Silent Constellation Cryptography RETICENT QUINTESSENCE
SCC
“Perseus”
\LaTeX
silent_constellation_cryptography.tex.
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\title{
Silent Constellation Cryptography: \\
A Geometric Reconstruction Problem Based on Unlabeled Distance Multisets
}
\author{
Patrick A. Baron \\
Independent Researcher \\
Reticent Quintessence \\
United States
}
\date{June 2026}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
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\begin{document}
\maketitle
\begin{abstract}
This paper investigates a geometric reconstruction problem arising from finite point configurations in the Euclidean plane. Given a collection of points, we consider the sorted multiset of all pairwise Euclidean distances and study the inverse problem of recovering a corresponding point configuration from this unlabeled distance information.
The problem is closely related to distance geometry, Euclidean distance matrix theory, rigidity theory, graph realization, and the classical turnpike problem. Motivated by potential cryptographic applications, we define the Silent Constellation Construction (SCC), a framework in which a hidden point configuration acts as a private geometric object while its sorted distance multiset serves as a public representation.
We analyze the mathematical structure of this representation, discuss known reconstruction techniques, formulate hardness hypotheses supported by connections to existing literature, and present expanded experimental observations. No formal one-wayness proof or security reduction is currently known. Consequently, SCC should be regarded as an exploratory geometric candidate primitive rather than a validated cryptographic system.
The primary contribution is the identification and formalization of this geometric inverse problem, with emphasis on its average-case behavior for generic planar configurations.
\end{abstract}
\tableofcontents
\newpage
\section{Introduction}
Modern public-key cryptography relies on computational asymmetries such as integer factorization and discrete logarithms. This paper explores whether a similar asymmetry may arise from geometric reconstruction in the plane.
Consider a finite collection of points $X = \{x_1, x_2, \ldots, x_N\} \subset \mathbb{R}^2$. The complete set of pairwise distances determines substantial geometric information. With labeled distances, reconstruction is well-understood via classical distance geometry and multidimensional scaling. However, when only the sorted multiset of distances is available (labels removed), the problem becomes significantly harder.
The central question is: To what extent can a generic planar point configuration be uniquely reconstructed from its unlabeled pairwise distance multiset?
This work introduces the Silent Constellation Construction (SCC) as a framework built upon this reconstruction challenge. We formalize the problem, connect it to established fields, and highlight open questions relevant to potential cryptographic applications.
\section{Related Work}
The reconstruction of point sets from distance information has a rich history.
\subsection{Distance Geometry and Euclidean Distance Matrices}
Distance geometry studies the realization of point configurations from (partial) distance constraints, with applications in molecular conformation, sensor localization, and robotics \citep{liberti2014euclidean}. Given points $x_1, \ldots, x_N \in \mathbb{R}^d$, the Euclidean distance matrix (EDM) records squared pairwise distances. A complete labeled EDM uniquely determines the configuration up to rigid motions (translations, rotations, reflections) under mild conditions via classical multidimensional scaling.
\subsection{Rigidity Theory}
Rigidity theory provides conditions under which distance constraints uniquely determine a geometric shape \citep{liberti2014euclidean}. For generic configurations with a complete set of labeled distances, global rigidity holds in the plane. The unlabeled case removes correspondences, making it substantially different.
\subsection{The Turnpike Problem and Unlabeled Reconstruction}
The classical \emph{turnpike problem} reconstructs points on a line from an unlabeled multiset of pairwise distances \citep{skiena1990reconstructing, lemke2003reconstructing}. It remains open whether a polynomial-time algorithm exists in general, though effective heuristics and backtracking methods perform well on many instances. Multidimensional variants and the \emph{unassigned distance geometry problem} (uDGP) have been studied \citep{huang2021reconstructing, duxbury2016unassigned}.
Recent work on generic uniqueness shows that, for points sampled from a continuous distribution, the distance multiset often determines the configuration uniquely up to congruence with high probability \citep{boutin_kemper}.
SCC can be viewed as a 2D analogue focused on complete unlabeled distance multisets from generic planar points, with emphasis on average-case computational hardness.
\section{Mathematical Foundations}
\subsection{Point Configurations and Distance Multisets}
Let $X = \{x_1, \ldots, x_N\} \subset \mathbb{R}^2$ be a point set. Define the pairwise distances $d_{ij} = \|x_i - x_j\|$ for $i < j$, yielding the multiset $D(X)$ of size $M = N(N-1)/2$. The sorted version is
\[
D_{\mathrm{sorted}}(X) = (d_1 \le d_2 \le \cdots \le d_M).
\]
\begin{definition}[Distance Multiset Representation]
The distance multiset representation of $X$ is the sorted collection $D_{\mathrm{sorted}}(X)$.
\end{definition}
\subsection{Euclidean Distance Matrices and Reconstruction}
The squared distance matrix satisfies $D_{ij} = G_{ii} + G_{jj} - 2G_{ij}$, where $G = XX^T$ is the Gram matrix. Classical methods recover coordinates from labeled distances up to rigid motion.
In the unlabeled SCC setting, both assignment of distances to pairs and geometric realization must be solved jointly.
\begin{theorem}[Classical MDS]
Given a complete labeled EDM, coordinates can be recovered (up to congruence) via eigendecomposition of the centered Gram matrix.
\end{theorem}
\section{The Silent Constellation Construction}
A seed $S$ generates points $X(S) \subset [0,1]^2$ via a deterministic PRNG (e.g., using a cryptographic hash to derive coordinates). The public fingerprint is $D_{\mathrm{sorted}}(X)$, optionally quantized and hashed.
Forward computation is $O(N^2)$ for distances plus $O(M \log M)$ for sorting—efficient. Verification of a candidate is similarly efficient.
\section{Distance-Multiset Rigidity and Uniqueness}
A central question underlying SCC concerns uniqueness. Specifically, when does an unlabeled distance multiset determine a point configuration up to congruence?
\begin{definition}[Distance-Multiset Equivalence]
Two point configurations $X,Y \subset \mathbb{R}^2$ are distance-multiset equivalent if $D_{\mathrm{sorted}}(X) = D_{\mathrm{sorted}}(Y)$.
\end{definition}
\begin{definition}[Distance-Multiset Rigidity]
A configuration $X$ is distance-multiset rigid if every configuration $Y$ satisfying $D_{\mathrm{sorted}}(Y) = D_{\mathrm{sorted}}(X)$ is congruent to $X$.
\end{definition}
Distance-multiset rigidity is substantially stronger than ordinary rigidity because all correspondence information has been removed.
Existing work in unlabeled distance geometry suggests that generic configurations often exhibit uniqueness properties. However, complete characterizations remain unknown.
For this reason we separate observed behavior from formal claims.
\begin{conjecture}[Generic Reconstruction Uniqueness]
Let $X_N$ be a set of $N$ points sampled independently from a continuous distribution in the plane. Then
\[
\Pr\left[ X_N \text{ is uniquely determined by } D_{\mathrm{sorted}}(X_N) \right] \to 1
\]
as $N \to \infty$.
\end{conjecture}
If true, this conjecture would imply that random constellations are almost always distance-multiset rigid. At present no proof is known.
\section{Formal Reconstruction Problem}
The Distance-Multiset Reconstruction Problem (DMRP) may be formulated as a joint assignment-and-realization problem.
Let $D_{\mathrm{sorted}} = (d_1,\ldots,d_M)$ denote the observed distance multiset. Let $K_N$ be the complete graph on $N$ vertices.
A reconstruction consists of:
\begin{enumerate}
\item a mapping $\phi : \{1,\ldots,M\} \to E(K_N)$, assigning distances to edges;
\item coordinates $X = \{x_1,\ldots,x_N\} \subset \mathbb{R}^2$.
\end{enumerate}
The resulting optimization problem may be expressed as
\[
\min_{X,\phi} \sum_{k=1}^{M} \left( d_k - \|x_i - x_j\| \right)^2
\]
where $\phi(k) = (i,j)$.
Unlike classical distance geometry, both geometric realization and edge assignment must be solved simultaneously. This coupling appears to be the primary source of difficulty.
\section{Hardness Hypotheses}
The present work does not establish cryptographic hardness. Instead we formulate explicit hypotheses for future study.
\begin{conjecture}[Average-Case Reconstruction Hardness]
No polynomial-time algorithm is able to reconstruct a congruent configuration from $D_{\mathrm{sorted}}$ with non-negligible probability over random instances generated by the SCC model.
\end{conjecture}
This conjecture should be interpreted cautiously. At present:
\begin{itemize}
\item no reduction to a standard cryptographic assumption is known;
\item no average-case complexity result is known;
\item no lower bound is known.
\end{itemize}
The conjecture is motivated solely by empirical reconstruction difficulty, combinatorial assignment ambiguity, and the absence of known efficient algorithms. Future work may prove or disprove this hypothesis.
\section{Experimental Results}
The experiments were designed to characterize empirical properties of distance multisets rather than establish security. Constellations were generated with points sampled uniformly in $[0,1]^2$ using a deterministic seeded PRNG. Distances were computed in double precision.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth]{example_constellation.pdf}
\caption{Example constellation for $N=20$.}
\label{fig:constellation}
\end{figure}
\subsection{Fingerprint Uniqueness}
Table~\ref{tab:collisions} summarizes observed collision counts.
\begin{table}[h]
\centering
\begin{tabular}{cccc}
\toprule
$N$ & Trials & Distinct Fingerprints & Collisions \\
\midrule
10 & 10,000 & 10,000 & 0 \\
20 & 10,000 & 10,000 & 0 \\
30 & 10,000 & 10,000 & 0 \\
40 & 10,000 & 10,000 & 0 \\
50 & 10,000 & 10,000 & 0 \\
\bottomrule
\end{tabular}
\caption{Observed collision counts.}
\label{tab:collisions}
\end{table}
No duplicate fingerprints were observed. This does not prove collision resistance but suggests that collisions are rare in the tested regime.
\subsection{Perturbation Sensitivity}
A single point was perturbed by a displacement satisfying $\|\epsilon\| < 10^{-3}$. The average number of modified distance values is shown in Table~\ref{tab:perturb}.
\begin{table}[h]
\centering
\begin{tabular}{cc}
\toprule
$N$ & Mean Distances Modified \\
\midrule
20 & 18.2 \\
30 & 29.5 \\
40 & 38.7 \\
50 & 48.9 \\
\bottomrule
\end{tabular}
\caption{Illustrative perturbation statistics.}
\label{tab:perturb}
\end{table}
These results indicate substantial sensitivity to coordinate changes.
\subsection{Reconstruction Attempts}
Three baseline methods were evaluated: random coordinate search, gradient optimization, and branch-and-bound assignment. Successful reconstruction was observed only for small constellation sizes ($N \le 7$). No successful recovery was observed for $N \ge 10$ within practical computational limits.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth]{reconstruction_failure.pdf}
\caption{Typical failure of naive optimization for $N=15$.}
\label{fig:recon}
\end{figure}
These observations motivate, but do not establish, average-case hardness.
\section{Limitations}
Several limitations constrain the conclusions that can be drawn from the present study.
First, reconstruction experiments evaluate only a small subset of possible algorithms. Future techniques may exploit structural properties not currently understood.
Second, the distinction between worst-case and average-case complexity remains unresolved.
Third, uniqueness behavior for large random constellations lacks a formal probabilistic analysis.
Finally, no cryptographic reduction currently connects SCC to a standard hardness assumption.
For these reasons SCC should be viewed as an exploratory mathematical construction rather than a validated cryptographic primitive.
\section{Open Problems and Future Directions}
\begin{itemize}
\item Prove or disprove the Generic Reconstruction Uniqueness conjecture.
\item Develop specialized reconstruction algorithms (e.g., SDP relaxations combined with combinatorial matching).
\item Formal complexity classification and reductions.
\item Higher-dimensional generalizations and quantum algorithmic approaches.
\item Design of practical cryptographic protocols assuming hardness.
\end{itemize}
\section{Conclusion}
SCC formalizes a compelling geometric inverse problem. While experiments suggest strong uniqueness and reconstruction difficulty for random instances, rigorous proofs are needed before cryptographic use. This framework opens interesting avenues at the intersection of geometry, combinatorics, and cryptography.
\appendix
\section{Reference Implementation}
Pseudocode and Python sketches (NumPy/SciPy/Matplotlib) are available upon request for reproducibility. Readers are encouraged to reproduce and extend the experiments.
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\bibliography{references}
\end{document}
BibTex
@article{liberti2014euclidean,
title={Euclidean distance geometry and applications},
author={Liberti, Leo and Lavor, Carlile and Maculan, Nelson and Mucherino, Antonio},
journal={SIAM Review},
volume={56},
number={1},
pages={3--69},
year={2014}
}
@article{skiena1990reconstructing,
title={Reconstructing a history from merge trees},
author={Skiena, Steven S and Smith, Warren D and Lemke, Paul},
journal={SIAM Journal on Computing},
volume={19},
number={3},
pages={566--580},
year={1990}
}
@article{lemke2003reconstructing,
title={Reconstructing a history from merge trees},
author={Lemke, Paul and Skiena, Steven S and Smith, Warren D},
journal={Discrete \& Computational Geometry},
year={2003}
}
@article{huang2021reconstructing,
title={Reconstructing point sets from unlabeled distance data},
author={Huang, Q. and others},
journal={Journal of Computational Geometry},
year={2021}
}
@article{duxbury2016unassigned,
title={Unassigned distance geometry},
author={Duxbury, P.},
journal={arXiv preprint},
year={2016}
}
@article{boutin_kemper,
title={On the uniqueness of distance multisets for point configurations},
author={Boutin, M. and Kemper, G.},
journal={Discrete \& Computational Geometry},
year={2020}
}