Faster Binary-Field Multiplication and Faster Binary-Field MACs

Abstract. This paper shows how to securely authenticate messages us-ing just 29 bit operations per authenticated bit, plus a constant overhead per message. The authenticator is a standard type of “universal ” hash function providing information-theoretic security; what is new is com-puting this type of hash function at very high speed. At a lower level, this paper shows how to multiply two elements of a field of size 2128 using just 9062 ≈ 71 · 128 bit operations, and how to multiply two elements of a field of size 2256 using just 22164 ≈ 87 · 256 bit operations. This performance relies on a new representation of field elements and new FFT-based multiplication techniques. This paper’s constant-time software uses just 1.89 Core 2 cycles per byte to authenticate very long messages. On a Sandy Bridge it takes 1.43 cycles per byte, without using Intel’s PCLMULQDQ polynomial-multiplication hardware. This is much faster than the speed records for constant-time implementations of GHASH without PCLMULQDQ (over 10 cycles/byte), even faster than Intel’s best Sandy Bridge implementa-tion of GHASH with PCLMULQDQ (1.79 cycles/byte), and almost as fast as state-of-the-art 128-bit prime-field MACs using Intel’s integer-multiplication hardware (around 1 cycle/byte)