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I am study public key cryptology and interested in elliptic curve cryplogical algorithms. I have some problem about elliptic curve equation. First I can't find inter process and transformation steps from general curve equation to elliptic curve equation weierstrass form, Second , why a1 and a3 scaler is left side, a2,a4,a6 scaler is right side of y^2+a1xy+a3y=x^3+a2x^2+a4x+a6 equation .

Thank you in advance

Aziz YÜCELEN
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  • I know this question was posed and closed on MathOverflow, but Snark left an answer there - did you not find it helpful? http://mathoverflow.net/questions/69519/about-how-to-elliptic-curve-equation-and-discriminant-closed – Gerry Myerson Jul 06 '11 at 07:13
  • Reducing to Weierstrass is just a matter of choosing suitable coordinates. About the last question:just to have all terms containing y on one side of the equation and the terms containing only x on the other. – Andrea Mori Jul 06 '11 at 08:45
  • In OP's defence I will say that it is not easy to find a source, where the required change of coordinates is done with elementary tools only. I've only seen it in lecture notes by a local number theorist. When I was interested in these things, Menezes' book Elliptic Curve Public Key Cryptosystems was a standard reference to cryptopeople. Menezes refers to Silverman at this point, and Silverman (for his very good reasons) does this the standard way of using Riemann-Roch to study the set of functions with a single pole of order at most 6. – Jyrki Lahtonen Jul 06 '11 at 09:59
  • A good place to go is also https://webs.ucm.es/BUCM/mat/doc8354.pdf, Ian Connell, Elliptic Curve Handbook, page 115, Cubic to Weierstraß, describing Nagell's algorithm. – dan_fulea Jul 11 '21 at 12:49

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