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 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 Pell’s cubic. The number of operations involved in the test grows linearly with respect to the bit length $$\log _2(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the input integer n . The algorithm is deterministic for integers less than $$2^{36}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>36</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>