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<p>The Polynomial Modular Number System (PMNS) is a non-positional number system designed for modular arithmetic. Its efficiency, both in software and hardware, has been demonstrated for integers commonly used in Elliptic Curve Cryptography&#xa0;[<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>]. In&#xa0;[<CitationRef CitationID="CR3">3</CitationRef>, <CitationRef CitationID="CR4">4</CitationRef>], the authors introduce specific prime forms that are particularly well-suited for PMNS arithmetic. In this work, we extend their results to a broader class of prime numbers. In practice, our approach yields performance that is competitive with, and in some cases superior to, Pseudo-Mersenne arithmetic. As a result, we expand the set of prime numbers that are well-suited for modular arithmetic. Furthermore, we contribute a database of proof of concept Elliptic Curves constructed with those primes that verify the Brainpool Standard.</p>

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Friendly primes for efficient modular arithmetic using the Polynomial Modular Number System

  • 期刊论文

  • 发表日期：

    2025年9月26日

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  • Fangan Yssouf Dosso,
  • Nadia El Mrabet,
  • Nicolas Méloni,
  • François Palma,
  • Pascal Véron

摘要

The Polynomial Modular Number System (PMNS) is a non-positional number system designed for modular arithmetic. Its efficiency, both in software and hardware, has been demonstrated for integers commonly used in Elliptic Curve Cryptography [1, 2]. In [3, 4], the authors introduce specific prime forms that are particularly well-suited for PMNS arithmetic. In this work, we extend their results to a broader class of prime numbers. In practice, our approach yields performance that is competitive with, and in some cases superior to, Pseudo-Mersenne arithmetic. As a result, we expand the set of prime numbers that are well-suited for modular arithmetic. Furthermore, we contribute a database of proof of concept Elliptic Curves constructed with those primes that verify the Brainpool Standard.