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Non-detectable patterns hidden within sequences of bits

David Allen, Jose J La Luz, Guarionex Salivia, Jonathan Hardwick

TL;DR

The article investigates whether bit sequences derived from combinatorial symmetry—via iterated cones on duals of simplicial complexes—are detectable as non-random by standard randomness tests. By linking $f$-vectors and $h$-vectors through dual polytopes and applying the $g$-theorem, the authors show that these sequences can be classified as random by NIST despite underlying symmetric structure, and that the construction can yield arbitrarily many such sequences. They provide polynomial-time algorithms to transform between $f$- and $h$-vectors, analyze duality to simple polytopes, and demonstrate non-detectability through three controlled experiments with comprehensive appendices and code updates. The results highlight limitations of the NIST suite in recognizing certain symmetry-driven randomness, with implications for combinatorial cryptology and randomness testing. The work also clarifies how Pascal's triangle relationships and Dehn-Sommerville-type constraints influence the generated bitstreams.

Abstract

In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by con- verting a collection of numbers encoding certain combinatorial nu- merics from objects exhibiting symmetry in various dimensions. Using the algorithms first described in [1] we show that the NIST testing suite described in publication 800-22 does not detect these symmetries hidden within these sequences.

Non-detectable patterns hidden within sequences of bits

TL;DR

The article investigates whether bit sequences derived from combinatorial symmetry—via iterated cones on duals of simplicial complexes—are detectable as non-random by standard randomness tests. By linking -vectors and -vectors through dual polytopes and applying the -theorem, the authors show that these sequences can be classified as random by NIST despite underlying symmetric structure, and that the construction can yield arbitrarily many such sequences. They provide polynomial-time algorithms to transform between - and -vectors, analyze duality to simple polytopes, and demonstrate non-detectability through three controlled experiments with comprehensive appendices and code updates. The results highlight limitations of the NIST suite in recognizing certain symmetry-driven randomness, with implications for combinatorial cryptology and randomness testing. The work also clarifies how Pascal's triangle relationships and Dehn-Sommerville-type constraints influence the generated bitstreams.

Abstract

In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by con- verting a collection of numbers encoding certain combinatorial nu- merics from objects exhibiting symmetry in various dimensions. Using the algorithms first described in [1] we show that the NIST testing suite described in publication 800-22 does not detect these symmetries hidden within these sequences.
Paper Structure (9 sections, 11 theorems, 16 equations, 16 figures)

This paper contains 9 sections, 11 theorems, 16 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Simplicial Complexes and Iterated Cones
  4. Dual Polyhedra and Symmetry
  5. Bit Conversions
  6. Algorithms
  7. Results
  8. Appendix A
  9. Appendix B

Key Result

Proposition 2.1

For a graph $G$ such that $f_1(G) \neq 0$, let $C^{j}(G) = \mathcal{K}^{j+1}$. For $j$ sufficiently large, the complexes $\mathcal{K}^{j+1}$ are not dual to a polytope that satisfies the $g$-theorem.

Figures (16)

  • Figure 1: Sparklines showing the difference between expected NIST results if bit-stream is identified as random (left) vs non-random (right).
  • Figure 2: Approximate Entropy NIST test $p$-values after applying between 0 - 99 coning operations on the dual of the standard 3751 simplex with h-vector consisting of all 1's
  • Figure 3: Linear Complexity NIST test $p$-values after applying between 0 - 99 coning operations on a vector of length 3751 with a pattern of all 1's
  • Figure 4: Rank NIST test $p$-values after applying between 0 - 99 coning operations on a vector of length 3751 with a pattern of all 1's
  • Figure 5: Approximate Entropy NIST test $p$-values on a vectors of length between 3750 and 3849 with a pattern of all 1's, without applying coning operations
  • ...and 11 more figures

Theorems & Definitions (32)

  • Proposition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Definition 3.5
  • Lemma 3.6
  • Definition 3.7
  • Example 3.8
  • Lemma 3.9
  • ...and 22 more