New Architectures for Digit-Level Single, Hybrid-Double, Hybrid-Triple Field Multiplications and Exponentiation Using Gaussian Normal Bases

Gaussian normal bases (GNBs) are special set of normal bases (NBs) which yield low complexity
$GF\left(2^{m}\right)$
arithmetic operations. In this paper, we present new architectures for the digit-level single, hybrid-double, and hybrid-triple multiplication of
$GF\left(2^{m}\right)$
elements based on the GNB representation for odd values of
$m > 1$
. The proposed fully-serial-in single multipliers perform multiplication of two field elements and offer high throughput when the data-path capacity for entering inputs is limited. The proposed hybrid-double and hybrid-triple digit-level GNB multipliers perform, respectively, two and three field multiplications using the same latency required for a single digit-level multiplier, at the expense of increased area. In addition, we present a new eight-ary field exponentiation architecture which does not require precomputed or stored intermediate values.