Linear equivalence between elliptic curves in Weierstrass and Hesse form

Alexander Rostovtsev

Paper 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.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. The paper was not published elsewhere
Keywords: elliptic curve cryptosystem
Contact author(s): rostovtsev @ ssl stu neva ru
History: 2008-10-20: received
Short URL: https://ia.cr/2008/439
License: CC BY

BibTeX

@misc{cryptoeprint:2008/439,
      author = {Alexander Rostovtsev},
      title = {Linear equivalence between elliptic curves in Weierstrass and Hesse form},
      howpublished = {Cryptology {ePrint} Archive, Paper 2008/439},
      year = {2008},
      url = {https://eprint.iacr.org/2008/439}
}