Novel Precomputation Schemes for Elliptic Curve Cryptosystems

We present an innovative technique to add elliptic curve points with the form P ±Q , and discuss its application to the generation of precomputed tables for the scalar multiplication. Our analysis shows that the proposed schemes offer, to the best of our knowledge, the lowest costs for precomputing points on both single and multiple scalar multiplication and for various elliptic curve forms, including the highly efficient Jacobi quartics and Edwards curves.

[1]  Ed Dawson,et al.  Faster Group Operations on Elliptic Curves , 2009, AISC.

[2]  Kimmo Järvinen,et al.  FPGA Design of Self-certified Signature Verification on Koblitz Curves , 2007, CHES.

[3]  J. Solinas.  Low-Weight Binary Representations for Pairs of Integers , 2001 .

[4]  Nicolas Meloni,et al.  New Point Addition Formulae for ECC Applications , 2007, WAIFI.

[5]  Yuefei Zhu,et al.  An Improved Algorithm for uP + vQ Using JSF13 , 2004, ACNS.

[6]  Tanja Lange,et al.  Faster Addition and Doubling on Elliptic Curves , 2007, ASIACRYPT.

[7]  Patrick Longa,et al.  New Composite Operations and Precomputation Scheme for Elliptic Curve Cryptosystems over Prime Fields , 2008, Public Key Cryptography.

[8]  Tibor Juhas.  The use of elliptic curves in cryptography , 2007 .

[9]  Alfred Menezes,et al.  Software Implementation of the NIST Elliptic Curves Over Prime Fields , 2001, CT-RSA.

[10]  Tsuyoshi Takagi,et al.  Efficient Representations on Koblitz Curves with Resistance to Side Channel Attacks , 2005, ACISP.

[11]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[12]  Ed Dawson,et al.  Jacobi Quartic Curves Revisited , 2009, ACISP.

[13]  Ricardo Dahab,et al.  Improved Algorithms for Elliptic Curve Arithmetic in GF(2n) , 1998, Selected Areas in Cryptography.

[14]  Naofumi Takagi,et al.  A fast addition algorithm for elliptic curve arithmetic in GF(2n) using projective coordinates , 2000, Inf. Process. Lett..

[15]  Tanja Lange,et al.  Inverted Edwards Coordinates , 2007, AAECC.

[16]  Serdar Boztas,et al.  Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , 2001, Lecture Notes in Computer Science.

[17]  Ricardo Dahaby.  Improved Algorithms for Elliptic Curve Arithmetic in Gf(2 N ) Improved Algorithms for Elliptic Curve Arithmetic in Gf (2 N ) , 1998 .

[18]  Chae Hoon Lim,et al.  Fast Implementation of Elliptic Curve Arithmetic in GF(pn) , 2000, Public Key Cryptography.

[19]  Bodo Möller.  Algorithms for Multi-exponentiation , 2001, Selected Areas in Cryptography.

[20]  Patrick Longa,et al.  Fast and Flexible Elliptic Curve Point Arithmetic over Prime Fields , 2008, IEEE Transactions on Computers.