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.
This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61350110536, 61361136003.
Chapter PDF
Similar content being viewed by others
References
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)
Chen, J., Yap, C.-K.: Reversal complexity. SIAM Journal on Computing 20(4), 622–638 (1991)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography by Cellular Automata or How Fast Can Complexity Emerge in Nature? In: ICS, pp. 1–19 (2010)
Impagliazzo, R., Levin, L.A., Luby, M.: In: Symposium on Theory of Computing (STOC), pp. 12–24 (1989)
Vadhan, S.P., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: Symposium on Theory of Computing (STOC), pp. 817–836 (2012)
Yu, X., Yung, M.: Space Lower-Bounds for Pseudorandom-Generators. In: Structure in Complexity Theory Conference, pp. 186–197 (1994)
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)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Symposium on Theory of Computing (STOC), pp. 84–93 (2005)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally Private Randomizing Polynomials and Their Applications. Computational Complexity 15(2), 115–162 (2006)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SIAM Journal of Computing (SICOMP) 36(4), 845–888 (2006)
Applebaum, B., Ishai, Y., Kushilevitz, E.: On pseudorandom generators with linear stretch in \({\rm NC}\sp 0\). Computational Complexity 17(1), 38–69 (2008)
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)
Kharitonov, M., Goldberg, A.V., Yung, M.: Lower Bounds for Pseudorandom Number Generators. In: Foundations of Computer Science (FOCS), pp. 242–247 (1989)
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)
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)
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)
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)
Hernich, A., Schweikardt, N.: Reversal complexity revisited. Theoretical Computer Science 401(1-3), 191–205 (2008)
Beame, P., Huynh, T.: The Value of Multiple Read/Write Streams for Approximating Frequency Moments. ACM Transactions on Computation Theory 3(2), 6 (2012)
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)
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)
Alekhnovich, M.: More on average case vs approximation complexity. In: Foundations of Computer Science (FOCS), pp. 298–307 (2003)
Kilian, J.: Founding cryptography on oblivious transfer. In: Symposium on Theory of Computing (STOC), pp. 20–31 (1988)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 International Association for Cryptologic Research
About this paper
Cite this paper
Papakonstantinou, P.A., Yang, G. (2014). Cryptography with Streaming Algorithms. In: Garay, J.A., Gennaro, R. (eds) Advances in Cryptology – CRYPTO 2014. CRYPTO 2014. Lecture Notes in Computer Science, vol 8617. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44381-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-662-44381-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44380-4
Online ISBN: 978-3-662-44381-1
eBook Packages: Computer ScienceComputer Science (R0)