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On Tightly Secure Primitives in the Multi-instance Setting

  • Dennis HofheinzEmail author
  • Ngoc Khanh Nguyen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11442)

Abstract

We initiate the study of general tight reductions in cryptography. There already exist a variety of works that offer tight reductions for a number of cryptographic tasks, ranging from encryption and signature schemes to proof systems. However, our work is the first to provide a universal definition of a tight reduction (for arbitrary primitives), along with several observations and results concerning primitives for which tight reductions have not been known.

Technically, we start from the general notion of reductions due to Reingold, Trevisan, and Vadhan (TCC 2004), and equip it with a quantification of the respective reduction loss, and a canonical multi-instance extension to primitives. We then revisit several standard reductions whose tight security has not yet been considered. For instance, we revisit a generic construction of signature schemes from one-way functions, and show how to tighten the corresponding reduction by assuming collision-resistance from the used one-way function. We also obtain tightly secure pseudorandom generators (by using suitable rerandomisable hard-core predicates), and tightly secure lossy trapdoor functions.

References

  1. 1.
    Abe, M., David, B., Kohlweiss, M., Nishimaki, R., Ohkubo, M.: Tagged one-time signatures: tight security and optimal tag size. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 312–331. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36362-7_20CrossRefGoogle Scholar
  2. 2.
    Abe, M., Hofheinz, D., Nishimaki, R., Ohkubo, M., Pan, J.: Compact structure-preserving signatures with almost tight security. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part II. LNCS, vol. 10402, pp. 548–580. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63715-0_19CrossRefGoogle Scholar
  3. 3.
    Attrapadung, N., Hanaoka, G., Yamada, S.: A framework for identity-based encryption with almost tight security. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015, Part I. LNCS, vol. 9452, pp. 521–549. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48797-6_22CrossRefGoogle Scholar
  4. 4.
    Auerbach, B., Cash, D., Fersch, M., Kiltz, E.: Memory-tight reductions. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 101–132. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_4CrossRefGoogle Scholar
  5. 5.
    Baecher, P., Brzuska, C., Fischlin, M.: Notions of black-box reductions, revisited. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part I. LNCS, vol. 8269, pp. 296–315. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-42033-7_16CrossRefGoogle Scholar
  6. 6.
    Bellare, M., Boldyreva, A., Micali, S.: Public-key encryption in a multi-user setting: security proofs and improvements. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 259–274. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-45539-6_18CrossRefzbMATHGoogle Scholar
  7. 7.
    Blazy, O., Kiltz, E., Pan, J.: (Hierarchical) identity-based encryption from affine message authentication. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 408–425. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_23CrossRefGoogle Scholar
  8. 8.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo random bits. In: 23rd FOCS, pp. 112–117. IEEE Computer Society Press, November 1982Google Scholar
  9. 9.
    Boyen, X., Mei, Q., Waters, B.: Direct chosen ciphertext security from identity-based techniques. In: Atluri, V., Meadows, C., Juels, A. (eds.) ACM CCS 2005, pp. 320–329. ACM Press, November 2005Google Scholar
  10. 10.
    Chen, J., Gong, J., Weng, J.: Tightly secure IBE under constant-size master public key. In: Fehr, S. (ed.) PKC 2017, Part I. LNCS, vol. 10174, pp. 207–231. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54365-8_9CrossRefGoogle Scholar
  11. 11.
    Chen, J., Wee, H.: Fully, (almost) tightly secure IBE and dual system groups. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 435–460. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40084-1_25CrossRefGoogle Scholar
  12. 12.
    Damgård, I.B.: Collision free hash functions and public key signature schemes. In: Chaum, D., Price, W.L. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 203–216. Springer, Heidelberg (1988).  https://doi.org/10.1007/3-540-39118-5_19CrossRefGoogle Scholar
  13. 13.
    Damgård, I.B.: On the randomness of Legendre and Jacobi sequences. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 163–172. Springer, New York (1990).  https://doi.org/10.1007/0-387-34799-2_13CrossRefGoogle Scholar
  14. 14.
    Dodis, Y., Oliveira, R., Pietrzak, K.: On the generic insecurity of the full domain hash. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 449–466. Springer, Heidelberg (2005).  https://doi.org/10.1007/11535218_27CrossRefGoogle Scholar
  15. 15.
    Dodis, Y., Reyzin, L., Smith, A.: Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 523–540. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_31CrossRefGoogle Scholar
  16. 16.
    Fischlin, M., Lehmann, A., Ristenpart, T., Shrimpton, T., Stam, M., Tessaro, S.: Random oracles with(out) programmability. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 303–320. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17373-8_18CrossRefGoogle Scholar
  17. 17.
    Gay, R., Hofheinz, D., Kiltz, E., Wee, H.: Tightly CCA-secure encryption without pairings. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part I. LNCS, vol. 9665, pp. 1–27. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49890-3_1CrossRefGoogle Scholar
  18. 18.
    Gay, R., Hofheinz, D., Kohl, L.: Kurosawa-Desmedt meets tight security. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part III. LNCS, vol. 10403, pp. 133–160. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63697-9_5CrossRefGoogle Scholar
  19. 19.
    Gay, R., Hofheinz, D., Kohl, L., Pan, J.: More efficient (almost) tightly secure structure-preserving signatures. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 230–258. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78375-8_8CrossRefGoogle Scholar
  20. 20.
    Goldreich, O.: Foundations of Cryptography: Basic Applications, vol. 2. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  21. 21.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: 21st ACM STOC, pp. 25–32. ACM Press, May 1989Google Scholar
  22. 22.
    Gong, J., Chen, J., Dong, X., Cao, Z., Tang, S.: Extended nested dual system groups, revisited. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016, Part I. LNCS, vol. 9614, pp. 133–163. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49384-7_6CrossRefGoogle Scholar
  23. 23.
    Haitner, I., Holenstein, T., Reingold, O., Vadhan, S., Wee, H.: Universal one-way hash functions via inaccessible entropy. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 616–637. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_31CrossRefGoogle Scholar
  24. 24.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hofheinz, D.: Algebraic partitioning: fully compact and (almost) tightly secure cryptography. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016, Part I. LNCS, vol. 9562, pp. 251–281. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49096-9_11CrossRefzbMATHGoogle Scholar
  26. 26.
    Hofheinz, D.: Adaptive partitioning. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part III. LNCS, vol. 10212, pp. 489–518. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_17CrossRefGoogle Scholar
  27. 27.
    Hofheinz, D., Jager, T.: Tightly secure signatures and public-key encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 590–607. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_35CrossRefGoogle Scholar
  28. 28.
    Hofheinz, D., Koch, J., Striecks, C.: Identity-based encryption with (almost) tight security in the multi-instance, multi-ciphertext setting. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 799–822. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_36CrossRefGoogle Scholar
  29. 29.
    Holenstein, T.: Pseudorandom generators from one-way functions: a simple construction for any hardness. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 443–461. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_23CrossRefGoogle Scholar
  30. 30.
    Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: 21st ACM STOC, pp. 44–61. ACM Press, May 1989Google Scholar
  31. 31.
    Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 8–26. Springer, New York (1990).  https://doi.org/10.1007/0-387-34799-2_2CrossRefGoogle Scholar
  32. 32.
    Katz, J., Koo, C.-Y.: On constructing universal one-way hash functions from arbitrary one-way functions. Cryptology ePrint Archive, Report 2005/328 (2005). http://eprint.iacr.org/2005/328
  33. 33.
    Lamport, L.: Constructing digital signatures from a one-way function. Technical report SRI-CSL-98, SRI International Computer Science Laboratory, October 1979Google Scholar
  34. 34.
    Libert, B., Joye, M., Yung, M., Peters, T.: Concise multi-challenge CCA-secure encryption and signatures with almost tight security. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 1–21. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45608-8_1CrossRefGoogle Scholar
  35. 35.
    Libert, B., Peters, T., Joye, M., Yung, M.: Compactly hiding linear spans. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015, Part I. LNCS, vol. 9452, pp. 681–707. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48797-6_28CrossRefGoogle Scholar
  36. 36.
    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences i: measure of pseudorandomness, the Legendre symbol. Acta Arith. 82(4), 365–377 (1997)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Merkle, R.C.: A digital signature based on a conventional encryption function. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 369–378. Springer, Heidelberg (1988).  https://doi.org/10.1007/3-540-48184-2_32CrossRefGoogle Scholar
  38. 38.
    Naor, M.: On cryptographic assumptions and challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45146-4_6CrossRefGoogle Scholar
  39. 39.
    Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of psuedo-random functions. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23–25 October 1995, pp. 170–181 (1995)Google Scholar
  40. 40.
    Pandey, O., Pass, R., Vaikuntanathan, V.: Adaptive one-way functions and applications. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 57–74. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85174-5_4CrossRefGoogle Scholar
  41. 41.
    Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. Cryptology ePrint Archive, Report 2007/279 (2007). http://eprint.iacr.org/2007/279
  42. 42.
    Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 187–196. ACM Press, May 2008Google Scholar
  43. 43.
    Reingold, O., Trevisan, L., Vadhan, S.: Notions of reducibility between cryptographic primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24638-1_1CrossRefzbMATHGoogle Scholar
  44. 44.
    Rompel, J.: One-way functions are necessary and sufficient for secure signatures. In: 22nd ACM STOC, pp. 387–394. ACM Press, May 1990Google Scholar
  45. 45.
    Simon, D.R.: Finding collisions on a one-way street: can secure hash functions be based on general assumptions? In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 334–345. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0054137CrossRefGoogle Scholar
  46. 46.
    van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for some hidden shift problems. In: 14th SODA, pp. 489–498. ACM-SIAM, January 2003Google Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.IBM Research ZurichRüschlikonSwitzerland

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