Abstract
Traditionally, computation within self-assembly models is hard to conceal because the self-assembly process generates a crystalline assembly whose computational history is inherently part of the structure itself. With no way to remove information from the computation, this computational model offers a unique problem: how can computational input and computation be hidden while still computing and reporting the final output? Designing such systems is inherently motivated by privacy concerns in biomedical computing and applications in cryptography. In this paper we propose the problem of performing “covert computation” within tile self-assembly that seeks to design self-assembly systems that “conceal” both the input and computational history of performed computations. We achieve these results within the growth-only restricted abstract Tile Assembly Model (aTAM) with positive and negative interactions. We show that general-case covert computation is possible by implementing a set of basic covert logic gates capable of simulating any circuit (functionally complete). To further motivate the study of covert computation, we apply our new framework to resolve an outstanding complexity question; we use our covert circuitry to show that the unique assembly verification problem within the growth-only aTAM with negative interactions is coNP-complete.
Similar content being viewed by others
Notes
It is important to note that the term covert has specific meaning in cryptography which does not apply here.
References
Adleman, L.M., Cheng, Q., Goel, A., Huang, M.D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)
Brun, Y.: Arithmetic computation in the tile assembly model: addition and multiplication. Theor. Comput. Sci. 378, 17–31 (2007)
Cantu, A.A., Luchsinger, A., Schweller, R., Wylie, T.: Covert computation in self-assembled circuits. In: Baier, C., Chatzigiannakis, I., Flocchini, P., Leonardi, S. (eds.) 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), Leibniz International Proceedings in Informatics (LIPIcs), vol. 132, pp. 31:1–31:14. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2019). https://doi.org/10.4230/LIPIcs.ICALP.2019.31. http://drops.dagstuhl.de/opus/volltexte/2019/10607
Chalk, C., Demiane, E.D., Demaine, M.L., Martinez, E., Schweller, R., Vega, L., Wylie, T.: Universal shape replicators via self-assembly with attractive and repulsive forces. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’17) (2017)
Chalk, C., Luchsinger, A., Schweller, R., Wylie, T.: Self-assembly of any shape with constant tile types using high temperature. In: Proceedings of the 26th Annual European Symposium on Algorithms, ESA’18 (2018)
Chaum, D., Crépeau, C., Damgård, I.B.: Multiparty unconditionally secure protocols (abstract). In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC’88), pp. 11–19 (1988)
Cheng, Q., Aggarwal, G., Goldwasser, M.H., Kao, M.Y., Schweller, R.T., de Espanés, P.M.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34, 1493–1515 (2005)
Claes, P., Liberton, D.K., Daniels, K.E.A.: Modeling 3d facial shape from dna. PLOS Genet. 10(3), 1–14 (2014). https://doi.org/10.1371/journal.pgen.1004224
De Cristofaro, E., Faber, S., Tsudik, G.: Secure genomic testing with size- and position-hiding private substring matching. In: Proceedings of the 12th ACM Workshop on Privacy in the Electronic Society, WPES’13, pp. 107–118. ACM (2013)
Doty, D.: Theory of algorithmic self-assembly. Commun. ACM 55(12), 78–88 (2012)
Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. Algorithmica 66(1), 153–172 (2013)
Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of the 53rd IEEE Conference on Foundations of Computer Science, FOCS ’12 (2012)
Dowlin, N., Gilad-Bachrach, R., Laine, K., Lauter, K., Naehrig, M., Wernsing, J.: Manual for using homomorphic encryption for bioinformatics. Proc. IEEE 105(3), 552–567 (2017)
Evans, C.: Crystals that count! physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, California Inst. of Tech. (2014)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences, 1st edn. W. H. Freeman, New York (1979)
Gymrek, M., McGuire, A.L., Golan, D., Halperin, E., Erlich, Y.: Identifying personal genomes by surname inference. Science 339(6117), 321–324 (2013)
Huang, Z., Ayday, E., Fellay, J., Hubaux, J., Juels, A.: Genoguard: Protecting genomic data against brute-force attacks. In: 2015 IEEE Symposium on Security and Privacy, pp. 447–462 (2015). https://doi.org/10.1109/SP.2015.34
Keenan, A., Schweller, R., Sherman, M., Zhong, X.: Fast arithmetic in algorithmic self-assembly. Nat. Comput. 15(1), 115–128 (2016)
Lander, E., Linton, L., et al.: Initial sequencing and analysis of the human genome. Nature 409(6822), 860–921 (2001)
Luchsinger, A., Schweller, R., Wylie, T.: Self-assembly of shapes at constant scale using repulsive forces. Nat. Comput. (2018). https://doi.org/10.1007/s11047-018-9707-9
Patitz, M.J.: An introduction to tile-based self-assembly and a survey of recent results. Nat. Comput. 13(2), 195–224 (2014)
Patitz, M.J., Rogers, T.A., Schweller, R., Summers, S.M., Winslow, A.: Resiliency to multiple nucleation in temperature-1 self-assembly. In: Proceedings of DNA Computing and Molecular Programming, DNA’16, pp. 98–113 (2016)
Patitz, M.J., Schweller, R.T., Summers, S.M.: Exact shapes and turing universality at temperature 1 with a single negative glue. DNA Comp. Mol. Prog. LNCS 6937, 175–189 (2011)
Reif, J.H., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. Theor. Comput. Sci. 412(17), 1592–1605 (2011)
Schweller, R., Sherman, M.: Fuel efficient computation in passive self-assembly. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’13, pp. 1513–1525. SIAM (2013)
Scott, A., Stege, U., van Rooij, I.: Minesweeper may not be np-complete but is hard nonetheless. Math. Intell. 33(4), 5–17 (2011)
Sheffer, H.M.: A set of five independent postulates for Boolean algebras, with application to logical constants. Trans. Am. Math. Soc. 14(4), 481–488 (1913)
Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Berlin (1999)
Winfree, E.: Algorithmic Self-assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)
Yang, J., Ma, J., Liu, S., Zhang, C.: A molecular cryptography model based on structures of DNA self-assembly. Chin. Sci. Bull. 59(11), 1192–1198 (2014). https://doi.org/10.1007/s11434-014-0170-4
Acknowledgements
This research was supported in part by National Science Foundation Grant CCF-1817602.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cantu, A.A., Luchsinger, A., Schweller, R. et al. Covert Computation in Self-Assembled Circuits. Algorithmica 83, 531–552 (2021). https://doi.org/10.1007/s00453-020-00764-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-020-00764-w