Low Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation

Abstract

We study Dickson bases for binary field representation. Such a representation seems interesting when no optimal normal basis exists for the field. We express the product of two field elements as Toeplitz or Hankel matrix-vector products. This provides a parallel multiplier which is subquadratic in space and logarithmic in time. Using the matrix-vector formulation of the field multiplication, we also present sequential multiplier structures with linear space complexity.

On Polynomial Multiplication in Chebyshev Basis

“…This is of particular interest since such medium size polynomials are used in validated computing to approximate functions using a Chebyshev Interpolation model [4]. This has also a practical interest since Chebyshev or Dickson polynomials can be used in cryptography applications [27,28], which often need medium size polynomials for defining extension field (e.g. F 2 160 ).…”

High‐throughput Dickson basis multiplier with a trinomial for lightweight cryptosystems

“…The Hasan-Negre Dickson basis multiplier in [22] employs sub-quadratic complexity design and thus is sub-quadratic in space and logarithmic in time. The Hasan-Negre Dickson basis multiplier in [25] utilises a one-dimensional array structure and thus has linear space and time complexities. The proposed Dickson basis multiplier uses a two-dimensional systolic array structure.…”

“…Some Dickson multipliers with sub-quadratic space complexity have been presented in [22][23][24]. Hasan and Negre [25] proposed the onedimensional array multiplier for Dickson basis with linear space complexity. In practice, we know that many multiplications and multiplicative inversions that are carried out by repeated multiplications are required in ECDSA [19].…”

“…Thus, a multiplier that can concurrently execute many multiplications is urgently needed to reduce execution time. However, the existing Dickson basis multipliers [22][23][24][25] have been proposed for performing only one multiplication at the same time. Therefore, herein we propose a systolic Dickson basis multiplier that can concurrently execute many multiplications at the same time.…”

Small FPGA Based Multiplication-Inversion Unit for Normal Basis Representation in GF(2m)

“…Additions and multiplications are polynomial operations modulo the irreducible polynomial f in GF(2)[x]. There exist other representations with particular properties such as Dickson basis [10] or dual basis [11].…”