Abstract
We prove several results relating injective one-way functions, time-bounded conditional Kolmogorov complexity, and time-bounded conditional entropy.
First we establish a connection between injective, strong and weak one-way functions and the expected value of the polynomial time-bounded Kolmogorov complexity, denoted here by E(K t(x|f(x))). These results are in both directions. More precisely, conditions on E(K t(x|f(x))) that imply that f is a weak one-way function, and properties of E(K t(x|f(x))) that are implied by the fact that f is a strong one-way function. In particular, we prove a separation result: based on the concept of time-bounded Kolmogorov complexity, we find an interval in which every function f is a necessarily weak but not a strong one-way function.
Then we propose an individual approach to injective one-way functions based on Kolmogorov complexity, defining Kolmogorov one-way functions and prove some relationships between the new proposal and the classical definition of one-way functions, showing that a Kolmogorov one-way function is also a deterministic one-way function. A relationship between Kolmogorov one-way functions and the conjecture of polynomial time symmetry of information is also proved.
Finally, we relate E(K t(x|f(x))) and two forms of time-bounded entropy, the unpredictable entropy H unp, in which “one-wayness” of a function can be easily expressed, and the Yao+ entropy, a measure based on compression/decompression schema in which only the decompressor is restricted to be time-bounded.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We are implicitly using the Linear Speedup Theorem, see [17].
When we say that ε(n) is \(1-\omega(\frac{1}{n})\), we mean that \((1-\varepsilon(n)) \in\omega(\frac{1}{n})\).
References
Barak, B., Shaltiel, R., Wigderson, A.: Computational analogues of entropy. In: Proceedings of International Conference on Random Structures and Algorithms, pp. 200–215 (2003)
Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13(4), 850–864 (1984)
Cachin, C.: Entropy Measures and Unconditional Security in Cryptography, 1st edn. ETH Series in Information Security and Cryptography. Hartung-Gorre, Konstanz (1997)
Chaitin, G.: On the length of programs for computing finite binary sequences. J. ACM 13(4), 547–569 (1966)
Cover, T., Thomas, J.: Elements of Information Theory. Wiley-Interscience, New York (1991)
Goldreich, O.: Foundations of Cryptography. Cambridge University Press, Cambridge (2001)
Goldwasser, S., Micali, S., Rivest, R.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM J. Comput. 17(2), 281–308 (1988)
Golshani, L., Pasha, E., Yari, G.: Some properties of Renyi entropy and Renyi entropy rate. Inf. Sci. 179, 2426–2433 (2009)
Impagliazzo, R., Levin, L., Luby, M.: Pseudo-random generation from one-way functions. In: Proceedings of Symposium on Theory of Computing’89, pp. 12–24. ACM, New York (1989)
Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography. In: Proceedings of Symposium on Foundations of Computer Science’89 (1989)
Jizba, P., Arimitsu, T.: The world according to Renyi: thermodynamics of multifractal systems. Ann. Phys. 312, 17 (2004)
Kolmogorov, A.: Three approaches to the quantitative definition of information. Probl. Inf. Transm. 1(1), 1–7 (1965)
Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Springer, Berlin (2008)
Longpré, L., Mocas, S.: Symmetry of information and one-way functions. Inf. Process. Lett. 46(2), 95–100 (1993)
Longpré, L., Watanabe, O.: On symmetry of information and polynomial time invertibility. Inf. Comput. 121(1), 14–22 (1995)
Lu, C., Reyzin, L.: Conditional computational entropy, or toward separating pseudoentropy from compressibility. In: Proceedings of EUROCRYPT’07, pp. 169–186 (2007)
Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)
Pinto, A.: Comparing notions of computational entropy. Theory Comput. Syst. 45, 944–962 (2009)
Renner, R., Wolf, S.: Simple and tight bounds for information reconciliation and privacy amplification. In: Advances in Cryptology—ASIACRYPT 2005. Lecture Notes in Computer Science, pp. 199–216. Springer, Berlin (2005)
Rompel, J.: One-way functions are necessary and sufficient for secure signatures. In: Proceedings of Symposium on Theory of Computing’90, pp. 387–394. ACM, New York (1990)
Solomonoff, R.: A formal theory of inductive inference, part I. Inf. Control 7(1), 1–22 (1964)
Yao, A.: Theory and applications of trapdoor functions (extended abstract). In: Proceedings of Symposium on Foundations of Computer Science, pp. 80–91 (1982)
Zvonkin, A., Levin, L.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematics Surveys 256, 83–124 (1970)
Acknowledgements
The authors thank to the anonymous reviewers for helpful comments.
The authors also thank EU FEDER and FCT projects CSI 2 (PTDC/EIA–CCO/099951/2008) and PEst-OE/EEI/LA0008/2011. They are also supported by grants of SQIG—Instituto de Telecomunicações and FCT grants SFRH/BD/33234/2007 and FRH/BPD/76231/2011.
Author information
Authors and Affiliations
Corresponding author
Additional information
A very preliminary version of this paper appeared in Proceedings of the 6th International Conference on Computability in Europe, Azores, Portugal, July 2010, pages 346–356.
Rights and permissions
About this article
Cite this article
Antunes, L., Matos, A., Pinto, A. et al. One-Way Functions Using Algorithmic and Classical Information Theories. Theory Comput Syst 52, 162–178 (2013). https://doi.org/10.1007/s00224-012-9418-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-012-9418-z