## Cryptology ePrint Archive: Report 2008/439

**Linear equivalence between elliptic curves in Weierstrass and Hesse form**

*Alexander Rostovtsev*

**Abstract: **Elliptic curves in Hesse form admit more suitable arithmetic than ones in Weierstrass form. But elliptic curve cryptosystems usually use Weierstrass form. It is known that both those forms are birationally equivalent. Birational equivalence is relatively hard to compute. We prove that elliptic curves in Hesse form and in Weierstrass form are linearly equivalent over initial field or its small extension and this equivalence is easy to compute. If cardinality of finite field q = 2 (mod 3) and Frobenius trace T = 0 (mod 3), then equivalence is defined over initial finite field. This linear equivalence allows multiplying of an elliptic curve point in Weierstrass form by passing to Hessian curve, computing product point for this curve and passing back. This speeds up the rate of point multiplication about 1,37 times.

**Category / Keywords: **public-key cryptography / elliptic curve cryptosystem

**Publication Info: **The paper was not published elsewhere

**Date: **received 13 Oct 2008

**Contact author: **rostovtsev at ssl stu neva ru

**Available format(s): **PDF | BibTeX Citation

**Version: **20081020:184432 (All versions of this report)

**Discussion forum: **Show discussion | Start new discussion

[ Cryptology ePrint archive ]