Some new results on binary polynomial multiplication

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CENK M., Hasan M. A.

JOURNAL OF CRYPTOGRAPHIC ENGINEERING, cilt.5, sa.4, ss.289-303, 2015 (ESCI) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 5 Sayı: 4
  • Basım Tarihi: 2015
  • Doi Numarası: 10.1007/s13389-015-0101-6
  • Dergi Adı: JOURNAL OF CRYPTOGRAPHIC ENGINEERING
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
  • Sayfa Sayıları: ss.289-303
  • Anahtar Kelimeler: Polynomial multiplication, Elliptic curve scalar multiplication, Binary fields, Karatsuba, Toom, Divide-and-conquer, KARATSUBA, 5-TERM, 6-TERM
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein's 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein's 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials over the finite field with four elements. Unlike Bernstein's algorithm, which has a linear delay complexity with respect to input size, the delay complexity of the new algorithm is logarithmic. The number of bit operations for the multiplication of polynomials over the finite field with four elements is also computed. Finally, all these new results are combined to obtain improved complexities.