One-Way Functions Using Algorithmic and Classical Information Theories

Abstract

We prove several results relating injective one-way functions, timebounded 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 oneway 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.

Information Distances versus Entropy Metric

“…Then, relations between entropy based cryptographic security and Kolmogorov complexity based cryptographic security have been studied [22][23][24][25]. One-way functions have been studied on both time-bounded entropy and time-bounded Kolmogorov complexity [26]. However, the relationship between information distance and entropy has not been studied.…”

Kolmogorov One-Way Functions Revisited

“…Furthermore, we only consider injective one-way functions. We denote by Σ the set with 0 and 1, thus making Σ * the alphabet of all finite binary strings and Σ n the set of strings of size n. In the next definitions, we use the adapt the definitions presented in [1] and that were considered in [11].…”

“…Definition 1 (Weak one-way function (as in [11]). A function f : Σ * → Σ * is a weak one-way function (wowf) if the following conditions hold:…”

“…Here, we are interested in the connection between Kolmogorov complexity and the study of one-way functions, a line of work first considered in [10,11]. In these works, the Kolmogorov complexity of x given f (x), K t (x| f (x)), is studied as a measure of how hard it is to invert f (x) to obtain x.…”

“…In this paper, we show in Proposition 4 that the expected value approach cannot be used to fully characterize the class of strong one-way functions. Moreover, we provide a sufficient condition under which Kolmogorov one-way functions (as defined in [11]) are weak one-way functions.…”