Low complexity bit-parallel polynomial basis multipliers over binary fields for special irreducible pentanomials

Finite field GF(2m)GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF(2m)GF(2m) multiplication is of special interest because it is considered the most important building block. GF(2m)GF(2m) multipliers present reduced space and time complexities when the field is generated by some special irreducible polynomials. Among these, irreducible pentanomials of degree m are specially important because they are abundant and there are several eligible candidates for a given m  . In this paper, we consider bit-parallel polynomial basis multipliers over the finite field GF(2m)GF(2m) generated using type 2 irreducible pentanomials, for which explicit formulas and algorithms for the computation of the products are given. In this contribution, two new subclasses of type 2 irreducible pentanomials are also introduced. The theoretical complexity analysis proves that the bit-parallel multipliers here presented have the lowest number of XOR gates known to date for similar polynomial basis multipliers based on this type of irreducible pentanomials, while the number of AND gates and the time complexity match the best known results found in the literature.