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U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes

Bowman Hall

TL;DR

The study addresses whether Arnault-type composites engineered to pass Miller–Rabin tests through base 11 can also evade the strong Lucas component of Baillie–PSW. It constructs 3-prime Arnault composites targeting ~350-bit numbers and measures Lucas degeneracy with the collapse metric delta(n,D) = log_2 n − log_2(U_d mod n) across discriminants with (D/n) = -1. Across 200 samples, all composites fail the strong Lucas test; the mean delta is 1.61 bits, with a median of 1.0 and a maximum of 8, and 26% show no collapse at all, indicating near independence between MR resistance and Lucas behavior. The results bolster the robustness of Baillie–PSW, showing that MR evasion via Arnault constructs does not translate into Lucas evasion, and suggest that exploring beyond Arnault constructions may be required to approach Lucas degeneracy.

Abstract

We present a computational study of 200 composite integers of approximately 350 bits, engineered using the Arnault framework to pass all Miller-Rabin tests up to base 11. Generated at a rate of approximately 20 per hour from a high-throughput construction process producing ~7,700 Carmichael numbers per minute, all samples fail the strong Lucas probable prime test. We introduce the U-bit collapse metric delta = log_2(n) - log_2(U_d mod n) to quantify deviation from the expected uniform distribution of Lucas sequence terms. Analysis reveals minimal collapse values: mean delta = 1.61 bits, median delta = 1.0 bits, maximum delta = 8 bits, with 26% showing no measurable collapse. We analyze correlations with prime residue classes modulo 35, Arnault construction parameters (k,M), and composite bit-sizes. Our results demonstrate that composites engineered for Miller-Rabin resistance exhibit negligible Lucas sequence degeneracy, providing strong empirical evidence for the statistical independence of these two primality test components and supporting the continued robustness of Baillie-PSW-type tests.

U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes

TL;DR

The study addresses whether Arnault-type composites engineered to pass Miller–Rabin tests through base 11 can also evade the strong Lucas component of Baillie–PSW. It constructs 3-prime Arnault composites targeting ~350-bit numbers and measures Lucas degeneracy with the collapse metric delta(n,D) = log_2 n − log_2(U_d mod n) across discriminants with (D/n) = -1. Across 200 samples, all composites fail the strong Lucas test; the mean delta is 1.61 bits, with a median of 1.0 and a maximum of 8, and 26% show no collapse at all, indicating near independence between MR resistance and Lucas behavior. The results bolster the robustness of Baillie–PSW, showing that MR evasion via Arnault constructs does not translate into Lucas evasion, and suggest that exploring beyond Arnault constructions may be required to approach Lucas degeneracy.

Abstract

We present a computational study of 200 composite integers of approximately 350 bits, engineered using the Arnault framework to pass all Miller-Rabin tests up to base 11. Generated at a rate of approximately 20 per hour from a high-throughput construction process producing ~7,700 Carmichael numbers per minute, all samples fail the strong Lucas probable prime test. We introduce the U-bit collapse metric delta = log_2(n) - log_2(U_d mod n) to quantify deviation from the expected uniform distribution of Lucas sequence terms. Analysis reveals minimal collapse values: mean delta = 1.61 bits, median delta = 1.0 bits, maximum delta = 8 bits, with 26% showing no measurable collapse. We analyze correlations with prime residue classes modulo 35, Arnault construction parameters (k,M), and composite bit-sizes. Our results demonstrate that composites engineered for Miller-Rabin resistance exhibit negligible Lucas sequence degeneracy, providing strong empirical evidence for the statistical independence of these two primality test components and supporting the continued robustness of Baillie-PSW-type tests.
Paper Structure (15 sections, 2 equations, 3 figures)

This paper contains 15 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Strong Lucas Probable Prime Test
  3. Methodology
  4. Composite Construction
  5. U-Bit Collapse Metric
  6. Experimental Results
  7. Sample Overview
  8. U-Bit Collapse Analysis
  9. Residue Modulo 35 Analysis
  10. Construction Parameters
  11. Discussion
  12. Residue Class Concentration
  13. Implications for Baillie--PSW
  14. Open Questions
  15. Conclusion

Figures (3)

  • Figure 1: Distribution of U-bit collapse $\delta$ across 200 measured composites. The mean collapse of 1.61 bits is negligible compared to the $\approx$350 bits required for a strong Lucas pseudoprime, demonstrating that Miller--Rabin resistance does not translate to Lucas resistance.
  • Figure 2: Frequency of the 15 most common prime residue patterns (mod 35) in the dataset. No single pattern dominates, with the most frequent appearing in only 4.5% of composites. This diversity suggests multiple residue configurations can satisfy the Miller--Rabin base-11 resistance requirements.
  • Figure 3: Scatter plots showing U-bit collapse $\delta$ versus Arnault construction parameters $k$ and $M$, and composite bit-size. Negligible correlations ($|\rho| < 0.09$ for all) indicate that parameters optimized for Miller--Rabin resistance have no meaningful influence on Lucas test performance.

Theorems & Definitions (1)

  • Definition 1