Flexible Composite Galois Field $GF((2^m)^2)$ Multiplier Designs

Abstract

Composite Galois Field $GF((2^m)^n)$ multiplications denote the multiplication with extension field over the ground field $GF(2^m)$, that are used in cryptography and error correcting codes. In this paper, composite versatile and vector $GF((2^m)^2)$ multipliers are proposed. The proposed versatile $GF((2^m)^2)$ multiplier design is used to perform the $GF((2^x)^2)$ multiplication, where $2\le x\le m$. The proposed vector $GF((2^m)^2)$ multiplier design is used to perform $2^k$ numbers of $GF((2^{\frac{m}{2^k}})^2)$ multiplications in parallel, where throughput is comparatively higher than other designs and $k\in \{0, 1, ...(log_{2}m)-1) \}$. In both the works, the hardware cost is the trade-off while the flexibility is high. The proposed and existing multipliers are synthesised and compared using 45 nm CMOS technology. The throughputs of the proposed parallel and serial vector $GF((2^8)^2)$ multipliers are $72.7\%$ and $53.62\%$ greater than Karatsuba based multiplier design [11] respectively.