New Table Look-Up Methods for Faster Frobenius Map Based Scalar Multiplication Over GF(pn)

We describe a new scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems. The algorithm is obtained by combining Koblitz’s idea of using Frobenius automorphism along with a very special kind of look-up table. In the case where the base point is unknown, we present an efficient algorithm to compute the look-up table online. Our algorithm applies to prime power fields GF(p n ). One important subclass of such fields are Optimal Extension Fields (OEF’s) which are believed to be ideal for efficient implementation of cryptographic primitives. Over prime power fields, our algorithm compares favourably to other known algorithms for scalar multiplication.

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

[2]  Kunio Kobayashi,et al.  Fast Elliptic Curve Algorithm Combining Frobenius Map and Table Reference to Adapt to Higher Characteristic , 1999, EUROCRYPT.

[3]  Jerome A. Solinas,et al.  Efficient Arithmetic on Koblitz Curves , 2000, Des. Codes Cryptogr..

[4]  Erik Woodward Knudsen,et al.  Elliptic Scalar Multiplication Using Point Halving , 1999, ASIACRYPT.

[5]  Tanja Lange,et al.  Improved Algorithms for Efficient Arithmetic on Elliptic Curves Using Fast Endomorphisms , 2003, EUROCRYPT.

[6]  D. Cantor.  Computing in the Jacobian of a hyperelliptic curve , 1987 .

[7]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[8]  Scott A. Vanstone,et al.  Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms , 2001, CRYPTO.

[9]  Neal Koblitz,et al.  CM-Curves with Good Cryptographic Properties , 1991, CRYPTO.

[10]  Aggelos Kiayias,et al.  Self Protecting Pirates and Black-Box Traitor Tracing , 2001, CRYPTO.

[11]  Volker Müller.  Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two , 1998, Journal of Cryptology.

[12]  Kazumaro Aoki,et al.  A Cyclic Window Algorithm for ECC Defined over Extension Fields , 2001, ICICS.

[13]  Nigel P. Smart.  Elliptic Curve Cryptosystems over Small Fields of Odd Characteristic , 1999, Journal of Cryptology.

[14]  Tanja Lange,et al.  Speeding up the Arithmetic on Koblitz Curves of Genus Two , 2000, Selected Areas in Cryptography.

[15]  Walter M. Lioen,et al.  Factorization of RSA-140 Using the Number Field Sieve , 1999, CRYPTO 1999.

[16]  Jean-Jacques Quisquater,et al.  Analysis of the Gallant-Lambert-Vanstone Method Based on Efficient Endomorphisms: Elliptic and Hyperelliptic Curves , 2002, Selected Areas in Cryptography.

[17]  Mun-Kyu Lee,et al.  Trading Inversions for Multiplications in Elliptic Curve Cryptography , 2006, Des. Codes Cryptogr..

[18]  YoungJu Choie,et al.  Speeding up the Scalar Multiplication in the Jacobians of Hyperelliptic Curves Using Frobenius Map , 2002, INDOCRYPT.

[19]  Neal Koblitz,et al.  Hyperelliptic cryptosystems , 1989, Journal of Cryptology.

[20]  Alfred Menezes,et al.  Progress in Cryptology — INDOCRYPT 2002 , 2002, Lecture Notes in Computer Science.

[21]  Jongin Lim,et al.  Information Security and Cryptology - ICISC 2003 , 2003, Lecture Notes in Computer Science.

[22]  Joan Feigenbaum,et al.  Advances in Cryptology-Crypto 91 , 1992 .

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

[24]  Mun-Kyu Lee,et al.  Efficient Scalar Multiplication in Hyperelliptic Curves Using A New Frobenius Expansion , 2003, ICISC.

[25]  Aggelos Kiayias,et al.  Polynomial Reconstruction Based Cryptography , 2001, Selected Areas in Cryptography.

[26]  Jerome A. Solinas.  An Improved Algorithm for Arithmetic on a Family of Elliptic Curves , 1997, CRYPTO.

[27]  Jacques Stern,et al.  Advances in Cryptology — EUROCRYPT ’99 , 1999, Lecture Notes in Computer Science.

[28]  R. Zuccherato,et al.  An elementary introduction to hyperelliptic curves , 1996 .