Efficient software implementation of binary field arithmetic using vector instruction sets

In this paper we describe an efficient software implementation of characteristic 2 fields making extensive use of vector instruction sets commonly found in desktop processors. Field elements are represented in a split form so performance-critical field operations can be formulated in terms of simple operations over 4-bit sets. In particular, we detail techniques for implementing field multiplication, squaring, square root extraction and present a constant-memory lookup-based multiplication strategy. Our representation makes extensive use of the parallel table lookup (PTLU) instruction recently introduced in popular desktop platforms and follows the trend of accelerating implementations of cryptography through PTLU-style instructions. We present timings for several binary fields commonly employed for curve-based cryptography and illustrate the presented techniques with executions of the ECDH and ECDSA protocols over binary curves at the 128-bit and 256-bit security levels standardized by NIST. Our implementation results are compared with publicly available benchmarking data.