Closed formulae for the Weil pairing inversion

Balasubramanian, 1998, The improbability that an elliptic curve has subexponential discrete log problem under the Menezes–Okamoto–Vanstone algorithm, J. Cryptology, 11, 141, 10.1007/s001459900040 Blake, 2007, Polynomial approximation of bilinear Diffie–Hellman maps, Finite Fields Appl. Blake, 1999, Elliptic Curves in Cryptography, vol. 265 Duursma, 2003, Tate pairing implementation for hyperelliptic curves y2=xp−x+d, vol. 2894, 111 Escofier, 2001, Galois Theory, vol. 204 Galbraith, 2005, Pairings, vol. 317 S. Galbraith, F. Hess, F. Vercautern, Aspects of pairing inversion, preprint, IACR e-print 2007/256, 2007 Galbraith, 2007, Simplified pairing computation and security implications, J. Math. Cryptol., 1, 267, 10.1515/JMC.2007.013 Hess Hess, 2006, The eta pairing revisited, IEEE Trans. Inform. Theory, 52, 4595, 10.1109/TIT.2006.881709 Hitt, 2007, On the minimal embedding field, vol. 4575, 294 Howe, 1996, The Weil pairing and the Hilbert symbol, Math. Ann., 305, 387, 10.1007/BF01444229 Joux, 2002, The Weil and Tate pairings as building blocks for public key cryptosystems (survey), vol. 2369, 20 Lange, 2003, Interpolation of the discrete logarithm in Fq by boolean functions and by polynomials in several variables modulo a divisor of q−1, Discrete Appl. Math., 128, 193, 10.1016/S0166-218X(02)00445-6 Luca, 2006, Elliptic curves with low embedding degree, J. Cryptology, 19, 553, 10.1007/s00145-006-0544-0 V.S. Miller, Short programs for functions on curves, preprint, 1986, available at http://crypto.stanford.edu/miller/miller.pdf Miller, 2004, The Weil pairing and its efficient calculation, J. Cryptology, 17, 235, 10.1007/s00145-004-0315-8 Mullen, 1986, A polynomial representation for logarithms in GF(q), Acta Arith., 47, 255, 10.4064/aa-47-3-255-261 Niederreiter, 1990, A short proof for explicit formulas for discrete logarithms in finite fields, Appl. Algebra Engrg. Comm. Comput., 1, 55, 10.1007/BF01810847 Satoh, 2006, On degrees of polynomial interpolations related to elliptic curves, vol. 3969, 155 Satoh, 2006, On polynomial interpolations related to Verheul homomorphisms, LMS J. Comput. Math., 9, 135, 10.1112/S1461157000001224 Silverman, 1986, The Arithmetic of Elliptic Curves, vol. 106 Vélu, 1971, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math., 273, 238 Verheul, 2001, Evidence that XTR is more secure than supersingular elliptic curve cryptosystem, 195 Weil, 1948, Variétés abéliennes et courbes algébriques, vol. 8 Wells, 1984, A polynomial form for logarithms modulo a prime, IEEE Trans. Inform. Theory, 30, 845, 10.1109/TIT.1984.1056986