1 | initial version |

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()[0]; 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)

2 | No.2 Revision |

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()[0]; P
(1/4 : 23/8 : 1)
sage: -P
(1/4 : -23/8 :
```~~1)~~1)
~~ ~~sage: E = EllipticCurve(GF(2017), [1,8])
sage: P = E.random_point(); P
(488 : 578 : 1)
sage: -P
(488 : 1439 : ~~1)~~

1)

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