• DOI:10.13016/M2KL6X-ZJYQ
• Corpus ID: 201107135

PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts

@article{Phatak2019PPTNL,
  title={PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts},
  author={Dhananjay S. Phatak and Alan T. Sherman and Steven D. Houston and Andrew Henry},
  journal={ArXiv},
  year={2019},
  volume={abs/1908.06964}
}

• D. Phatak, A. Sherman, +1 author Andrew Henry
• Published 20 August 2019
• Mathematics, Computer Science
• ArXiv

In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures.
In Part/Article 1, we start with the Baseline Primality Conjecture~(PBPC) which enables deterministic primality detection with a low complexity = O((log N)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N is available (which happens to be the case for an overwhelming majority = 11/12 = 91.67% of all odd integers). We then demonstrate Primality Lemma PL-1, which reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion. This Lemma, together with the Baseline Primality Conjecture enables a synergistic fusion of Miller-Rabin iterations and our method(s), resulting in hybrid algorithms that are substantially better than their components. Next, we illustrate how the requirement of an explicit value of a QNR can be circumvented by using relations of the form: Polynomial(x) mod N = 0 ; whose solutions implicitly specify Non Residues modulo-N. We then develop a method to derive low-degree canonical polynomials that together guarantee implicit Non Residues modulo-N ; which along with the Generalized Primality Conjectures enable algorithms that achieve a worst case deterministic polynomial complexity = O( (log N)^3 polylog(log N)) ; unconditionally ; for any/all values of N.
In Part/Article 2 , we present substantial experimental data that corroborate all the conjectures. No counter example has been found.
Finally in Part/Article 3, we present analytic proof(s) of the Baseline Primality Conjecture that we have been able to complete for some special cases.