS 1984, pp. 229–240. IEEE Computer Society (October 1984)

    Google Scholar 

  36. Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. In: Topics in Algebraic and Noncommutative Geometry: Proceedings in Memory of Ruth Michler. Contemporary Mathematics, vol. 324, pp. 71–90. American Mathematical Society (2003)

    Google Scholar 

  37. Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices and applications. Cryptology ePrint Archive, Report 2012/610 (2012)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, San Diego, USA

    Karyn Benson & Hovav Shacham

  2. University of Texas at Austin, USA

    Brent Waters

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  1. Karyn Benson
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  2. Hovav Shacham
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  3. Brent Waters
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  1. Institute for Future Enviroments, Queensland University of Technology, GPO Box 2434, 4000, Brisbane, QLD, Australia

    Ed Dawson

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Benson, K., Shacham, H., Waters, B. (2013). The k-BDH Assumption Family: Bilinear Map Cryptography from Progressively Weaker Assumptions. In: Dawson, E. (eds) Topics in Cryptology – CT-RSA 2013. CT-RSA 2013. Lecture Notes in Computer Science, vol 7779. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36095-4_20

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  • DOI: https://doi.org/10.1007/978-3-642-36095-4_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36094-7

  • Online ISBN: 978-3-642-36095-4

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