The performance of public-key cryptosystems is mainly appointed by the underlying finite field arithmetic. Among the basic arithmetic operations over finite field, the multiplicative inversion is the most time consuming operation. In this paper, a fast inversion algorithm over GF(2~m) with the polynomial basis representation is proposed. The proposed algorithm executes in about 27.5% or 45.6% less iterations than the extended binary gcd algorithm (EBGA) or the montgomery inverse algorithm (MIA) over GF(2~(163)), respectively. In addition, we propose a new hardware architecture to apply for low-complexity systems. The proposed architecture takes approximately 48.3% or 24.9% less the number of reduction operations than or over GF(2~(239)), respectively. Furthermore, it executes in about 21.8% less the number of addition operations than over GF(2~(163)).
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