Two normal basis multiplication algorithms for GF(2n)

For software implementations, word-level normal basis multiplication algorithms utilize the full data-path of the processor, and hence are more efficient than the bit-level multiplication algorithm presented in the IEEE standard P1363–2000. In this paper, two word-level normal basis multiplication algorithms are proposed for GF(2
ⁿ
). The first algorithm is suitable for high complexity normal bases, while the second algorithm is fast for type-I optimal normal bases and low complexity normal bases. Theoretical analyses and experimental results both indicate that the presented algorithms are efficient in GF(2
²³³
), GF(2
²⁸³
), GF(2
⁴⁰⁹
), and GF(2
⁵⁷¹
), which are four of the five binary fields recommended by the National Institute of Standards and Technology (NIST) for the elliptic curve digital signature algorithm (ECDSA) applications.