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Novel performant primality test on a Pell's cubic

Luca Di Domenico, Nadir Murru

Abstract

Primality testing is an especially useful topic for public-key cryptography. In this paper, a novel primality test algorithm based on the Pell's cubic will be introduced, and its necessary primality conditions will be proved using three integer sequences connected to operations applied in the projectivization of the Pell's cubic. The number of operations involved in the test grows linearly with respect to the bit-length $\log_2(n)$ of the input integer $n$. The algorithm is deterministic for integers less than $2^{32}$.

Novel performant primality test on a Pell's cubic

Abstract

Primality testing is an especially useful topic for public-key cryptography. In this paper, a novel primality test algorithm based on the Pell's cubic will be introduced, and its necessary primality conditions will be proved using three integer sequences connected to operations applied in the projectivization of the Pell's cubic. The number of operations involved in the test grows linearly with respect to the bit-length of the input integer . The algorithm is deterministic for integers less than .

Paper Structure

This paper contains 11 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. On the Pell's cubic
  3. The projectivization of the Pell's cubic over a finite field
  4. Isomorphisms regarding the projectivization of the Pell's cubic
  5. Some integer sequences
  6. Binet's formulae for the Pell-X, Pell-Y and Pell-Z sequences
  7. A necessary condition for primality
  8. The primality test
  9. Choosing the parameters
  10. The Square-and-Multiply operation
  11. The novel primality test

Key Result

Lemma 2

For any prime $p>3$ and for $r \not= 0, -1$, the element $[1:1:0]$ lies in $(\mathbb{P}_p, \otimes_r)$.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • Definition 7
  • Proposition 8
  • Proposition 9
  • Theorem 10
  • ...and 10 more