# elliptic curve point multiplicative inverse

is that possible to calculate the multiplicative inverse for an elliptic curve point . for example X is a point i need to calculate inverse of X such that X*X^-1 give me unity.

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Please share with us an example and the own tries. Please give reference for the multiplicative structure on an elliptic curve.

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from ell_point.html sagemath help : division_points(m, poly_only=False) Return a list of all points Q such that mQ=P where P = self.

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Please define the "multiplication" on an elliptic curve, and also define a "unity" on it.

Usually, the operation on an elliptic curve $E$ is denoted additively, with the plus sign. For a (rational) point, usually denoted by $P$ on the curve, its inverse is denoted by $-P$ and computed in the same manner. For instance:

sage: E = EllipticCurve(QQ, [1,8])
sage: E.gens()
[(1/4 : 23/8 : 1)]
sage: P = E.gens(); P
(1/4 : 23/8 : 1)
sage: -P
(1/4 : -23/8 : 1)

sage: E = EllipticCurve(GF(2017), [1,8])
sage: P = E.random_point(); P
(488 : 578 : 1)
sage: -P
(488 : 1439 : 1)

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sorry if the question is not clear. i will explain based on your example. given the point P. if i perform scalar multiplication 3*P that will give a new point say P2. my request is to how to calculate P from P2 .. can i make inverse scalar multiplication of P2 by 3 to get the original point P .