No abstract

No abstract

Massey-Omura multiplier of qp P m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible all-one polynomials, the parallel Massey-Omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multipliers contain redundancy in that special class of finite fields, but it also has redundancy in fields qp P m defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original Massey-Omura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible all-one polynomials, the multiplier's circuit complexity matches the best result available in the open literature. Index TermsÐFinite field, Massey-Omura multiplier, all-one polynomial, optimal normal bases.

Efficient implementation of point multiplication is crucial for elliptic curve cryptographic systems. This paper presents the implementation results of an elliptic curve crypto-processor over binary fields(2 ) on binary Edwards and generalized Hessian curves using Gaussian normal basis (GNB). We demonstrate how parallelization in higher levels can be performed by full resource utilization of computing point addition and point-doubling formulas for both binary Edwards and generalized Hessian curves. Then, we employ the -coordinate differential formulations for computing point multiplication. Using a lookup-table (LUT)-based pipelined and efficient digit-level GNB multiplier, we evaluate the LUT complexity and time-area tradeoffs of the proposed crypto-processor on an FPGA. We also compare the implementation results of point multiplication on these curves with the ones on the traditional binary generic curve. To the best of the authors' knowledge, this is the first FPGA implementation of point multiplication on binary Edwards and generalized Hessian curves represented by -coordinates. Index Terms-Binary Edwards curves (BECs), elliptic curve cryptography (ECC), Gaussian normal basis (GNB), generalizedHessian curves (GHCs).

No abstract