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We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ ofthe form $f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1$. Let $m=2b+c$ and use $f$to define the finite field extension of degree $m$. We give the exact number ofoperations required for computing the reduction modulo $f$. We also provide amultiplier based on Karatsuba algorithm in $\mathbb{F}_2[x]$ combined with ourreduction process. We give the total cost of the multiplier and found that thebit-parallel multiplier defined by this new class of polynomials has improvedXOR and AND complexity. Our multiplier has comparable time delay when comparedto other multipliers based on Karatsuba algorithm.

A new class of irreducible pentanomials for polynomial based multipliers in binary fields