New Bit Parallel Multiplier With Low Space Complexity for All Irreducible Trinomials Over $GF(2^{n})$

Abstract

Koç and Sunar proposed an architecture of the Mastrovito multiplier for the irreducible trinomial <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$f(x)=x^{n}+x^{k}+1$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k\neq n/2$</tex></formula> to reduce the time complexity. Also, many multipliers based on the Karatsuba-Ofman algorithm (KOA) was proposed that sacrificed time efficiency for low space complexity. In this paper, a new multiplication formula which is a variant of KOA presented. We also provide a straightforward architecture of a non-pipelined bit-parallel multiplier using the new formula. The proposed multiplier has lower space complexity than and comparable time complexity to previous Mastrovito multipliers' for all irreducible trinomials.