Faster implementation of ηt pairing over GF(3m) using minimum number of logical instructions for GF(3)-addition

Abstract

The η T pairing in characteristic three is implemented by arithmetic in GF(3)={0,1,2}. Harrison et al. reported an efficient implementation of the GF(3)-addition by using seven logical instructions (consisting of AND, OR, and XOR) with the two-bit encoding { (0,0) 0, (0,1) 1, (1,0) 2 }. It has not yet been proven whether seven is the minimum number of logical instructions for the GF(3)-addition. In this paper, we show many implementations of the GF(3)-addition using only six logical instructions with different encodings such as { (1,1) 0, (0,1) 1, (1,0) 2 } or { (0,0) 0, (0,1) 1, (1,1) 2 }. We then prove that there is no implementation of the GF(3)-addition using five logical instructions with any encoding of GF(3) by two bits. Moreover, we apply the new GF(3)-additions to an efficient software implementation of the η T pairing. The running time of the η T pairing over GF(3509), that is considered to be realized as 128-bit security, using the new GF(3)-addition with the encoding { (0,0) 0, (0,1) 1, (1,1) 2 } is 16.3 milliseconds on an AMD Opteron 2.2-GHz processor. This is approximately 7% faster than the implementation using the previous GF(3)-addition with seven logical instructions. © Springer-Verlag Berlin Heidelberg 2008.