Improvement of Miller Loop for a Pairing on FK12 Curve and Evaluation with other STNFS Curves

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Pairing is carried out by two steps, Miller loop and final exponentiation. In this manuscript, the authors propose an efficient Miller loop for a pairing on the FK12 curve. A Hamming weight and bit-length of loop parameter have a great effect on the computational cost of the Miller loop. Optimal-ate pairing is used as the most efficient pairing on the FK12 curve currently. The loop parameter of optimal-ate pairing is 6z + 2 where z is the integer to make the FK12 curve parameter. Our method uses z which has a shorter bit-length than the previous optimal-ate pairing as the loop parameter. Usually, z has a low Hamming weight to make final exponentiation efficient. Therefore, the loop parameter in our method has a lower Hamming weight than the loop parameter of the previous one in many cases. The authors evaluate our method by the number of multiplications and execution time. As a result, the proposed algorithm leads to a 3.71% reduction in the number of multiplications and a 3.03% reduction in the execution time. In addition, the authors implement other STNFS secure curves and evaluate these curves from viewpoint of execution time.

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