A New Look-Up Table Approach for High-Speed Finite Field Multiplication

This paper presents a new high-speed multiplier over GF(2^m) based on look-up table (LUT) approach. A straight-forward LUT-based multiplication requires a table of size (m x 2^m) bits for the Galois field of order m which is quite large for the fields of large orders recommended by the National Institute of Standards and Technology (NIST). Therefore, in this paper, we propose a digit-serial LUT-based technique, where certain number of operand bits are grouped into digits, and multiplication is performed in serial/parallel manner. We restrict the digit-size to 4 to store only 16 words in the LUT. We have also proposed a digit-parallel design to achieve higher speed than its digit-serial counterpart, which is very much useful for high-speed applications. We have chosen m=233 to satisfy the security requirements in elliptic curve cryptography, but our method can be used for other prime extensions, as well. We have estimated the area-time complexity of our designs in terms of LUT access-time and XOR-delay. The proposed LUT-based implementation will be useful for high-speed applications in elliptic curve cryptography and error control coding.