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Minicrypt Primitives with Algebraic Structure and Applications

  • Navid AlamatiEmail author
  • Hart Montgomery
  • Sikhar Patranabis
  • Arnab Roy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11477)

Abstract

Algebraic structure lies at the heart of Cryptomania as we know it. An interesting question is the following: instead of building (Cryptomania) primitives from concrete assumptions, can we build them from simple Minicrypt primitives endowed with some additional algebraic structure? In this work, we affirmatively answer this question by adding algebraic structure to the following Minicrypt primitives:
  • One-Way Function (OWF)

  • Weak Unpredictable Function (wUF)

  • Weak Pseudorandom Function (wPRF)

The algebraic structure that we consider is group homomorphism over the input/output spaces of these primitives. We also consider a “bounded” notion of homomorphism where the primitive only supports an a priori bounded number of homomorphic operations in order to capture lattice-based and other “noisy” assumptions. We show that these structured primitives can be used to construct many cryptographic protocols. In particular, we prove that:
  • (Bounded) Homomorphic OWFs (HOWFs) imply collision-resistant hash functions, Schnorr-style signatures and chameleon hash functions.

  • (Bounded) Input-Homomorphic weak UFs (IHwUFs) imply CPA-secure PKE, non-interactive key exchange, trapdoor functions, blind batch encryption (which implies anonymous IBE, KDM-secure and leakage-resilient PKE), CCA2 deterministic PKE, and hinting PRGs (which in turn imply transformation of CPA to CCA security for ABE/1-sided PE).

  • (Bounded) Input-Homomorphic weak PRFs (IHwPRFs) imply PIR, lossy trapdoor functions, OT and MPC (in the plain model).

In addition, we show how to realize any CDH/DDH-based protocol with certain properties in a generic manner using IHwUFs/IHwPRFs, and how to instantiate such a protocol from many concrete assumptions.

We also consider primitives with substantially richer structure, namely Ring IHwPRFs and L-composable IHwPRFs. In particular, we show the following:

  • Ring IHwPRFs with certain properties imply FHE.

  • 2-composable IHwPRFs imply (black-box) IBE, and L-composable IHwPRFs imply non-interactive \((L+1)\)-party key exchange.

Our framework allows us to categorize many cryptographic protocols based on which structured Minicrypt primitive implies them. In addition, it potentially makes showing the existence of many cryptosystems from novel assumptions substantially easier in the future.

References

  1. [AD97]
    Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: 29th ACM STOC, pp. 284–293. ACM Press, May 1997Google Scholar
  2. [Ajt96]
    Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: 28th ACM STOC, pp. 99–108. ACM Press, May 1996Google Scholar
  3. [AMG07]
    Aguilar-Melchor, C., Gaborit, P.: A lattice-based computationally-efficient private information retrieval protocol. In: Western European Workshop on Research in Cryptology. Citeseer (2007)Google Scholar
  4. [AS15]
    Asharov, G., Segev, G.: Limits on the power of indistinguishability obfuscation and functional encryption. In: Guruswami, V. (ed.) 56th FOCS, pp. 191–209. IEEE Computer Society Press, October 2015Google Scholar
  5. [Bar17]
    Barak, B.: The complexity of public-key cryptography. Cryptology ePrint Archive, Report 2017/365 (2017). https://eprint.iacr.org/2017/365
  6. [BBF13]
    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
  7. [BDRV18]
    Berman, I., Degwekar, A., Rothblum, R.D., Vasudevan, P.N.: From laconic zero-knowledge to public-key cryptography - extended abstract. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 674–697. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96878-0_23CrossRefGoogle Scholar
  8. [BDV17]
    Bitansky, N., Degwekar, A., Vaikuntanathan, V.: Structure vs. hardness through the obfuscation lens. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 696–723. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_23CrossRefGoogle Scholar
  9. [BGI+01]
    Barak, B., Goldreich, O., Impagliazzo, R., Rudich, S., Sahai, A., Vadhan, S.P., Yang, K.: On the (im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44647-8_1CrossRefGoogle Scholar
  10. [BH08]
    Boneh, D., Hamburg, M.: Generalized identity based and broadcast encryption schemes. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 455–470. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-89255-7_28CrossRefGoogle Scholar
  11. [BHY09]
    Bellare, M., Hofheinz, D., Yilek, S.: Possibility and impossibility results for encryption and commitment secure under selective opening. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 1–35. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01001-9_1CrossRefGoogle Scholar
  12. [BLMR13]
    Boneh, D., Lewi, K., Montgomery, H., Raghunathan, A.: Key homomorphic PRFs and their applications. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 410–428. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_23CrossRefGoogle Scholar
  13. [BLSV18]
    Brakerski, Z., Lombardi, A., Segev, G., Vaikuntanathan, V.: Anonymous IBE, leakage resilience and circular security from new assumptions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part I. LNCS, vol. 10820, pp. 535–564. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78381-9_20CrossRefGoogle Scholar
  14. [BM82]
    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
  15. [BR17]
    Bogdanov, A., Rosen, A.: Pseudorandom functions: three decades later. Cryptology ePrint Archive, Report 2017/652 (2017). https://eprint.iacr.org/2017/652
  16. [BSW11]
    Boneh, D., Sahai, A., Waters, B.: Functional encryption: definitions and challenges. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 253–273. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19571-6_16CrossRefGoogle Scholar
  17. [CGW15]
    Chen, J., Gay, R., Wee, H.: Improved dual system ABE in prime-order groups via predicate encodings. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part II. LNCS, vol. 9057, pp. 595–624. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46803-6_20CrossRefGoogle Scholar
  18. [DG17a]
    Döttling, N., Garg, S.: From selective IBE to full IBE and selective HIBE. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part I. LNCS, vol. 10677, pp. 372–408. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70500-2_13CrossRefGoogle Scholar
  19. [DG17b]
    Döttling, N., Garg, S.: Identity-based encryption from the Diffie-Hellman assumption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 537–569. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_18CrossRefGoogle Scholar
  20. [DGHM18]
    Döttling, N., Garg, S., Hajiabadi, M., Masny, D.: New constructions of identity-based and key-dependent message secure encryption schemes. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part I. LNCS, vol. 10769, pp. 3–31. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-76578-5_1CrossRefzbMATHGoogle Scholar
  21. [DH76]
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theor. 22(6), 644–654 (1976)MathSciNetCrossRefGoogle Scholar
  22. [DHS15]
    Derler, D., Hanser, C., Slamanig, D.: Revisiting cryptographic accumulators, additional properties and relations to other primitives. In: Nyberg, K. (ed.) CT-RSA 2015. LNCS, vol. 9048, pp. 127–144. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-16715-2_7CrossRefGoogle Scholar
  23. [ElG84]
    ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_2CrossRefGoogle Scholar
  24. [FH18]
    Fischlin, M., Harasser, P.: Invisible sanitizable signatures and public-key encryption are equivalent. In: Preneel, B., Vercauteren, F. (eds.) ACNS 2018. LNCS, vol. 10892, pp. 202–220. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-93387-0_11CrossRefzbMATHGoogle Scholar
  25. [Fis12]
    Fischlin, M.: Black-box reductions and separations in cryptography. In: Mitrokotsa, A., Vaudenay, S. (eds.) AFRICACRYPT 2012. LNCS, vol. 7374, pp. 413–422. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31410-0_26CrossRefGoogle Scholar
  26. [FMV18]
    Friolo, D., Masny, D., Venturi, D.: Secure multi-party computation from strongly uniform key agreement. Cryptology ePrint Archive, Report 2018/473 (2018). http://eprint.iacr.org/
  27. [Gen09]
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st ACM STOC, pp. 169–178. ACM Press, May/June (2009)Google Scholar
  28. [GGH13a]
    Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_1CrossRefGoogle Scholar
  29. [GGH+13b]
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, pp. 40–49. IEEE Computer Society Press, October 2013Google Scholar
  30. [GGH18]
    Garg, S., Gay, R., Hajiabadi, M.: New techniques for efficient trapdoor functions and applications. Cryptology ePrint Archive, Report 2018/872 (2018). http://eprint.iacr.org/
  31. [GGM84]
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions (extended abstract). In: 25th FOCS, pp. 464–479. IEEE Computer Society Press, October 1984Google Scholar
  32. [GH18]
    Garg, S., Hajiabadi, M.: Trapdoor Functions from the computational Diffie-Hellman assumption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 362–391. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96881-0_13CrossRefGoogle Scholar
  33. [GHMM18]
    Garg, S., Hajiabadi, M., Mahmoody, M., Mohammed, A.: Limits on the power of garbling techniques for public-key encryption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 335–364. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96878-0_12CrossRefGoogle Scholar
  34. [GL89]
    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
  35. [GPR16]
    Goyal, V., Pandey, O., Richelson, S.: Textbook non-malleable commitments. In: STOC (2016)Google Scholar
  36. [GPSW06]
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Juels, A., Wright, R.N., Vimercati, S. (ed.) ACM CCS 2006, pp. 89–98. ACM Press, October/November 2006. Available as Cryptology ePrint Archive Report 2006/309Google Scholar
  37. [GPSZ17]
    Garg, S., Pandey, O., Srinivasan, A., Zhandry, M.: Breaking the sub-exponential barrier in obfustopia. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part III. LNCS, vol. 10212, pp. 156–181. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_6CrossRefzbMATHGoogle Scholar
  38. [GPV08]
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 197–206. ACM Press, May 2008Google Scholar
  39. [GW11]
    Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Fortnow, L., Vadhan, S.P. (eds.) 43rd ACM STOC, pp. 99–108. ACM Press, June 2011Google Scholar
  40. [HHRS07]
    Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols - a tight lower bound on the round complexity of statistically-hiding commitments. In: 48th FOCS, pp. 669–679. IEEE Computer Society Press, October 2007Google Scholar
  41. [HILL99]
    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
  42. [HKS16]
    Hajiabadi, M., Kapron, B.M., Srinivasan, V.: On generic constructions of circularly-secure, leakage-resilient public-key encryption schemes. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016, Part II. LNCS, vol. 9615, pp. 129–158. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49387-8_6CrossRefzbMATHGoogle Scholar
  43. [HO12]
    Hemenway, B., Ostrovsky, R.: On homomorphic encryption and chosen-ciphertext security. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 52–65. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-30057-8_4CrossRefGoogle Scholar
  44. [IKLP06]
    Ishai, Y., Kushilevitz, E., Lindell, Y., Petrank, E.: Black-box constructions for secure computation. In: Kleinberg, J.M. (ed.) 38th ACM STOC, pp. 99–108. ACM Press, May 2006Google Scholar
  45. [IKO05]
    Ishai, Y., Kushilevitz, E., Ostrovsky, R.: Sufficient conditions for collision-resistant hashing. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 445–456. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-30576-7_24CrossRefGoogle Scholar
  46. [Imp95]
    Impagliazzo, R.: A personal view of average-case complexity. In: Tenth Annual IEEE Conference on Proceedings of Structure in Complexity Theory, pp. 134–147, June 1995. ISSN 1063-6870Google Scholar
  47. [IZ89]
    Impagliazzo, R., Zuckerman, D.: How to recycle random bits. In: 30th FOCS, pp. 248–253. IEEE Computer Society Press, October/November 1989Google Scholar
  48. [KO97]
    Kushilevitz, E., Ostrovsky, R.: Replication is NOT needed: SINGLE database, computationally-private information retrieval. In: FOCS, pp. 364–373 (1997)Google Scholar
  49. [KW18]
    Koppula, V., Waters, B.: Realizing chosen ciphertext security generically in attribute-based encryption and predicate encryption. Cryptology ePrint Archive, Report 2018/847 (2018). http://eprint.iacr.org/
  50. [MM16]
    Mahmoody, M., Mohammed, A.: On the power of hierarchical identity-based encryption. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 243–272. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_9CrossRefGoogle Scholar
  51. [MMN+16]
    Mahmoody, M., Mohammed, A., Nematihaji, S., Pass, R., Shelat, A.: Lower bounds on assumptions behind indistinguishability obfuscation. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016, Part I. LNCS, vol. 9562, pp. 49–66. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49096-9_3CrossRefGoogle Scholar
  52. [OSV15]
    Ostrovsky, R., Scafuro, A., Venkitasubramanian, M.: Resettably sound zero-knowledge arguments from OWFs - the (semi) black-box way. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part I. LNCS, vol. 9014, pp. 345–374. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46494-6_15CrossRefGoogle Scholar
  53. [PS08]
    Pietrzak, K., Sjödin, J.: Weak pseudorandom functions in Minicrypt. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 423–436. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-70583-3_35CrossRefGoogle Scholar
  54. [PW08]
    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
  55. [Reg05]
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th ACM STOC, pp. 84–93. ACM Press, May 2005Google Scholar
  56. [Rom90]
    Rompel, J.: One-way functions are necessary and sufficient for secure signatures. In: 22nd ACM STOC, pp. 387–394. ACM Press, May 1990Google Scholar
  57. [Rot11]
    Rothblum, R.: Homomorphic encryption: from private-key to public-key. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 219–234. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19571-6_14CrossRefGoogle Scholar
  58. [RSA78]
    Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetCrossRefGoogle Scholar
  59. [RTV04]
    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
  60. [Sha84]
    Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_5CrossRefGoogle Scholar
  61. [SW05]
    Sahai, A., Waters, B.R.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005).  https://doi.org/10.1007/11426639_27CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  • Navid Alamati
    • 1
    • 2
    Email author
  • Hart Montgomery
    • 2
  • Sikhar Patranabis
    • 2
    • 3
  • Arnab Roy
    • 2
  1. 1.University of MichiganAnn ArborUSA
  2. 2.Fujitsu Laboratories of AmericaSunnyvaleUSA
  3. 3.IIT KharagpurKharagpurIndia

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