Subquadratic space complexity Gaussian normal basis multipliers overGF(2^m) based on Dickson–Karatsuba decomposition

Abstract

Gaussian normal basis (GNB) of the even-type is popularly used in elliptic curve cryptosystems. Efficient GNB multipliers could be realised by Toeplitz matrix-vector decomposition to realise subquadratic space complexity architectures. In this study, Dickson polynomial representation is proposed as an alternative way to represent an GNB of characteristic two. The authors have derived a novel recursive Dickson-Karatsuba decomposition to achieve a subquadratic space-complexity parallel GNB multiplier. By theoretical analysis, it is shown that the proposed subquadratic multiplier saves about 50% bit-multiplications compared with the corresponding subquadratic GNB multiplication using Toeplitz matrix-vector product approach.

Efficient and low‐complexity hardware architecture of Gaussian normal basis multiplication over GF(2 m ) for elliptic curve cryptosystems

“…However, in [20] the number of clock cycles is relatively reduced. In [24] Dickson polynomial representation as an alternative way to represent the GNB of characteristic 2 is proposed. Here a novel recursive Dickson-Karatsuba decomposition to achieve a subquadratic space-complexity parallel GNB multiplier is presented.…”

“…There are several architectures of the normal basis and Gaussian normal basis (GNB) multiplication presented in recent years [2–24]. For example, in [4] a novel scalable multiplication algorithm is presented for a GNB using Hankel matrix–vector representation.…”