On One-way Functions and Kolmogorov Complexity

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial
$t(n)\geq(1+\varepsilon)n,\ \varepsilon > 0$
, 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); •
$t$
-time bounded Kolmogorov Complexity,
$K^{t}$
, is mildly hard-on-average (i.e., there exists a polynomial
$p(n) > 0$
such that no PPT algorithm can compute
$K^{t}$
, for more than a
$1-\frac{1}{p(n)}$
fraction of
$n$
-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.