On the Power of Multidoubling in Speeding Up Elliptic Scalar Multiplication

We discuss multidoubling methods for efficient elliptic scalar multiplication. The methods allows computation of 2kP directly from P without computing the intermediate points, where P denotes a randomly selected point on an elliptic curve. We introduce algorithms for elliptic curves with Montgomery form and Weierstrass form defined over finite fields with characteristic greater than 3 in terms of affine coordinates. These algorithms are faster than k repeated doublings. Moreover, we apply the algorithms to scalar multiplication on elliptic curves and analyze computational complexity. As a result of our implementation with respect to the Montgomery and Weierstrass forms in terms of affine coordinates, we achieved running time reduced by 28% and 31%, respectively, in the scalar multiplication of an elliptic curve of size 160-bit over finite fields with characteristic greater than 3.

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

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

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

[4]  Kazuo Ohta,et al.  Advances in Cryptology — ASIACRYPT’98 , 2002, Lecture Notes in Computer Science.

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

[6]  Kouichi Sakurai,et al.  Efficient Scalar Multiplications on Elliptic Curves with Direct Computations of Several Doublings , 2001 .

[7]  Bart Preneel,et al.  On the Performance of Signature Schemes Based on Elliptic Curves , 1998, ANTS.

[8]  Ricardo Dahab,et al.  Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation , 1999, CHES.

[9]  Gordon B. Agnew,et al.  An Implementation of Elliptic Curve Cryptosystems Over F2155 , 1993, IEEE J. Sel. Areas Commun..

[10]  Burton S. Kaliski.  Advances in Cryptology - CRYPTO '97 , 1997 .

[11]  Y. Han.  Direct computation for elliptic curve cryptosystems , 1999 .

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

[13]  Kouichi Sakurai,et al.  Efficient Scalar Multiplications on Elliptic Curves without Repeated Doublings and Their Practical Performance , 2000, ACISP.

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

[15]  Yvo Desmedt.  Public Key Cryptography — PKC 2003 , 2002, Lecture Notes in Computer Science.

[16]  Atsuko Miyaji,et al.  Efficient elliptic curve exponentiation , 1997, ICICS.

[17]  Yukio Tsuruoka,et al.  Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method , 1992, CRYPTO.

[18]  Volker Müller.  Efficient Algorithms for Multiplication on Elliptic Curves , 1998 .

[19]  H. W. Lenstra,et al.  Factoring integers with elliptic curves , 1987 .

[20]  Kouichi Sakurai,et al.  Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications , 2000, Public Key Cryptography.

[21]  Michael Wiener,et al.  Advances in Cryptology — CRYPTO’ 99 , 1999 .

[22]  Daniel M. Gordon,et al.  A Survey of Fast Exponentiation Methods , 1998, J. Algorithms.

[23]  P. L. Montgomery.  Speeding the Pollard and elliptic curve methods of factorization , 1987 .

[24]  Ernest F. Brickell,et al.  Advances in Cryptology — CRYPTO’ 92 , 2001, Lecture Notes in Computer Science.