Elliptic Curves and Cryptography

The subject of elliptic curves encompasses a vast amount of mathematics. Our aim in this section is to summarize just enough of the basic theory for cryptographic applications. For additional reading, there are a number of survey articles and books devoted to elliptic curve cryptography [14, 68, 81, 135], and many others that describe the number theoretic aspects of the theory of elliptic curves, including [25, 65, 73, 74, 136, 134, 138].

[1]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

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

[3]  Berit Skjernaa,et al.  Satoh's algorithm in characteristic 2 , 2003, Math. Comput..

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

[5]  J. Silverman,et al.  Rational Points on Elliptic Curves , 1992 .

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

[7]  Neal Koblitz,et al.  Algebraic aspects of cryptography , 1998, Algorithms and computation in mathematics.

[8]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[9]  Nigel P. Smart,et al.  The Discrete Logarithm Problem on Elliptic Curves of Trace One , 1999, Journal of Cryptology.

[10]  Igor A. Semaev,et al.  Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p , 1998, Math. Comput..

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

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

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

[14]  Takakazu Satoh,et al.  Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves , 1998 .

[15]  J. Cassels.  Lectures on elliptic curves , 1991 .

[16]  R. Schoof.  Journal de Theorie des Nombres de Bordeaux 7 (1995), 219{254 , 2022 .

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

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

[19]  Victor S. Miller,et al.  The Weil Pairing, and Its Efficient Calculation , 2004, Journal of Cryptology.

[20]  R. Harley,et al.  An extension of Satoh's algorithm and its implementation , 2000 .

[21]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

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

[23]  J. Silverman.  Advanced Topics in the Arithmetic of Elliptic Curves , 1994 .

[24]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1993, IEEE Trans. Inf. Theory.