  We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ of
the 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 of
operations required for computing the reduction modulo $f$. We also provide a
multiplier based on Karatsuba algorithm in $\mathbb{F}_2[x]$ combined with our
reduction process. We give the total cost of the multiplier and found that the
bit-parallel multiplier defined by this new class of polynomials has improved
XOR and AND complexity. Our multiplier has comparable time delay when compared
to other multipliers based on Karatsuba algorithm.
