International Workshop on Cryptographic Hardware and Embedded Systems

CHES 2010: Cryptographic Hardware and Embedded Systems, CHES 2010 pp 65-79

Co-Z Addition Formulæ and Binary Ladders on Elliptic Curves

(Extended Abstract)
  • Raveen R. Goundar
  • Marc Joye
  • Atsuko Miyaji
Conference paper

DOI: 10.1007/978-3-642-15031-9_5

Volume 6225 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Goundar R.R., Joye M., Miyaji A. (2010) Co-Z Addition Formulæ and Binary Ladders on Elliptic Curves. In: Mangard S., Standaert FX. (eds) Cryptographic Hardware and Embedded Systems, CHES 2010. CHES 2010. Lecture Notes in Computer Science, vol 6225. Springer, Berlin, Heidelberg


Meloni recently introduced a new type of arithmetic on elliptic curves when adding projective points sharing the same Z-coordinate. This paper presents further co-Z addition formulæ for various point additions on Weierstraß elliptic curves. It explains how the use of conjugate point addition and other implementation tricks allow one to develop efficient scalar multiplication algorithms making use of co-Z arithmetic. Specifically, this paper describes efficient co-Z based versions of Montgomery ladder and Joye’s double-add algorithm. Further, the resulting implementations are protected against a large variety of implementation attacks.


Elliptic curvesMeloni’s techniqueJacobian coordinatesregular binary laddersimplementation attacksembedded systems
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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Raveen R. Goundar
    • 1
  • Marc Joye
    • 2
  • Atsuko Miyaji
    • 1
  1. 1.Japan Advanced Institute of Science and TechnologyIshikawaJapan
  2. 2.Technicolor, Security & Content Protection LabsCesson-Sévigné CedexFrance