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.
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Kawahara, Y., Aoki, K., Takagi, T. (2008). Faster Implementation of η T Pairing over GF(3m) Using Minimum Number of Logical Instructions for GF(3)-Addition. In: Galbraith, S.D., Paterson, K.G. (eds) Pairing-Based Cryptography – Pairing 2008. Pairing 2008. Lecture Notes in Computer Science, vol 5209. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85538-5_19
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DOI: https://doi.org/10.1007/978-3-540-85538-5_19
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