2 research outputs found
On One-way Functions and Kolmogorov Complexity
We prove that the equivalence of two fundamental problems in the theory of
computing. For every polynomial , the
following are equivalent:
- One-way functions exists (which in turn is equivalent to the existence of
secure private-key encryption schemes, digital signatures, pseudorandom
generators, pseudorandom functions, commitment schemes, and more);
- -time bounded Kolmogorov Complexity, , is mildly hard-on-average
(i.e., there exists a polynomial such that no PPT algorithm can
compute , for more than a fraction of -bit strings).
In doing so, we present the first natural, and well-studied, computational
problem characterizing the feasibility of the central private-key primitives
and protocols in Cryptography
Ker-I Ko and the study of resource-bounded Kolmogorov complexity
Ker-I Ko was among the first people to recognize the importance of resource-bounded Kolmogorov complexity as a tool for better understanding the structure of complexity classes. In this brief informal reminiscence, I review the milieu of the early 1980’s that caused an up-welling of interest in resource-bounded Kolmogorov complexity, and then I discuss some more recent work that sheds additional light on the questions related to Kolmogorov complexity that Ko grappled with in the 1980’s and 1990’s.
In particular, I include a detailed discussion of Ko’s work on the question of whether it is NP-hard to determine the time-bounded Kolmogorov complexity of a given string. This problem is closely connected with the Minimum Circuit Size Problem (MCSP), which is central to several contemporary investigations in computational complexity theory.Peer reviewe