PMNS for Efficient Arithmetic and Small Memory Cost

TL;DR: The Polynomial Modular Number System (PMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime.

Abstract: The Polynomial Modular Number System (PMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime <inline-formula><tex-math notation="LaTeX">$p$</tex-math><alternatives><mml:math><mml:mi>p</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq1-3187786.gif"/></alternatives></inline-formula>. Such a system is defined by a tuple <inline-formula><tex-math notation="LaTeX">$(p, n, \gamma, \rho, E)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq2-3187786.gif"/></alternatives></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$p$</tex-math><alternatives><mml:math><mml:mi>p</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq3-3187786.gif"/></alternatives></inline-formula>, <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq4-3187786.gif"/></alternatives></inline-formula>, <inline-formula><tex-math notation="LaTeX">$\gamma$</tex-math><alternatives><mml:math><mml:mi>γ</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq5-3187786.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math><alternatives><mml:math><mml:mi>ρ</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq6-3187786.gif"/></alternatives></inline-formula> are positive integers, <inline-formula><tex-math notation="LaTeX">$E\in \mathbb {Z}[X]$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>E</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq7-3187786.gif"/></alternatives></inline-formula>, with <inline-formula><tex-math notation="LaTeX">$E(\gamma) \equiv 0 \pmod p$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:mi>γ</mml:mi><mml:mo>)</mml:mo><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width="4.44443pt"/><mml:mo>(</mml:mo><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.277778em"/><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq8-3187786.gif"/></alternatives></inline-formula>. In (Didier, <italic>et al.</italic> 2020) conditions required to build efficient AMNS (PMNS with <inline-formula><tex-math notation="LaTeX">$E(X)=X^{n} - \lambda$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>-</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq9-3187786.gif"/></alternatives></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\lambda \in \mathbb {Z}\setminus \lbrace 0\rbrace$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>λ</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo>∖</mml:mo><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq10-3187786.gif"/></alternatives></inline-formula>) are provided. In this paper, we generalise their approach for any monic polynomial <inline-formula><tex-math notation="LaTeX">$E\in \mathbb {Z}[X]$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>E</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="veron-ieq11-3187786.gif"/></alternatives></inline-formula> of degree <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq12-3187786.gif"/></alternatives></inline-formula>. We present new bounds and highlight a set of polynomials <inline-formula><tex-math notation="LaTeX">$E$</tex-math><alternatives><mml:math><mml:mi>E</mml:mi></mml:math><inline-graphic xlink:href="veron-ieq13-3187786.gif"/></alternatives></inline-formula> for very efficient operations in the PMNS and low memory requirement. We also provide AMNS and PMNS modular multiplication implementations, for a prime of size 256 bits, in classic C. We also provide, for the same prime, the first implementation taking advantage of the SIMD <monospace>AVX512</monospace> instruction set. The <monospace>AVX512</monospace> PMNS is 72 % faster than its AMNS counterpart (classical C version). This version presents a more than 60 % speed-up in comparison with the state-of-the-art Montgomery-CIOS modular multiplication of the <monospace>GMP</monospace> library.