  Galois Field arithmetic blocks are the key components in many security
applications, such as Elliptic Curve Cryptography (ECC) and the S-Boxes of the
Advanced Encryption Standard (AES) cipher. This paper introduces a novel
hardware intellectual property (IP) protection technique by obfuscating
arithmetic functions over Galois Field (GF), specifically, focusing on
obfuscation of GF multiplication that underpins complex GF arithmetic and
elliptic curve point arithmetic functions. Obfuscating GF multiplication
circuits is important because the choice of irreducible polynomials in GF
multiplication has the great impact on the performance of the hardware designs,
and because the significant effort is spent on finding an optimum irreducible
polynomial for a given field, which can provide one company a competitive
advantage over another.
