Advertisement

Cryptography with Streaming Algorithms

  • Periklis A. Papakonstantinou
  • Guang Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8617)

Abstract

We put forth the question of whether cryptography is feasible using streaming devices. We give constructions and prove lower bounds. In streaming cryptography (not to be confused with stream-ciphers) everything—the keys, the messages, and the seeds—are huge compared to the internal memory of the device. These streaming algorithms have small internal memory size and make a constant number of passes over big data maintained in a constant number of read/write external tapes. Typically, the internal memory size is O(logn) and we use 2 external tapes; whereas 1 tape is provably insufficient. In this setting we cannot compute instances of popular intractability assumptions. Nevertheless, we base cryptography on these assumptions by employing non-black-box techniques, and study its limitations.

We introduce new techniques to obtain unconditional lower bounds showing that no super-linear stretch pseudorandom generator exists, and no Public Key Encryption (PKE) exists with private-keys of size sub-linear in the plaintext length.

For possibility results, assuming the existence of one-way functions computable in NC1—e.g. factoring, lattice assumptions—we obtain streaming algorithms computing one-way functions and pseudorandom generators. Given the Learning With Errors (LWE) assumption we construct PKE where both the encryption and decryption are streaming algorithms. The starting point of our work is the groundbreaking work of Applebaum-Ishai-Kushilevitz on Cryptography in NC0. In the end, our developments are technically orthogonal to their work; e.g. there is a PKE where the decryption is a streaming algorithm, whereas no PKE decryption can be in NC0.

Keywords

streaming lower bound big data randomized encoding non-black-box PRG PKE 

References

  1. 1.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography with constant input locality. Journal of Cryptology, 429–469; In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 92–110. Springer, Heidelberg (2007)Google Scholar
  2. 2.
    Chen, J., Yap, C.-K.: Reversal complexity. SIAM Journal on Computing 20(4), 622–638 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography by Cellular Automata or How Fast Can Complexity Emerge in Nature? In: ICS, pp. 1–19 (2010)Google Scholar
  4. 4.
    Impagliazzo, R., Levin, L.A., Luby, M.: In: Symposium on Theory of Computing (STOC), pp. 12–24 (1989)Google Scholar
  5. 5.
    Vadhan, S.P., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: Symposium on Theory of Computing (STOC), pp. 817–836 (2012)Google Scholar
  6. 6.
    Yu, X., Yung, M.: Space Lower-Bounds for Pseudorandom-Generators. In: Structure in Complexity Theory Conference, pp. 186–197 (1994)Google Scholar
  7. 7.
    Micciancio, D., Peikert, C.: Trapdoors for lattices: Simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Symposium on Theory of Computing (STOC), pp. 84–93 (2005)Google Scholar
  9. 9.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally Private Randomizing Polynomials and Their Applications. Computational Complexity 15(2), 115–162 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SIAM Journal of Computing (SICOMP) 36(4), 845–888 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: On pseudorandom generators with linear stretch in \({\rm NC}\sp 0\). Computational Complexity 17(1), 38–69 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bronson, J., Juma, A., Papakonstantinou, P.A.: Limits on the stretch of non-adaptive constructions of pseudo-random generators. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 504–521. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Kharitonov, M., Goldberg, A.V., Yung, M.: Lower Bounds for Pseudorandom Number Generators. In: Foundations of Computer Science (FOCS), pp. 242–247 (1989)Google Scholar
  14. 14.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A Pseudorandom Generator from any One-way Function. SIAM Journal of Computing (SICOMP) 28(4), 1364–1396 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bar-Yossef, Z., Reingold, O., Shaltiel, R., Trevisan, L.: Streaming Computation of Combinatorial Objects. In: Annual IEEE Conference on Computational Complexity (CCC), vol. 17 (2002)Google Scholar
  16. 16.
    Haitner, I., Reingold, O., Vadhan, S.: Efficiency improvements in constructing pseudorandom generators from one-way functions. In: Symposium on Theory of Computing (STOC), pp. 437–446 (2010)Google Scholar
  17. 17.
    Grohe, M., Hernich, A., Schweikardt, N.: Lower bounds for processing data with few random accesses to external memory. Journal of the ACM 56(3): Art. 12, 58 (2009)Google Scholar
  18. 18.
    Hernich, A., Schweikardt, N.: Reversal complexity revisited. Theoretical Computer Science 401(1-3), 191–205 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Beame, P., Huynh, T.: The Value of Multiple Read/Write Streams for Approximating Frequency Moments. ACM Transactions on Computation Theory 3(2), 6 (2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in \({\rm NC}\sp 1\). Journal of Computer and System Sciences 38(1), 150–164 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Goldwasser, S., Micali, S.: Probabilistic Encryption and How to Play Mental Poker Keeping Secret All Partial Information. In: Symposium on Theory of Computing (STOC), pp. 365–377 (1982)Google Scholar
  22. 22.
    Alekhnovich, M.: More on average case vs approximation complexity. In: Foundations of Computer Science (FOCS), pp. 298–307 (2003)Google Scholar
  23. 23.
    Kilian, J.: Founding cryptography on oblivious transfer. In: Symposium on Theory of Computing (STOC), pp. 20–31 (1988)Google Scholar
  24. 24.
    Grohe, M., Schweikardt, N.: Lower bounds for sorting with few random accesses to external memory. In: Symposium on Principles of Database Systems (PODS), pp. 238–249 (2005)Google Scholar

Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Periklis A. Papakonstantinou
    • 1
  • Guang Yang
    • 1
  1. 1.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingChina

Personalised recommendations