Optimizing Galois Field Arithmetic for Diverse Processor Architectures and Applications

Abstract

Galois field implementations are central to the design of many reliable and secure systems, with many systems implementing them in software. The two most common Galois field operations are addition and multiplication; typically, multiplication is far more expensive than addition. In software, multiplication is generally done with a look-up to a pre-computed table, limiting the size of the field and resulting in uneven performance across architectures and applications. In this paper, we first anaylze existing table-based implementation and optimization techniques for multiplication in fields of the form GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ). Next, we propose the use of techniques in composite fields: extensions of GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ) in which multiplications are performed in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ) and efficiently combined. The composite field technique trades computation for storage space, which prevents eviction of look-up tables from the CPU cache and allows for arbitrarily large fields. Most Galois field optimizations are specific to a particular implementation; our technique is general and may be applied in any scenario requiring Galois fields. A detailed performance study across five architectures shows that the relative performance of each approach varies with architecture, and that CPU, memory limitations and fields size must be considered when selecting an appropriate Galois field implementation. We also find that the use of our composite field implementation is often faster and less memory intensive than traditional algorithms for GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ).