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A deterministic algorithm for discrete logarithm on some special elliptic curves over rational numbers

This paper provides a polynomial time algorithm for finding integer number d such that T = dS in the group of ℚ-rational points on a given elliptic curve E defined over ℚ (ECDLP) where T, S ϵ E (ℚ). In this method, first we determine the biggest integer number n such that S ϵ 2<sup>n</sup>E(ℚ) and then using halving and another algorithms, we solve this problem.