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- W4285194476 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></inline-formula> . Such a system is defined by a tuple <inline-formula><tex-math notation=LaTeX>$(p, n, gamma, rho, E)$</tex-math></inline-formula> , where <inline-formula><tex-math notation=LaTeX>$p$</tex-math></inline-formula> , <inline-formula><tex-math notation=LaTeX>$n$</tex-math></inline-formula> , <inline-formula><tex-math notation=LaTeX>$gamma$</tex-math></inline-formula> and <inline-formula><tex-math notation=LaTeX>$rho$</tex-math></inline-formula> are positive integers, <inline-formula><tex-math notation=LaTeX>$Ein mathbb {Z}[X]$</tex-math></inline-formula> , with <inline-formula><tex-math notation=LaTeX>$E(gamma) equiv 0 pmod p$</tex-math></inline-formula> . In (Didier, <i>et al.</i> 2020) conditions required to build efficient AMNS (PMNS with <inline-formula><tex-math notation=LaTeX>$E(X)=X^{n} - lambda$</tex-math></inline-formula> , where <inline-formula><tex-math notation=LaTeX>$lambda in mathbb {Z}setminus lbrace 0rbrace$</tex-math></inline-formula> ) are provided. In this paper, we generalise their approach for any monic polynomial <inline-formula><tex-math notation=LaTeX>$Ein mathbb {Z}[X]$</tex-math></inline-formula> of degree <inline-formula><tex-math notation=LaTeX>$n$</tex-math></inline-formula> . We present new bounds and highlight a set of polynomials <inline-formula><tex-math notation=LaTeX>$E$</tex-math></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." @default.
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- W4285194476 date "2022-07-01" @default.
- W4285194476 modified "2023-10-09" @default.
- W4285194476 title "PMNS for Efficient Arithmetic and Small Memory Cost" @default.
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- W4285194476 doi "https://doi.org/10.1109/tetc.2022.3187786" @default.
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