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Let $E$ be an elliptic curve; $\alpha,\beta$ two points of $E$; and $a$ a private key such that $\beta=a\cdot\alpha$. We choose random integer $k$ and plain text $x\in E$. Encryption and decryption methods are as follows:

$\operatorname{Enc}(x,k)=(k\cdot\alpha,x+k\cdot\beta)$

$\operatorname{Dec}(y_1,y_2)=y_2-a\cdot y_1$

How can we convert a message such as "Hello" to a point of $E$?

edit: In answer to question "Mapping of message onto elliptic curve and reverse it", we are given a general probabilistic method, and methods that are easy for special cases; for example when the curve is super-singular, or for ordinary curves defined over $\mathbb F_q$ with $q=3\bmod 4$ and group order divisible by $4$. I am looking for methods which are not probabilistic and are applicable to general cases. Do such methods exist?

I will be so thankful for any helpful comments and answers.

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Meysam Ghahramani
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  • I'm pretty sure the answer is: "There's no known way to map arbitrary integers to points on arbitrary curves in an easily reversible way." – SEJPM Jan 06 '16 at 13:11

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