International Association
for Cryptologic Research

In this paper, we present a new bit-parallel Montgomery multiplier for $GF(2^m)$ generated with irreducible trinomials. A newly proposed divide-and-conquer approach is applied to simplify the polynomial multiplication while the Montgomery squaring is induced to simplify the modular reduction. Meanwhile, this design effectively exploits the overlapped elements in squaring and reduction operation to reduce the space complexity. As a result, the proposed multiplier has about 25\\% reduced space complexity compared with previous multipliers, with a slight increase of time complexity. Among five binary fields recommended by NIST for the ECDSA (Elliptic Curve Digital Signature Algorithm), there exist two fields, i.e., $GF(2^{409})$, $GF(2^{233})$,

defined by trinomials. For these two fields, we show that our proposal outperforms the previous best known results if the space and time complexities are both considered.