Physical One-Way Functions for Decentralized Consensus Via Proof of Physical Work

Abstract

Decentralized consensus on the state of the Bitcoin blockchain is ensured by proof of work. It relies on digital one-way functions and is associated with an enormous environmental impact. This paper conceptualizes a physical one-way function that aims to transform a digital, electricity-consuming consensus mechanism into a physical process. Boundary conditions for the security requirements are established and discussed as well as experimentally investigated for a specific setup based on printing and optical analysis of pigment-carrier composites. In the context of the applied methods, this setup promises to be mathematically unclonable, steady, reproducible, collision resistant and non-invertible and illustrates the feasibility of a physical one-way function. Based on this, a framework for proof of physical work is conceptualized, which has the potential of a drastically lower CO₂ footprint. This work initiates a progressive, interdisciplinary field of research and demands further investigations with regards to alternative setups, security definitions and strategies for challenging them.

Keywords: cryptography; decentralized consensus; physical one‐way function; proof of work.

Figure 1

Illustration of a digital one‐way function (d‐OWF, top), physical unclonable function (PUF, middle) and physical one‐way function (p‐OWF, bottom). Dashed lines mark unwanted paths to either inverse the function or to bypass the physical aspect of a p‐OWF with a digital twin.

Figure 2

Measured optical signals at multiple wavelengths for both sample set 4L‐P81‐$\mathcal{E}$ (left) and $\{$6L‐PM90‐$\mathcal{E}$, 6L‐PM90‐$\mathcal{R}\}$ (right). The standard deviation from the triplicate measurement is given above each plot. Samples are colored based on the total amount of cyan magenta and yellow (perceptible color). For six layers, all samples appear grayish.

Figure 3

Illustration of the strictly upper triangular matrix for each sample index pair (i, j) (top left) and collision events for each of the three analyzed permutation sets (top right, bottom left and right). The average collisions per sample Z and applied safety factor S are shown as annotations in the respective plots.

Figure 4

Evolution of collision probability Z with increasing flat noise factor S, which favors the overlap of confidence intervals of spectra pairs during iterative comparison.

Figure 5

Bar chart showing proportion of variance explained by the first five principle components (PCs) in all three analyzed datasets. Scattered points indicate the cumulative variance explained by the transformed input data.

Figure 6

Score plots of transformed spectroscopic data by principle component analysis (PCA). Top row shows scoreplot of first and second principal component (PC), bottom row shows scoreplots of second and third PC. Each column is associated with one dataset. In subfigure a and d, each sample is represented by a point colored with the samples perceptible color. In subfigure b,c,e and f each sample is colored after the pigment type of the last layer.

Figure 7

Box plot comparing model success rate of datasets 6L‐PM90‐$\mathcal{R}$ a) and 6L‐PM90‐$\mathcal{E}$ b) for three different sizes of test data after 100 iterations of randomized train test splits. Each model is shown separately. White crosses mark the average success rate when randomly guessing the corresponding pigment type(s).

Figure 8

Schematic explanation of hash parsing and conversion into RGB colors saved in ${\mathbb{C}}_{\mathrm{B}}$ (block colors) / ${\mathbb{C}}_{\mathrm{T}}$ (target colors) and light intensity values for four lamps saved in tuple ${\mathbb{I}}_{\mathrm{T}}$.

Figure 9

Visualization of the printing procedure using computer‐generated print pages with individual color layers for extinction (left) and reflection (right). All dimensions are in mm.

Figure 10

Overview of samples included in permutation set 4L‐P81‐$\mathcal{E}$, 6L‐PM90‐$\mathcal{E}$, 6L‐PM90‐$\mathcal{R}$. Color sequence (print order) is depicted from left to right.