Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains

In this paper, we propose a efficient and secure point multiplication algorithm, based on double-base chains. This is achieved by taking advantage of the sparseness and the ternary nature of the so-called double-base number system (DBNS). The speed-ups are the results of fewer point additions and improved formulae for point triplings and quadruplings in both even and odd characteristic. Our algorithms can be protected against simple and differential side-channel analysis by using side-channel atomicity and classical randomization techniques. Our numerical experiments show that our approach leads to speed-ups compared to windowing methods, even with window size equal to 4, and other SCA resistant algorithms.

[1]  Alfred Menezes,et al.  Guide to Elliptic Curve Cryptography , 2004, Springer Professional Computing.

[2]  Marc Joye,et al.  Protections against Differential Analysis for Elliptic Curve Cryptography , 2001, CHES.

[3]  Graham A. Jullien,et al.  An Algorithm for Modular Exponentiation , 1998, Inf. Process. Lett..

[4]  Marc Joye,et al.  Low-cost solutions for preventing simple side-channel analysis: side-channel atomicity , 2004, IEEE Transactions on Computers.

[5]  Graham A. Jullien,et al.  Theory and Applications of the Double-Base Number System , 1999, IEEE Trans. Computers.

[6]  Paul C. Kocher,et al.  Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems , 1996, CRYPTO.

[7]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[8]  Kristin E. Lauter,et al.  Fast Elliptic Curve Arithmetic and Improved Weil Pairing Evaluation , 2003, CT-RSA.

[9]  Tsuyoshi Takagi,et al.  Fast Elliptic Curve Multiplications Resistant against Side Channel Attacks , 2005, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[10]  Tsuyoshi Takagi,et al.  A Fast Parallel Elliptic Curve Multiplication Resistant against Side Channel Attacks , 2002, Public Key Cryptography.

[11]  Christof Paar,et al.  Efficient Algorithms for Elliptic Curve Cryptosystems , 1997, CRYPTO.

[12]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[13]  Paul C. Kocher,et al.  Differential Power Analysis , 1999, CRYPTO.

[14]  Nigel P. Smart,et al.  Advances in Elliptic Curve Cryptography (London Mathematical Society Lecture Note Series) , 2005 .

[15]  Atsuko Miyaji,et al.  Efficient Elliptic Curve Exponentiation Using Mixed Coordinates , 1998, ASIACRYPT.

[16]  Alfred Menezes,et al.  Software Implementation of Elliptic Curve Cryptography over Binary Fields , 2000, CHES.

[17]  Ian F. Blake,et al.  Advances in Elliptic Curve Cryptography: Preface , 2005 .

[18]  R. Tijdeman,et al.  On the maximal distance between integers composed of small primes , 1974 .

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

[20]  Kouichi Itoh,et al.  Fast Implementation of Public-Key Cryptography ona DSP TMS320C6201 , 1999, CHES.

[21]  Laurent Imbert,et al.  On converting numbers to the double-base number system , 2004, SPIE Optics + Photonics.

[22]  Alfred Menezes,et al.  Field inversion and point halving revisited , 2004, IEEE Transactions on Computers.

[23]  Servaas Vandenberghe,et al.  A Fast Software Implementation for Arithmetic Operations in GF(2n) , 1996, ASIACRYPT.

[24]  Marc Joye,et al.  Trading Inversions for Multiplications in Elliptic Curve Cryptography , 2006, Des. Codes Cryptogr..

[25]  Tanja Lange,et al.  Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .