GF(2/sup m/) multiplication and division over the dual basis

Abstract

In this paper an algorithm for GF(2") multiplication/division is presented and a new, more generalized definition of duality is proposed. From these the bit-serial Berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers. Furthermore, it is shown that hardware efficient, bit-parallel dual basis multipliers can also be designed. These multipliers have a regular structure, are easily extended to different GF(2") and hence suitable for VLSI implementations. As in the bit-serial case these bit-parallel multipliers can also be hardwired to carry out constant multiplication. These constant multipliers have reduced hardware requirements and are also simple to design. In addition, the multiplication/division algorithm also allows a bit-serial systolic finite field divider to be designed. This divider is modular, independent of the defining irreducible polynomial for the field, easily expanded to different GF(2m) and its longest delay path is independent of m.

Parallel Montgomery Multiplication in GF (2^k) Using Trinomial Residue Arithmetic

“…The most known general methods are an adaptation of Montgomery's multiplication [15] to binary fields [2], and the approach described by E. Mastrovito [12], where the multiplication is expressed as a matrix-vector product. However, the most efficient implementations are specific algorithms which use features of the extension fields, such as the type of the base [16,4,21,8], or the form of the irreducible polynomial defining the field. In his Ph.D.…”

High-Speed Polynomial Basis Multipliers Over GF(2m) for Special Pentanomials

“…Then the number of XOR gates represented by the above shared groups is given by the number of 1's (Hamming Weight) in the binary configuration of for even or of for odd plus the Hamming Weight of for even or of for odd . Therefore the number of XOR gates given by the shared groups that appear in the product coefficients is computed by (24) that matches (11). In (24), represents the limit of the summatory for even represents the limit for odd , represents the Hamming Weight of to be computed for even and the Hamming Weight of for odd .…”

“…where is the Hamming Weight of and where is given as: (10) • The number of XOR gates given by the shared groups that appear in the product coefficients is: (11) where represents the limit of the summatory for even represents the limit for odd represents the Hamming Weight of to be computed for even and the Hamming Weight of for odd . Therefore, the XOR gates of the multiplier given by the addition will be (12) A more compact expression for (12) could not be found.…”

“…For this reason, several bit-parallel polynomial basis (PB) multipliers have been proposed. Polynomial basis is the most widely used, although normal [10] or dual [11] basis can also be considered. The complexity of the multiplier depends on the generating irreducible polynomial selected for the finite field.…”

Hardware architectures for public key cryptography

“…Fenn et al proposed bit-serial and parallel multipliers and gave delay and area comparisons of them in [55]. Wu, Hasan and Blake proposed parallel multipliers in [218,219].…”