An Efficient Look-up Table-based Approach for Multiplication over GF(2^m) Generated by Trinomials

Abstract

In this paper, we present an efficient look-up table (LUT)-based approach to design multipliers for GF(2^m) generated by irreducible trinomials. A straightforward LUT-based multiplication requires a table of size (m×2^m) bits for the Galois field of degree m. The LUT size, therefore, becomes quite large for the fields of large degrees recommended by the National Institute of Standards and Technology (NIST). Keeping that in view, we have proposed a digit-serial LUT-based design, where operand bits are grouped into digits of fixed width, and multiplication is performed in serial/parallel manner. We restrict the digit size to 4 to store only 16 words in the LUT to have lower area-delay complexity. We have also proposed a digit-parallel LUT-based design for high-speed applications, using the same LUT as the digit-serial design, at the cost of some additional multiplexors and combinational logic for parallel modular reductions and additions. We have presented a simple circuit for the initialization of LUT content, which can be used to update the LUT in three cycles whenever required. The proposed digit-serial design involves less area-complexity and less time-complexity than those of the existing LUT-based designs. The proposed digit-parallel design offers nearly 28 % improvement in area-delay product over the best of the existing LUT-based designs. NIST has recommended five binary finite fields for elliptic curve cryptography, out of which two are generated by the trinomials Q(x)=x ²³³+x ⁷⁴+1 and Q(x)=x ⁴⁰⁹+x ⁸⁷+1. In this paper, we have designed a reconfigurable multiplier that can be used for both these fields. The proposed reconfigurable multiplier is shown to have a negligible reconfiguration overhead and would be useful for cryptographic applications.