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

Abstract

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.