Kolmogorov One-Way Functions Revisited
Abstract
:1. Introduction
2. Preliminaries
2.1. One-Way Functions
- There is a deterministic polynomial time algorithm such that on input x, outputs , i.e.,
- For any polynomial , there is a polynomial such that, for any probabilistic t-time bounded algorithm and sufficiently large n,
- There is a deterministic polynomial time algorithm such that on input x, outputs , i.e., .
- For any polynomial , for every positive polynomial , for any probabilistic t-time bounded algorithm , and for sufficiently large n,
2.2. Kolmogorov Complexity
3. Results
3.1. Kolmogorov Characterization of One-Way Functions
3.2. Expected Value Approach
- given input x, its output is given either by the identity function or by g, which are both polynomial time computable;
- we can easily deterministically invert half of the inputs and so there is an algorithm, e.g., the first projection such that,
- on the other hand, since is a strong one-way function, for any polynomial and any algorithm ,
3.3. Kolmogorov -Characterization of One-Way Functions
4. Conclusions
- Develop a specialized Kolmogorov complexity approximation (similar to a zip compressor) that allows the analysis of this function towards the definition proposed in this paper. This would allow to give, not just for this particular function, but also to any other possible proposal for a one-way function, a practical security analysis regarding its one-wayness.
- The analyses driven in [14] proves that quantum one-way functions exist if and only if classical one-way functions exist and the techniques used to derive the (quantum) security of such functions are different form the classical ones. In the literature, there are several definitions of (bounded) quantum Kolmogorov complexity [15,16,17,18,19,20] . One can study the adaptation of the results presented in this paper to address directly the characterization of one way-functions that are quantum resilient and provide insight regarding some (quantum) one-way function candidates such as the one presented in [14].
Author Contributions
Acknowledgments
Conflicts of Interest
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Casal, F.; Rasga, J.; Souto, A. Kolmogorov One-Way Functions Revisited. Cryptography 2018, 2, 9. https://doi.org/10.3390/cryptography2020009
Casal F, Rasga J, Souto A. Kolmogorov One-Way Functions Revisited. Cryptography. 2018; 2(2):9. https://doi.org/10.3390/cryptography2020009
Chicago/Turabian StyleCasal, Filipe, João Rasga, and André Souto. 2018. "Kolmogorov One-Way Functions Revisited" Cryptography 2, no. 2: 9. https://doi.org/10.3390/cryptography2020009