@techreport{oai:ipsj.ixsq.nii.ac.jp:00195789,
 author = {Yaoan, Jin and Atsuko, Miyaji and Yaoan, Jin and Atsuko, Miyaji},
 issue = {17},
 month = {May},
 note = {Elliptic curve cryptosystems (ECCs) are widely used because of their short key size． They can ensure enough security with shorter keys， and use less memory space to reduce parameters． Hence， an elliptic curve is typically used in embedded systems． The dominant computation of an ECC is scalar multiplication Q = kP， P ∈ E(Fq)． Thus， the security and efficiency of scalar multiplication are paramount． To render secure ECCs， complete addition formulae can be employed for a secure scalar multiplication． However， this requires significant memory and is thus not suitable for compact devices． Several coordinates exist for elliptic curves such as affine， Jacobian， projective． The complete addition formulae are not based on affine coordinates and thus require considerable memory． In this study， we achieved a compact ECC by focusing on affine coordinates． In fact， affine coordinates are highly advantageous in terms of memory but require many if statements for scalar multiplication owing to exceptional points． We improve the scalar multiplication and reduce the limitations for input k． Furthermore， we extend the affine addition formulae to delete some exceptional inputs for scalar multiplication． Our compact ECC reduces memory complexity up to 26 % and is much more efficient compared to Joye's RL 2-ary algorithm with the complete addition of formulae when the ratio I/M of computational complexity of inversion (I) to multiplication (M) is less than 7．2．, Elliptic curve cryptosystems (ECCs) are widely used because of their short key size. They can ensure enough security with shorter keys, and use less memory space to reduce parameters. Hence, an elliptic curve is typically used in embedded systems. The dominant computation of an ECC is scalar multiplication Q = kP, P ∈ E(Fq). Thus, the security and efficiency of scalar multiplication are paramount. To render secure ECCs, complete addition formulae can be employed for a secure scalar multiplication. However, this requires significant memory and is thus not suitable for compact devices. Several coordinates exist for elliptic curves such as affine, Jacobian, projective. The complete addition formulae are not based on affine coordinates and thus require considerable memory. In this study, we achieved a compact ECC by focusing on affine coordinates. In fact, affine coordinates are highly advantageous in terms of memory but require many if statements for scalar multiplication owing to exceptional points. We improve the scalar multiplication and reduce the limitations for input k. Furthermore, we extend the affine addition formulae to delete some exceptional inputs for scalar multiplication. Our compact ECC reduces memory complexity up to 26 % and is much more efficient compared to Joye's RL 2-ary algorithm with the complete addition of formulae when the ratio I/M of computational complexity of inversion (I) to multiplication (M) is less than 7.2.},
 title = {Secure and Compact Elliptic Curve Cryptosystems},
 year = {2019}
}