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Pairing based cryptography is one of the best security solution that devote a lot of attention. So, to make pairing practical, secure and computationally effcient, we choose to work with extension finite field of the form 𝔽 p k with k ≥ 12. In this paper, we focus on the case of curves with embedding degree 18. We use the tower building technique, and study the case of degree 2 or 3 twist to carry out most arithmetics operations in 𝔽 p 2 , 𝔽 p 3, 𝔽 p 6, 𝔽 p 9 and 𝔽 p 18, thus we speed up the computation in optimal ate pairing.

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We propose a method to speed-up Pollard's Rho algorithm using cycle detection with stack on elliptic curves. The Pollard's Rho using cycle detection with stack requires less time than Pollard's Rho original in reaching collision and it is guaranteed to stop within the second loop through the cycle. In practice, our new rudimentary implementation of proposed method decreases the running time and the number of evaluations of the iteration function on the cost of the memory space used.