Generalized one-way function and its application

TL;DR

This work tackles the existence and construction of one-way functions that remain secure against unlimited computational resources by introducing a generalized one-way function built from a provable quantum one-way function via a random mapping rule. It maps inputs through a randomly chosen $f_k$ from the set $\mathcal{K}_m$, producing output bits with probabilities $P(y_i=0)=\frac{1+\cos\left(\frac{2\pi}{m} f_k(\vec{x}_i)\right)}{2}$ and $P(y_i=1)=\frac{1-\cos\left(\frac{2\pi}{m} f_k(\vec{x}_i)\right)}{2}$, and formalizes one-wayness using density-matrix arguments and the posterior indistinguishability ${\rm Pr}[\vec{X}|\vec{Y}] \sim {\rm Pr}[\vec{X}]$ for large $m$. This enables Probability Key Distribution (PKD), a fully classical-data-processing protocol that achieves unconditional security by exchanging classical strings and applying an $n\times(n\log_{2} m)$ input, a Toeplitz-matrix-based masking $\mathbf{H}$, and error-correcting/privacy-amplification steps, with a secret-key bound $\ell \le n - \lambda - \log_2\frac{2}{\varepsilon_{\rm cor}} - 2\log_2\frac{3}{2\varepsilon_{\rm sec}}$. The approach leverages quantum randomness and the density-matrix framework to counter unlimited adversaries, achieving a raw-key error rate of about $18.2\%$ and enabling unconditionally secure encryption and signatures when combined with one-time pads or hashing-based schemes. Overall, the work links information-theoretic randomness, quantum state characterization, and classical processing to broaden unconditional cryptographic primitives and practical deployment.

Abstract

One-way functions are fundamental to classical cryptography and their existence remains a longstanding problem in computational complexity theory. Recently, a provable quantum one-way function has been identified, which maintains its one-wayness even with unlimited computational resources. Here, we extend the mathematical definition of functions to construct a generalized one-way function by virtually measuring the qubit of provable quantum one-way function and randomly assigning the corresponding measurement outcomes with identical probability. Remarkably, using this generalized one-way function, we have developed an unconditionally secure key distribution protocol based solely on classical data processing, which can then utilized for secure encryption and signature. Our work highlights the importance of information in characterizing quantum systems and the physical significance of the density matrix. We demonstrate that probability theory and randomness are effective tools for countering adversaries with unlimited computational capabilities.

Figures (3)

• Figure 1: Pure state and mixed state of a two-dimensional quantum system. (A) Bloch sphere representation of a density matirx. A superposition state, when measured in the $\mathcal{X}$ basis--which is realized by a Hadamard gate $H$ followed by a $\mathcal{Z}$ basis measurement $M_{z}$--will collapse randomly into one of the eigenstates of the measurement operator with a certain probability. (B) A quantum system can be represented as a pure state ${\lvert\phi(\frac{\pi}{2},\varphi)\rangle}$ on the periphery of the $\textrm{x}-\textrm{y}$ circle if one has the phase information $\varphi$ and $\theta=\frac{\pi}{2}$. (C) A quantum system can only be represented as the maximally mixed state $\hat{\textbf{I}}/2$ if $\theta=\frac{\theta}{2}$ and no phase information $\varphi$ is available. (D) The random mapping rule $f_{k}:j\rightarrow j'$. There are $m!$ (with $m=3$ as an example) possible random mappings from a finite domain of size $m$ to a finite codomain of size $m$.
• Figure 2: Comparison of several one-way functions. (A) Traditional one-way function based on the computational complexity assumptions. The forward computation is straightforward, while the reverse problem is very difficult. Once the trapdoor information is acquired, the solution becomes easily accessible. (B) Provable quantum one-way function. The one-wayness is rigorously maintained even with unlimited computational resources. The forward and backward quantum states are entirely distinct, as the quantum state of a system evolves with the information acquired. (C) Generalized one-way function. (D) Provable quantum one-way function combined with the $\mathcal{X}$ basis measurement. For each input $\vec{x}$ (where the random mapping rule transforms $\vec{x}$ to $\vec{x}'$), there is a probability $p(\vec{x}')$ of obtaining output 0 and $1 - p(\vec{x}')$ of obtaining output 1.
• Figure 3: Schematic diagram of PKD protocol. In each session, Alice uses the generalized one-way function to generate a bit string $\vec{Y}$ based on the input random bit string $\vec{X}$. Alice and Bob exchange the random bit string $\vec{X}$ using a data string $\vec{D}$, which is produced from the pre-shared bit string $\vec{K}$ and a Toeplitz matrix $\textbf{H}$. Alice then announces the XOR results $\vec{Y} \oplus \vec{R}_{a}$, where $\vec{R}_{a}$ is her raw key. Bob guesses the bit string $\vec{Y}$ to obtain $\vec{Z}$ based on the received $\vec{X}$. To deduce Alice's raw key $\vec{R}_{a}$, Bob computes his own raw key $\vec{R}_{b}$ by performing an XOR operation between $\vec{Y} \oplus \vec{R}_{a}$ and $\vec{Z}$. Finally, Alice and Bob apply error correction and privacy amplification to derive an identical and secret key bit string $\vec{\Gamma}$.