A new class of irreducible pentanomials for polynomial based multipliers in binary fields
This work addresses the need for more efficient polynomial-based multipliers in cryptography and coding theory, representing an incremental improvement over prior methods.
The authors tackled the problem of efficient multiplication in binary fields by introducing a new class of irreducible pentanomials, resulting in a bit-parallel multiplier with improved XOR and AND complexity and comparable time delay to existing methods.
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.