Low Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation

Subquadratic Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation

We study Dickson bases for binary field representation. Such representation seems interesting when no optimal normal basis exists for the field. We express the product of two elements as Toeplitz or Hankel matrix vector product. This provides a parallel multiplier which is subquadratic in space and logarithmic in time.

Efficient implementation of low time complexity and pipelined bit-parallel polynomial basis multiplier over binary finite fields

This paper presents two efficient implementations of fast and pipelined bit-parallel polynomial basis multipliers over GF (2m) by irreducible pentanomials and trinomials. The architecture of the first multiplier is based on a parallel and independent computation of powers of the polynomial variable. In the second structure only even powers of the polynomial variable are used. The par...

A Lower Bound on the Complexity of Polynomial Multiplication Over Finite Fields

Factoring Dickson polynomials over finite fields

We derive the factorizations of the Dickson polynomials Dn(X, a) and En(X, a), and of the bivariate Dickson polynomials Dn(X, a) − Dn(Y, a), over any finite field. Our proofs are significantly shorter and more elementary than those previously known.

Multiplication over Arbitrary Fields

We prove a lower bound of 52n2 3n for the rank of n n–matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.