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 ]
and use these polynomials to define finite field extensions of degree ( m = 2b + c ).

Highlights:

• Exact operation count for reduction modulo ( f(x) )
• Efficient Karatsuba-based multiplier in ( \mathbb{F}_2[x] )
• Lower XOR and AND gate complexity in hardware
• Competitive time delay compared to existing Karatsuba multipliers

This work supports the design of high-performance hardware for cryptographic binary field arithmetic.