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Cryptographic Assumptions: A Position Paper

  • Shafi Goldwasser
  • Yael Tauman Kalai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9562)

Abstract

The mission of theoretical cryptography is to define and construct provably secure cryptographic protocols and schemes. Without proofs of security, cryptographic constructs offer no guarantees whatsoever and no basis for evaluation and comparison. As most security proofs necessarily come in the form of a reduction between the security claim and an intractability assumption, such proofs are ultimately only as good as the assumptions they are based on. Thus, the complexity implications of every assumption we utilize should be of significant substance, and serve as the yard stick for the value of our proposals.

Lately, the field of cryptography has seen a sharp increase in the number of new assumptions that are often complex to define and difficult to interpret. At times, these assumptions are hard to untangle from the constructions which utilize them.

We believe that the lack of standards of what is accepted as a reasonable cryptographic assumption can be harmful to the credibility of our field. Therefore, there is a great need for measures according to which we classify and compare assumptions, as to which are safe and which are not. In this paper, we propose such a classification and review recently suggested assumptions in this light. This follows the footsteps of Naor (Crypto 2003).

Our governing principle is relying on hardness assumptions that are independent of the cryptographic constructions.

Keywords

Random Oracle Model Cryptographic Primitive Auxiliary Input Cryptographic Scheme Complexity Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We would like to thank many colleagues for their comments on an earlier draft of this work, including Ducas, Goldreich Micianccio, Peikert, Regev, Sahai, and Vaikuntanathan. In particular, we are grateful to Micciancio for extensive illuminating discussions on many aspects of hardness assumptions on lattices, and to Sahai and Vaikuntanathan for clarifying discussions on the strength of the sub-group elimination assumption used for IO obfuscation. Thanks to Bernstein for pointing out a long overlooked worst-case to average-case reduction for discrete-log in fields of small characteristic.

References

  1. 1.
    Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, 22–24 May 1996, Philadelphia, Pennsylvania, USA, pp. 99–108 (1996)Google Scholar
  2. 2.
    Alekhnovich, M.: More on average case vs approximation complexity. In: Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS 2003), 11–14 October 2003, Cambridge, MA, USA, pp. 298–307 (2003)Google Scholar
  3. 3.
    Barak, B.: How to go beyond the black-box simulation barrier. In: 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 106–115 (2001)Google Scholar
  4. 4.
    Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  5. 5.
    Barbulescu, R., Gaudry, P., Joux, A., Thomé, E.: A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 1–16. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  6. 6.
    Barić, N., Pfitzmann, B.: Collision-free accumulators and fail-stop signature schemes without trees. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 480–494. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  7. 7.
    Bernstein, D.J.: Private communication (2015)Google Scholar
  8. 8.
    Bitansky, N., Canetti, R., Chiesa, A., Goldwasser, S., Lin, H., Rubinstein, A., Tromer, E.: The hunting of the SNARK. IACR Cryptology ePrint Archive 2014, p. 580 (2014)Google Scholar
  9. 9.
    Bitansky, N., Garg, S., Lin, H., Pass, R., Telang, S.: Succinct randomized encodings and their applications. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, 14–17 June 2015, Portland, OR, USA, pp. 439–448 (2015)Google Scholar
  10. 10.
    Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21–23, 2000, Portland, OR, USA, pp. 435–440 (2000)Google Scholar
  11. 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) CrossRefGoogle Scholar
  12. 12.
    Boneh, D., Lipton, R.J.: Algorithms for black-box fields and their application to cryptography. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 283–297. Springer, Heidelberg (1996) Google Scholar
  13. 13.
    Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Symposium on Theory of Computing Conference, STOC 2013, 1–4 June 2013, Palo Alto, CA, USA, pp. 575–584 (2013)Google Scholar
  14. 14.
    Brakerski, Z., Rothblum, G.N.: Virtual black-box obfuscation for all circuits via generic graded encoding. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 1–25. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  15. 15.
    Cachin, C., Micali, S., Stadler, M.A.: Computationally private information retrieval with polylogarithmic communication. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 402–414. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  16. 16.
    Canetti, R.: Towards realizing random oracles: hash functions that hide all partial information. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 455–469. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  17. 17.
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. J. ACM 51(4), 557–594 (2004). http://doi.acm.org/10.1145/1008731.1008734 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015) Google Scholar
  19. 19.
    Cheon, J.H., Lee, C., Ryu, H.: Cryptanalysis of the new CLT multilinear maps. IACR Cryptology ePrint Archive 2015, p. 934 (2015)Google Scholar
  20. 20.
    Chung, K., Lin, H., Pass, R.: Constant-round concurrent zero knowledge from p-certificates. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26–29 October, 2013, Berkeley, CA, USA, pp. 50–59 (2013)Google Scholar
  21. 21.
    Di Crescenzo, G., Lipmaa, H.: Succinct NP proofs from an extractability assumption. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 175–185. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  22. 22.
    Damgård, I.B.: 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) Google Scholar
  23. 23.
    Damgård, I., Faust, S., Hazay, C.: Secure two-party computation with low communication. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 54–74. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  24. 24.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987) CrossRefGoogle Scholar
  26. 26.
    Fujisaki, E., Okamoto, T.: Statistical zero knowledge protocols to prove modular polynomial relations. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  27. 27.
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26–29 October, 2013, Berkeley, CA, USA, pp. 40–49 (2013)Google Scholar
  28. 28.
    Gentry, C., Lewko, A.B., Sahai, A., Waters, B.: Indistinguishability obfuscation from the multilinear subgroup elimination assumption. IACR Cryptology ePrint Archive 2014, p. 309 (2014)Google Scholar
  29. 29.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 17–20 May 2008, Victoria, British Columbia, Canada, pp. 197–206 (2008)Google Scholar
  30. 30.
    Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: STOC, pp. 99–108 (2011)Google Scholar
  31. 31.
    Goldreich, O.: The Foundations of Cryptography: Basic Techniques, vol. 1. Cambridge University Press, Cambridge (2001) CrossRefMATHGoogle Scholar
  32. 32.
    Goldreich, O.: Randomness and computation. In: Goldreich, O. (ed.) Studies in Complexity and Cryptography. LNCS, vol. 6650, pp. 507–539. Springer, Heidelberg (2011) Google Scholar
  33. 33.
    Goldreich, O., Goldwasser, S.: On the limits of nonapproximability of lattice problems. J. Comput. Syst. Sci. 60(3), 540–563 (2000)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 14–17 May 1989, Seattle, Washigton, USA, pp. 25–32 (1989)Google Scholar
  35. 35.
    Goldwasser, S., Kalai, Y.T.: On the (in)security of the Fiat-Shamir paradigm. In: Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS 2003), 11–14 October 2003, Cambridge, MA, USA, pp. 102–113 (2003)Google Scholar
  36. 36.
    Goldwasser, S., Kalai, Y.T., Peikert, C., Vaikuntanathan, V.: Robustness of thelearning with errors assumption. In: Proceedings of the Innovations in Computer Science, ICS 2010, 5–7 January 2010, Tsinghua University, Beijing, China, pp. 230–240 (2010). http://conference.itcs.tsinghua.edu.cn/ICS2010/content/papers/19.html
  37. 37.
    Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Heninger, N., Shacham, H.: Reconstructing RSA private keys from random key bits. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 1–17. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  39. 39.
    Impagliazzo, R., Jaiswal, R., Kabanets, V., Wigderson, A.: Uniform direct product theorems: simplified, optimized, and derandomized. SIAM J. Comput. 39(4), 1637–1665 (2010)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Khot, S.: On the unique games conjecture. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23–25 October 2005, Pittsburgh, PA, USA, p. 3 (2005)Google Scholar
  41. 41.
    Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: A Cryptographic Perspective, vol. 671. Springer Science & Business Media, New York (2012)MATHGoogle Scholar
  42. 42.
    Minaud, B., Fouque, P.: Cryptanalysis of the new multilinear map over the integers. IACR Cryptology ePrint Archive 2015, p. 941 (2015)Google Scholar
  43. 43.
    Naor, M.: On cryptographic assumptions and challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  44. 44.
    Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  45. 45.
    Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation from semantically-secure multilinear encodings. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 500–517. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  46. 46.
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, May 31 - June 2, 2009, Bethesda, MD, USA, pp. 333–342 (2009)Google Scholar
  47. 47.
    Rabin, M.O.: Digitalized signatures and public-key functions as intractable as factorization. Technical report, MIT Laboratory for Computer Science (1979)Google Scholar
  48. 48.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 22–24 May 2005, Baltimore, MD, USA, pp. 84–93 (2005)Google Scholar
  49. 49.
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Shoup, V.: New algorithms for finding irreducible polynomials over finite fields. In: 29th Annual Symposium on Foundations of Computer Science, 24–26 October 1988, White Plains, New York, USA, pp. 283–290 (1988)Google Scholar
  51. 51.
    Shoup, V.: Searching for primitive roots in finite fields. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 13–17 May 1990, Baltimore, Maryland, USA, pp. 546–554 (1990)Google Scholar
  52. 52.
    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) CrossRefGoogle Scholar
  53. 53.
    Stehlé, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  54. 54.
    Wee, H.: On obfuscating point functions. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 22–24 May 2005, Baltimore, MD, USA, pp. 523–532 (2005). http://doi.acm.org/10.1145/1060590.1060669

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.MITCambridgeUSA
  2. 2.Weizmann InstituteRehovotIsrael
  3. 3.Microsoft ResearchCambridgeUSA

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