International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

Graham A. Jullien

Publications

Year
Venue
Title
2004
EPRINT
Parallel Montgomery Multiplication in $GF(2^k)$ using Trinomial Residue Arithmetic
We propose the first general multiplication algorithm in $\mathrm{GF}(2^k)$ with a subquadratic area complexity of $\mathcal{O}(k^{8/5}) = \mathcal{O}(k^{1.6})$. Using the Chinese Remainder Theorem, we represent the elements of $\mathrm{GF}(2^k)$; i.e. the polynomials in $\mathrm{GF}(2)[X]$ of degree at most $k-1$, by their remainder modulo a set of $n$ pairwise prime trinomials, $T_1,\dots,T_{n}$, of degree $d$ and such that $nd \geq k$. Our algorithm is based on Montgomery's multiplication applied to the ring formed by the direct product of the trinomials.