Galois Field Instructions in the Sandblaster 2.0 Architectrue

Abstract

This paper presents a novel approach to implementing multiplication of Galois Fields with . Elements of GF() can be represented as polynomials of degree less than N over GF(2). Operations are performed modulo an irreducible polynomial of degree n over GF(2). Our approach splits a Galois Field multiply into two operations, polynomial-multiply and polynomial-remainder over GF(2). We show how these two operations can be implemented using the same hardware. Further, we show that in many cases several polynomial-multiply operations can be combined before needing to a polynomial-remainder. The Sandblaster 2.0 is a SIMD architecture. It has SIMD variants of the poly-multiply and poly-remainder instructions. We use a Reed-Solomon encoder and decoder to demonstrate the performance of our approach. Our new approach achieves speedup of 11.5x compared to the standard SIMD processor of 8x.

Implementation of the Chien search algorithm on a baseband processor

“…In the column basis computation approach, (2) can be reformatted as shown in (3). In this equation, the ⊗ notation represents a Kronecker product operator, where…”

“…In the case of a short BCH code, the memory space required for the corresponding Galois field primitive element and its power terms is not a critical design issue. Sandbridge's Sandblaster [3] does not address the decoder of long BCH codes, but instead, the efficient implementation of Galois multiplications in their baseband processor.…”