Novel Trace-Based Construction of the Parallel-In-Serial-Out Multiplication for Elliptic Curve Digital Signature Algorithm Like Binary Extension Fields

Abstract

The Gaussian normal bases (GNBs) recommended by the National Institute of Standards and Technology (NIST) for the five binary extension fields of the Elliptic Curve Digital Signature Algorithm (ECDSA), are self-dual. Multiplication, is a complex, and critical, field operation for the hardware-oriented implementations of field inversion - an important operation in the elliptic curve based algorithms - in the GNB representation. This paper, introduces a novel construction for the bit-serial parallel-in-serial-out (BS-PISO) multiplication over the NIST-like fields, based on the Trace mapping. The new formulations, are presented for the first time in literature as far as we know, and provide a new and simple understanding of the Trace-based mathematical model behind such a BS-PISO multiplication. In addition, derivation of existing formulations for the PISO GNB field multiplication, based on the new Trace-based construction, is presented. A digit-serial most-significant-digit first PISO field multiplication architecture is constructed, based on the new Trace-based BS-PISO formulations, and the corresponding theoretical space and time complexities are derived and compared to other counterparts. Moreover, post synthesis space and time readings are reported, based on Application specific integrated circuit (ASIC) realizations, using the NANGate 45nm open cell libraries.