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The safety measures of “Elliptic Curve Cryptography” (ECC) mostly rely on the quality of solving “Elliptic Curve Discrete Logarithm Problem” (ECDLP). Known P and Q on the curve satisfying Q = kP, the integer k is the discrete logarithm of Q to the base P. Although P and Q are known, the group order of the curve is so huge that it is unable to compute k. It was observed in the reference [8] that elliptic curve general attacks obviously would resolve ECDLP if the group order of the curve was small enough to compute k.

Figure 4.4 Experiment on encryption.

Figure 4.5 Experiment on decryption.

The implementation and the analysis on the mathematical properties of elliptic curve arithmetic based on the integration of complex number arithmetic with modular arithmetic are described in the reference [3]. The group orders of the curves in the fields: GF(p), GF(^2m), GF(2^m) Z(GF(p)), and Z(GF(2^m)) were observed in the reference [2] in which the group order of the curve in Z(GF(p)) exposed in Appendix (C) of the reference [2] is 47 that exists between and in the case of and q = p² in accordance with Hasses’s theorem [11]. Hence, the security rank is approximately squared in terms of the group order. The attempt of resolving ECDLP on complex plane is so more computationally difficult that time duration will be long. Thus, the safety rank of ECC is significantly enhanced by using the curve of Z(GF(p)).