Cryptology ePrint Archive: Report 2004/279

Parallel Montgomery Multiplication in $GF(2^k)$ using Trinomial Residue Arithmetic

Jean-Claude Bajard and Laurent Imbert and Graham A. Jullien

Abstract: 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.

Category / Keywords: implementation / finite field arithmetic

Publication Info: Submitted to ARITH17, the 17th IEEE symposium on computer arithmetic

Date: received 28 Oct 2004, last revised 10 Mar 2005

Contact author: Laurent Imbert at lirmm fr

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Version: 20050310:223709 (All versions of this report)

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