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The Algebraic Group Model and its Applications

  • Georg Fuchsbauer
  • Eike Kiltz
  • Julian Loss
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10992)

Abstract

One of the most important and successful tools for assessing hardness assumptions in cryptography is the Generic Group Model (GGM). Over the past two decades, numerous assumptions and protocols have been analyzed within this model. While a proof in the GGM can certainly provide some measure of confidence in an assumption, its scope is rather limited since it does not capture group-specific algorithms that make use of the representation of the group.

To overcome this limitation, we propose the Algebraic Group Model (AGM), a model that lies in between the Standard Model and the GGM. It is the first restricted model of computation covering group-specific algorithms yet allowing to derive simple and meaningful security statements. To prove its usefulness, we show that several important assumptions, among them the Computational Diffie-Hellman, the Strong Diffie-Hellman, and the interactive LRSW assumptions, are equivalent to the Discrete Logarithm (DLog) assumption in the AGM. On the more practical side, we prove tight security reductions for two important schemes in the AGM to DLog or a variant thereof: the BLS signature scheme and Groth’s zero-knowledge SNARK (EUROCRYPT 2016), which is the most efficient SNARK for which only a proof in the GGM was known. Our proofs are quite simple and therefore less prone to subtle errors than those in the GGM.

Moreover, in combination with known lower bounds on the Discrete Logarithm assumption in the GGM, our results can be used to derive lower bounds for all the above-mentioned results in the GGM.

Keywords

Algebraic algorithms Generic group model Security reductions Cryptographic assumptions 

Notes

Acknowledgments

We thank Dan Brown for valuable comments and Pooya Farshim for discussions on polynomials. We also thank Helger Lipmaa for sharing with us his independent security proof for Groth’s SNARK. The first author is supported by the French ANR EfTrEC project (ANR-16-CE39-0002). The second author was supported in part by ERC Project ERCC (FP7/615074) and by DFG SPP 1736 Big Data. The third author was supported by ERC Project ERCC (FP7/615074).

References

  1. [ABM15]
    Abdalla, M., Benhamouda, F., MacKenzie, P.: Security of the J-PAKE password-authenticated key exchange protocol. In: 2015 IEEE Symposium on Security and Privacy, pp. 571–587. IEEE Computer Society Press, May 2015. 35Google Scholar
  2. [ABR01]
    Abdalla, M., Bellare, M., Rogaway, P.: The oracle Diffie-Hellman assumptions and an analysis of DHIES. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 143–158. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45353-9_12. 35, 45CrossRefGoogle Scholar
  3. [ABS16]
    Ambrona, M., Barthe, G., Schmidt, B.: Automated unbounded analysis of cryptographic constructions in the generic group model. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 822–851. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_29. 39CrossRefMATHGoogle Scholar
  4. [ACdM05]
    Ateniese, G., Camenisch, J., de Medeiros, B.: Untraceable RFID tags via insubvertible encryption. In: Atluri, V., Meadows, C., Juels, A. (eds.) ACM CCS 2005, pp. 92–101. ACM Press, November 2005. 36Google Scholar
  5. [ACHdM05]
    Ateniese, G., Camenisch, J., Hohenberger, S., de Medeiros, B.: Practical group signatures without random oracles. Cryptology ePrint Archive, Report 2005/385 (2005). http://eprint.iacr.org/2005/385. 36
  6. [AGO11]
    Abe, M., Groth, J., Ohkubo, M.: Separating short structure-preserving signatures from non-interactive assumptions. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 628–646. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25385-0_34. 35CrossRefGoogle Scholar
  7. [AM09]
    Aggarwal, D., Maurer, U.: Breaking RSA generically is equivalent to factoring. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 36–53. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01001-9_2. 34, 37CrossRefGoogle Scholar
  8. [BB08]
    Boneh, D., Boyen, X.: Short signatures without random oracles and the SDH assumption in bilinear groups. J. Cryptol. 21(2), 149–177 (2008). 58MathSciNetCrossRefGoogle Scholar
  9. [BCI+13]
    Bitansky, N., Chiesa, A., Ishai, Y., Paneth, O., Ostrovsky, R.: Succinct non-interactive arguments via linear interactive proofs. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 315–333. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36594-2_18. 52CrossRefGoogle Scholar
  10. [BCL04]
    Bangerter, E., Camenisch, J., Lysyanskaya, A.: A cryptographic framework for the controlled release of certified data. In: Security Protocols Workshop, pp. 20–24 (2004). 36Google Scholar
  11. [BCPR16]
    Bitansky, N., Canetti, R., Paneth, O., Rosen, A.: On the existence of extractable one-way functions. SIAM J. Comput. 45(5), 1910–1952 (2016). 38MathSciNetCrossRefGoogle Scholar
  12. [BCS05]
    Backes, M., Camenisch, J., Sommer, D.: Anonymous yet accountable access control. In: WPES, pp. 40–46 (2005). 36Google Scholar
  13. [BDZ03]
    Bao, F., Deng, R.H., Zhu, H.F.: Variations of Diffie-Hellman problem. In: Qing, S., Gollmann, D., Zhou, J. (eds.) ICICS 2003. LNCS, vol. 2836, pp. 301–312. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-39927-8_28. 43CrossRefGoogle Scholar
  14. [BFF+14]
    Barthe, G., Fagerholm, E., Fiore, D., Mitchell, J.C., Scedrov, A., Schmidt, B.: Automated analysis of cryptographic assumptions in generic group models. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 95–112. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_6. 37, 39CrossRefMATHGoogle Scholar
  15. [BFW16]
    Bernhard, D., Fischlin, M., Warinschi, B.: On the hardness of proving CCA-security of signed ElGamal. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016, Part I. LNCS, vol. 9614, pp. 47–69. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49384-7_3. 35CrossRefGoogle Scholar
  16. [BG04]
    Brown, D.R.L., Gallant, R.P.: The static Diffie-Hellman problem. Cryptology ePrint Archive, Report 2004/306 (2004). http://eprint.iacr.org/2004/306. 37
  17. [BL96]
    Boneh, D., Lipton, R.J.: Algorithms for black-box fields and their application to cryptography (extended abstract). In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 283–297. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-68697-5_22. 33CrossRefGoogle Scholar
  18. [BLS04]
    Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. J. Cryptol. 17(4), 297–319 (2004). 34, 48MathSciNetCrossRefGoogle Scholar
  19. [BMV08]
    Bresson, E., Monnerat, J., Vergnaud, D.: Separation results on the “one-more” computational problems. In: Malkin, T. (ed.) CT-RSA 2008. LNCS, vol. 4964, pp. 71–87. Springer, Heidelberg (2008). 35Google Scholar
  20. [Boy08]
    Boyen, X.: The uber-assumption family (invited talk). In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 39–56. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85538-5_3. 38, 39CrossRefGoogle Scholar
  21. [BR93]
    Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Ashby, V. (ed.) ACM CCS 1993, pp. 62–73. ACM Press, November 1993. 41Google Scholar
  22. [BR04]
    Bellare, M., Rogaway, P.: Code-based game-playing proofs and the security of triple encryption. Cryptology ePrint Archive, Report 2004/331 (2004). http://eprint.iacr.org/2004/331. 39
  23. [BV98]
    Boneh, D., Venkatesan, R.: Breaking RSA may not be equivalent to factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 59–71. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0054117. 34, 35, 40CrossRefGoogle Scholar
  24. [CGH98]
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited (preliminary version). In: 30th ACM STOC, pp. 209–218. ACM Press, May 1998. 38Google Scholar
  25. [Che06]
    Cheon, J.H.: Security analysis of the strong Diffie-Hellman problem. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 1–11. Springer, Heidelberg (2006).  https://doi.org/10.1007/11761679_1. 37, 47CrossRefGoogle Scholar
  26. [CHK+06]
    Camenisch, J., Hohenberger, S., Kohlweiss, M., Lysyanskaya, A., Meyerovich, M.: How to win the clonewars: efficient periodic n-times anonymous authentication. In: Juels, A., Wright, R.N., De Capitani di Vimercati, S. (eds.) ACM CCS 2006, pp. 201–210. ACM Press, October/November 2006. 36Google Scholar
  27. [CHL05]
    Camenisch, J., Hohenberger, S., Lysyanskaya, A.: Compact e-cash. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 302–321. Springer, Heidelberg (2005).  https://doi.org/10.1007/11426639_18. 36CrossRefGoogle Scholar
  28. [CHP07]
    Camenisch, J., Hohenberger, S., Pedersen, M.Ø.: Batch verification of short signatures. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 246–263. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-72540-4_14. 36CrossRefGoogle Scholar
  29. [CL04]
    Camenisch, J., Lysyanskaya, A.: Signature schemes and anonymous credentials from bilinear maps. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 56–72. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-28628-8_4. 35, 45CrossRefGoogle Scholar
  30. [CM14]
    Chase, M., Meiklejohn, S.: Déjà Q: using dual systems to revisit q-type assumptions. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 622–639. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-55220-5_34. 38CrossRefGoogle Scholar
  31. [Cor02]
    Coron, J.-S.: Optimal security proofs for PSS and other signature schemes. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 272–287. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-46035-7_18. 35, 36CrossRefGoogle Scholar
  32. [Dam92]
    Damgård, I.: Towards practical public key systems secure against chosen ciphertext attacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 445–456. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-46766-1_36. 35, 38CrossRefGoogle Scholar
  33. [Den02]
    Dent, A.W.: Adapting the weaknesses of the random oracle model to the generic group model. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 100–109. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-36178-2_6. 38CrossRefGoogle Scholar
  34. [DH76]
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976). 35, 43MathSciNetCrossRefGoogle Scholar
  35. [GBL08]
    Garg, S., Bhaskar, R., Lokam, S.V.: Improved bounds on security reductions for discrete log based signatures. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 93–107. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85174-5_6. 35CrossRefGoogle Scholar
  36. [GG17]
    Ghadafi, E., Groth, J.: Towards a classification of non-interactive computational assumptions in cyclic groups. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 66–96. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70697-9_3. 38CrossRefMATHGoogle Scholar
  37. [GGPR13]
    Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic span programs and succinct NIZKs without PCPs. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 626–645. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_37. 51CrossRefGoogle Scholar
  38. [Gro16]
    Groth, J.: On the size of pairing-based non-interactive arguments. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 305–326. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_11. 36, 50, 52, 53, 54, 56, 58CrossRefGoogle Scholar
  39. [HP78]
    Hellman, M.E., Pohlig, S.C.: An improved algorithm for computing logarithms over \({GF}(p)\) and its cryptographic significance. IEEE Trans. Inf. Theory 24(1), 106–110 (1978). 34MathSciNetCrossRefGoogle Scholar
  40. [JR10]
    Jager, T., Rupp, A.: The semi-generic group model and applications to pairing-based cryptography. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 539–556. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17373-8_31. 34, 38CrossRefGoogle Scholar
  41. [JR15]
    Joux, A., Rojat, A.: Security ranking among assumptions within the Uber Assumption framework. In: Desmedt, Y. (ed.) ISC 2013. LNCS, vol. 7807, pp. 391–406. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-27659-5_28. 38CrossRefGoogle Scholar
  42. [JS09]
    Jager, T., Schwenk, J.: On the analysis of cryptographic assumptions in the generic ring model. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 399–416. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10366-7_24. 37CrossRefMATHGoogle Scholar
  43. [Kil01]
    Kiltz, E.: A tool box of cryptographic functions related to the Diffie-Hellman function. In: Rangan, C.P., Ding, C. (eds.) INDOCRYPT 2001. LNCS, vol. 2247, pp. 339–349. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45311-3_32. 38CrossRefGoogle Scholar
  44. [KK12]
    Kakvi, S.A., Kiltz, E.: Optimal security proofs for full domain hash, revisited. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 537–553. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_32. 36CrossRefGoogle Scholar
  45. [KMP16]
    Kiltz, E., Masny, D., Pan, J.: Optimal security proofs for signatures from identification schemes. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part II. LNCS, vol. 9815, pp. 33–61. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53008-5_2. 35CrossRefGoogle Scholar
  46. [KSW08]
    Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78967-3_9. 39CrossRefGoogle Scholar
  47. [LR06]
    Leander, G., Rupp, A.: On the equivalence of RSA and factoring regarding generic ring algorithms. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 241–251. Springer, Heidelberg (2006).  https://doi.org/10.1007/11935230_16. 34CrossRefGoogle Scholar
  48. [LRSW99]
    Lysyanskaya, A., Rivest, R.L., Sahai, A., Wolf, S.: Pseudonym systems. In: Heys, H.M., Adams, C.M. (eds.) SAC 1999. LNCS, vol. 1758, pp. 184–199. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-46513-8_14. 35, 37, 45CrossRefGoogle Scholar
  49. [Mau05]
    Maurer, U.M.: Abstract models of computation in cryptography (invited paper). In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 1–12. Springer, Heidelberg (2005).  https://doi.org/10.1007/11586821_1. 33, 37, 39, 42CrossRefMATHGoogle Scholar
  50. [MRV16]
    Morillo, P., Ràfols, C., Villar, J.L.: The kernel matrix Diffie-Hellman assumption. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part I. LNCS, vol. 10031, pp. 729–758. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_27. 38CrossRefGoogle Scholar
  51. [MW98]
    Maurer, U.M., Wolf, S.: Lower bounds on generic algorithms in groups. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 72–84. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0054118. 33, 39CrossRefGoogle Scholar
  52. [MW99]
    Maurer, U., Wolf, S.: The relationship between breaking the Diffie-Hellman protocol and computing discrete logarithms. SIAM J. Comput. 28(5), 1689–1721 (1999). 43MathSciNetCrossRefGoogle Scholar
  53. [Nec94]
    Nechaev, V.I.: Complexity of a determinate algorithm for the discrete logarithm. Math. Notes 55(2), 165–172 (1994). 33MathSciNetCrossRefGoogle Scholar
  54. [Pol78]
    Pollard, J.M.: Monte Carlo methods for index computation mod \(p\). Math. Comput. 32, 918–924 (1978). 34MathSciNetMATHGoogle Scholar
  55. [PV05]
    Paillier, P., Vergnaud, D.: Discrete-log-based signatures may not be equivalent to discrete log. In: Roy, B.K. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 1–20. Springer, Heidelberg (2005). 34, 35, 40CrossRefGoogle Scholar
  56. [Riv04]
    Rivest, R.L.: On the notion of pseudo-free groups. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 505–521. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24638-1_28. 34CrossRefMATHGoogle Scholar
  57. [RLB+08]
    Rupp, A., Leander, G., Bangerter, E., Dent, A.W., Sadeghi, A.-R.: Sufficient conditions for intractability over black-box groups: generic lower bounds for generalized DL and DH problems. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 489–505. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-89255-7_30. 39CrossRefGoogle Scholar
  58. [Sho97]
    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).  https://doi.org/10.1007/3-540-69053-0_18. 33, 37, 38, 39, 42CrossRefGoogle Scholar
  59. [Sho04]
    Shoup, V.: Sequences of games: a tool for taming complexity in security proofs. Cryptology ePrint Archive, Report 2004/332 (2004). http://eprint.iacr.org/2004/332
  60. [SS01]
    Sadeghi, A.-R., Steiner, M.: Assumptions related to discrete logarithms: why subtleties make a real difference. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 244–261. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44987-6_16. 38CrossRefGoogle Scholar
  61. [FKL17]
    Fuchsbauer, G., Kiltz, E., Loss, J.: The algebraic group model and its applications. Cryptology ePrint Archive, Report 2017/620 (2017). http://eprint.iacr.org/2004/332. 45, 47

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  1. 1.Inria, ENS, CNRS, PSLParisFrance
  2. 2.Ruhr University BochumBochumGermany

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