Cryptographic Hardware and Embedded Systems, CHES 2010

Volume 6225 of the series Lecture Notes in Computer Science pp 65-79

Co-Z Addition Formulæ and Binary Ladders on Elliptic Curves

(Extended Abstract)
  • Raveen R. GoundarAffiliated withJapan Advanced Institute of Science and Technology
  • , Marc JoyeAffiliated withTechnicolor, Security & Content Protection Labs
  • , Atsuko MiyajiAffiliated withJapan Advanced Institute of Science and Technology


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 curves Meloni’s technique Jacobian coordinates regular binary ladders implementation attacks embedded systems