Galois Field Instructions in the Sandblaster 2.0 Architectrue

Mayan Moudgill,Daniel Iancu,Andrei Iancu

doi:10.1155/2009/129698

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.

Highlights

Galois Field arithmetic is widely used in applications such as error-correcting codes and cryptography
We describe an approach that implements Galois Field multiplication (GFM) using two instructions, one of which implements polynomial-multiply over GF(2), and the other of which implements the polynomial-remainder over GF(2)
The unified block that implements gfnormi and gfmul in the SB3500 consists of some setup followed by 8 stages of a computational kernel

Summary

Introduction

Galois Field arithmetic is widely used in applications such as error-correcting codes and cryptography. Since most instruction-sets do not provide for 4 inputs fields, a GFM instruction generally uses a special-purpose register that provides either the length or the prime, or both. The Galois Field sum of several GFM operations can be simplified to the polynomial sum (i.e., xor) of several polynomial-multiplies, followed by a single polynomial remainder. This is quite common in several algorithms that use Galois Field arithmetic. In those cases, we can implement the sum of N GFM operations using N polynomialmultiplies and 1 polynomial-remainder, incurring a 1 instruction overhead because of our split implementation.

Galois Field Multiply Instructions

Implementation

Reed-Solomon

Conclusions