# Kolmogorov One-Way Functions Revisited

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. One-Way Functions

**Definition**

**1**

**.**A function $f:{\Sigma}^{*}\to {\Sigma}^{*}$ is aweak one-way function ($\mathsf{wowf}$)if the following conditions hold:

- There is a deterministic polynomial time algorithm $\mathsf{A}$ such that on input x, $\mathsf{A}$ outputs $f\left(x\right)$, i.e., $\mathsf{A}\left(x\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}f\left(x\right).$
- For any polynomial $t(\xb7)$, there is a polynomial $q(\xb7)$ such that, for any probabilistic t-time bounded algorithm $\mathsf{B}$ and sufficiently large n,$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f\left(\mathsf{B}\right(f\left(x\right),r,n\left)\right)\ne f\left(x\right)\right]>\frac{1}{q\left(n\right)}\phantom{\rule{5.0pt}{0ex}}\xb7$$

**Definition**

**2**

**.**A function $f:{\Sigma}^{*}\to {\Sigma}^{*}$ is astrong one-way function ($\mathsf{sowf}$)if the following conditions hold:

- There is a deterministic polynomial time algorithm $\mathsf{A}$ such that on input x, $\mathsf{A}$ outputs $f\left(x\right)$, i.e., $\mathsf{A}\left(x\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)$.
- For any polynomial $t(\xb7)$, for every positive polynomial $q(\xb7)$, for any probabilistic t-time bounded algorithm $\mathsf{B}$, and for sufficiently large n,$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f\left(\mathsf{B}\right(f\left(x\right),r,n\left)\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)\right]<\frac{1}{q\left(n\right)}\phantom{\rule{5.0pt}{0ex}}\xb7$$

#### 2.2. Kolmogorov Complexity

**Definition**

**3**(Kolmogorov complexity of a string)

**.**

**Theorem**

**1**

**([11]).**

**Theorem**

**2**

**([11]).**

## 3. Results

#### 3.1. Kolmogorov Characterization of One-Way Functions

**Definition**

**4**(Kolmogorov weak one-way function)

**.**

**Proposition**

**1.**

**Proof.**

**Definition**

**5**(Kolmogorov strong one-way function)

**.**

**Proposition**

**2.**

**Proof.**

#### 3.2. Expected Value Approach

**Definition**

**6**(First-bit secure function)

**.**

- 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 $\Pi =\lambda x,y,z.x$ such that,$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f(\Pi (f\left(x\right),r,n\left)\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)\right]\ge 1/2\phantom{\rule{5.0pt}{0ex}},$$
- on the other hand, since $g(\xb7)$ is a strong one-way function, for any polynomial $q(\xb7)$ and any algorithm $\mathsf{B}$,$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f\left(\mathsf{B}\right(g\left(x\right),r,n\left)\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}g\left(x\right)\right]<\frac{1}{q\left(n\right)}\phantom{\rule{0.166667em}{0ex}},$$$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f(\Pi (f\left(x\right),r,n\left)\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)\right]<\frac{1}{2}+\frac{1}{2q\left(n\right)}\phantom{\rule{0.166667em}{0ex}},$$$${\mathbb{P}}_{(x,r)\in {\Sigma}^{n}\times {\Sigma}^{t\left(n\right)}}\left[f(\Pi (f\left(x\right),r,n\left)\right)\ne f\left(x\right)\right]>1-\left(\frac{1}{2}+\frac{1}{2q\left(n\right)}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}p\left(n\right)\phantom{\rule{5.0pt}{0ex}},$$

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**4.**

#### 3.3. Kolmogorov $\Delta $-Characterization of One-Way Functions

**Definition**

**7**

**.**Let $t(\xb7)$ be a polynomial, $f:{\Sigma}^{n}\to {\Sigma}^{m}$ an injective and polynomial time computable function and $\delta (\xb7)$ a positive function. We say that an instance $x\in {\Sigma}^{n}$ is $(t,\delta )$-secure relative to a random string $r\in {\Sigma}^{t\left(n\right)}$ and f if

**Corollary**

**2**

**.**Let $t(\xb7)$ be a polynomial. If f is a $(t,\epsilon ,\delta )$-secure Kolmogorov one-way function such that

**Definition**

**8**

**.**Let $f:{\Sigma}^{*}\to {\Sigma}^{*}$ be an injective polynomial time computable function. We say that f is a Kolmogorov one-way function if for every polynomial $t(\xb7)$, positive integer c, sufficiently large n, and x of length n,

**Proposition**

**5.**

**Proof.**

**Corollary**

**3.**

**Definition**

**9**(Kolmogorov Δ-weak one-way function)

**.**

**Proposition**

**6.**

**Definition**

**10**(Kolmogorov Δ-strong one-way function)

**.**

**Corollary**

**4.**

## 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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**MDPI and ACS Style**

Casal, F.; Rasga, J.; Souto, A.
Kolmogorov One-Way Functions Revisited. *Cryptography* **2018**, *2*, 9.
https://doi.org/10.3390/cryptography2020009

**AMA Style**

Casal F, Rasga J, Souto A.
Kolmogorov One-Way Functions Revisited. *Cryptography*. 2018; 2(2):9.
https://doi.org/10.3390/cryptography2020009

**Chicago/Turabian Style**

Casal, Filipe, João Rasga, and André Souto.
2018. "Kolmogorov One-Way Functions Revisited" *Cryptography* 2, no. 2: 9.
https://doi.org/10.3390/cryptography2020009