Use of Elliptic Curves in Cryptography

We discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

[1]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[2]  Douglas H. Wiedemann.  Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[3]  J. Cassels,et al.  Diophantine Equations with Special Reference To Elliptic Curves , 1966 .

[4]  É. Fouvry,et al.  Théorème de Brun-Titchmarsh; application au théorème de Fermat , 1985 .

[5]  Leonard M. Adleman,et al.  A subexponential algorithm for the discrete logarithm problem with applications to cryptography , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[6]  Horst G. Zimmer,et al.  On the difference of the Weil height and the Néron-Tate height , 1976 .

[7]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[8]  Joseph H. Silverman,et al.  Lower bound for the canonical height on elliptic curves , 1981 .

[9]  J. W. S. Cassels,et al.  On the equation , 1984 .

[10]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[11]  S. Lang,et al.  Elliptic Curves: Diophantine Analysis , 1978 .

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

[13]  Andrzej Schinzel,et al.  On the equation $y^m = P(x)$ , 1976 .

[14]  Andrew Bremner,et al.  On the equation , 1984 .

[15]  Andrew M. Odlyzko,et al.  Discrete Logarithms in Finite Fields and Their Cryptographic Significance , 1985, EUROCRYPT.

[16]  H. Swinnerton-Dyer,et al.  Notes on elliptic curves. II. , 1963 .

[17]  Don Zagier,et al.  On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3 , 1985 .