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

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.