The Weil Pairing, and Its Efficient Calculation

Abstract The Weil Pairing, first introduced by André Weil in 1940, plays an important role in the theoretical study of the arithmetic of elliptic curves and Abelian varieties. It has also recently become extremely useful in cryptologic constructions related to those objects. This paper gives the definition of the Weil Pairing, describes efficient algorithms to calculate it, gives two applications, and describes the motivation to considering it.

[1]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[2]  C. Hooley.  On Artin's conjecture. , 1967 .

[3]  W. Fulton.  Algebraic curves , 1969 .

[4]  A. Weil.  Lettre à Artin , 1979 .

[5]  A. Weil.  Sur les fonctions algébriques à corps de constantes fini , 1979 .

[6]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

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

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

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

[10]  Burton S. Kaliski,et al.  Elliptic curves and cryptography: a pseudorandom bit generator and other tools , 1988 .

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

[12]  B. M. Fulk.  MATH , 1992 .

[13]  G. Frey,et al.  A remark concerning m -divisibility and the discrete logarithm in the divisor class group of curves , 1994 .

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

[15]  R. Balasubramanian,et al.  The Improbability That an Elliptic Curve Has Subexponential Discrete Log Problem under the Menezes—Okamoto—Vanstone Algorithm , 1998, Journal of Cryptology.

[16]  Gerhard Frey,et al.  The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems , 1999, IEEE Trans. Inf. Theory.

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

[18]  P. Stevenhagen,et al.  ELLIPTIC FUNCTIONS , 2022 .