Efficient Algorithms for Elliptic Curve Cryptosystems

This contribution describes three algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm is an entirely new approach which accelerates the multiplications of points which is the core operation in elliptic curve public-key systems. The algorithm works in conjunction with the k-ary or sliding window method. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. This approach reduces the number of inversions in the underlying finite field at the cost of extra multiplications. For many practical implementations, where field inversion is at least four times as costly as field multiplication, the new approach proofs to be faster than traditional point multiplication methods. The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)n). Based on an idea of Itoh and Tsujii, we optimize the algorithm for software implementation of elliptic curves. The algorithm reduced inversion in the composite field to inversion in the subfield GF(2n). The third algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in GF((2n)n). We provide a detailed complexity analysis of the algorithm for the case that subfield arithmetic is performed through table look-up. We apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide absolute performance measures for the field operations and for an entire point multiplication.

[1]  Christof Paar,et al.  Some remarks on efficient inversion in finite fields , 1995, Proceedings of 1995 IEEE International Symposium on Information Theory.

[2]  Hilarie K. Orman,et al.  Fast Key Exchange with Elliptic Curve Systems , 1995, CRYPTO.

[3]  Neal Koblitz,et al.  Constructing Elliptic Curve Cryptosystems in Characteristic 2 , 1990, CRYPTO.

[4]  Christof Paar,et al.  Fast Arithmetic Architectures for Public-Key Algorithms over Galois Fields GF((2n)m) , 1997, EUROCRYPT.

[5]  T. Elgamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, CRYPTO 1984.

[6]  Alfred Menezes,et al.  The Implementation of Elliptic Curve Cryptosystems , 1990, AUSCRYPT.

[7]  Alfred Menezes,et al.  Elliptic curve public key cryptosystems , 1993, The Kluwer international series in engineering and computer science.

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

[9]  R. Zuccherato,et al.  Counting Points on Elliptic Curves Over F2m , 1993 .

[10]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[11]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1991, STOC '91.

[12]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[13]  D. J. Beauregard,et al.  Efficient Algorithms for Implementing Elliptic Curve Public-Key Schemes , 1996 .

[14]  Thomas Beth,et al.  Arithmetic Operations in GF(2 m ). , 1993 .

[15]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

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

[17]  R. McEliece.  Finite Fields for Computer Scientists and Engineers , 1986 .

[18]  Shu Lin,et al.  Error control coding : fundamentals and applications , 1983 .

[19]  N. Koblitz,et al.  A Fast Software Implementation for Arithmetic Operations in Gf(2 N ) (preprint) , 1996 .

[20]  T. Itoh,et al.  A Fast Algorithm for Computing Multiplicative Inverses in GF(2^m) Using Normal Bases , 1988, Inf. Comput..

[21]  Taher El Gamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, IEEE Trans. Inf. Theory.

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

[23]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[24]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[25]  Richard J. Fateman.  Polynomial Multiplication, Powers and Asymptotic Analysis: Some Comments , 1974, SIAM J. Comput..

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

[27]  Ernest F. Brickell,et al.  Fast Exponentiation with Precomputation (Extended Abstract) , 1992, EUROCRYPT.

[28]  Alfred Menezes,et al.  Public-Key Cryptosystems with Very Small Key Length , 1992, EUROCRYPT.

[29]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[30]  Christof Paar,et al.  A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields , 1996, IEEE Trans. Computers.

[31]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[32]  Ç. Koç.  Analysis of sliding window techniques for exponentiation , 1995 .