PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts
This work addresses primality testing, a fundamental problem in number theory and cryptography, with potential applications in secure communications, but it appears incremental as it builds on existing methods like Miller-Rabin and Euler-Criterion.
The authors tackled the problem of deterministic primality testing by introducing new algorithms that achieve low complexity, specifically O((log N)^2) for most odd integers when an explicit quadratic non-residue is available, and worst-case O((log N)^3 polylog(log N)) unconditionally using implicit non-residues.
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.