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New constructions of optimal frequency hopping sequences with new parameters
Generalizations of Verheul's theorem to asymmetric pairings
1. | Department of Mathematics, Bilkent University, Bilkent, Ankara 06800, Turkey |
2. | Google, Inc., 1600 Amphitheater Parkway, Mountain View, California 94043, United States |
3. | Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1 |
References:
[1] |
D. Aranha, K. Karabina, P. Longa, C. Gebotys and J. López, Faster explicit formulas for computing pairings over ordinary curves, in "Advances in Cryptology - EUROCRYPT 2011,'' (2011), 48-68. |
[2] |
P. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in "Selected Areas in Cryptography - SAC 2005,'' (2006), 319-331. |
[3] |
D. Boneh, B. Lynn, and H. Shacham, Short signatures from the Weil pairing, J. Cryptology, 17 (2004), 297-319.
doi: 10.1007/s00145-004-0314-9. |
[4] |
D. Brown and R. Gallant, The static Diffie-Hellman problem, available online at http://eprint.iacr.org/2004/306 |
[5] |
J. Cheon, Security analysis of the Strong Diffie-Hellman problem, in "Advances in Cryptology - EUROCRYPT 2006,'' (2006), 1-11. |
[6] |
D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two, IEEE Trans. Inform. Theory, 30 (1984), 587-594. |
[7] |
T. ElGamal, A subexponential-time algorithm for computing discrete logarithms over $GF(p^{2})$, IEEE Trans. Inform. Theory, 31 (1985), 473-481. |
[8] |
G. Frey and H. Rück, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comput., 62 (1994), 865-874. |
[9] |
S. Galbraith, Pairings, in "Advances in Elliptic Curve Cryptography'' (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005. |
[10] |
S. Galbraith, C. Ó hÉigeartaigh and C. Sheedy, Simplified pairing computation and security implications, J. Math. Crypt., 1 (2007), 267-281. |
[11] |
S. Galbraith, F. Hess and F. Vercauteren, Aspects of pairing inversion, IEEE Trans. Inform. Theory, 54 (2008), 5719-5728.
doi: 10.1109/TIT.2008.2006431. |
[12] |
S. Galbraith, K. Paterson and N. Smart, Pairings for cryptographers, Discr. Appl. Math., 156 (2008), 3113-3121.
doi: 10.1016/j.dam.2007.12.010. |
[13] |
Z. Hu, M. Xu and Z. Zhou, A generalization of Verheul's theorem for some ordinary curves, in "Inscrypt 2010,'' (2011), 105-114. |
[14] |
N. Koblitz and A. Menezes, Pairing-based cryptography at high security levels, in "Cryptography and Coding: 10th IMA International Conference,'' (2005), 13-36. |
[15] |
A. Lenstra and E. Verheul, The XTR public key system, in "Advances in Cryptology - CRYPTO 2000,'' (2000), 1-19. |
[16] |
A. Menezes, T. Okamoto and S. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Trans. Inform. Theory, 39 (1993), 1639-1646.
doi: 10.1109/18.259647. |
[17] |
A. Menezes and S. Vanstone, ECSTR (XTR): Elliptic curve singular trace representation, in "Rump Session of Crypto 2000''. |
[18] |
D. Mireles Morales, Cheon's algorithm, pairing inversion and the discrete logarithm problem, available online at http://eprint.iacr.org/2008/300 |
[19] |
D. Moody, The Diffie-Hellman problem and generalization of Verheul's theorem, Des. Codes Crypt., 52 (2009), 381-390. |
[20] |
J. Pollard, Monte Carlo methods for index computation mod $p$, Math. Comput., 32 (1978), 918-924. |
[21] |
T. Satoh, On degrees of polynomial interpolations related to elliptic curve cryptography, in "WCC 2005,'' (2006), 155-163. |
[22] |
T. Satoh, On polynomial interpolations related to Verheul homomorphisms, LMS J. Comput. Math., 9 (2006), 135-158. |
[23] |
T. Satoh, Closed formulae for the Weil pairing inversion, Finite Fields Appl., 14 (2008), 743-765.
doi: 10.1016/j.ffa.2007.12.003. |
[24] |
E. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems, J. Cryptology, 17 (2004), 277-296.
doi: 10.1007/s00145-004-0313-x. |
show all references
References:
[1] |
D. Aranha, K. Karabina, P. Longa, C. Gebotys and J. López, Faster explicit formulas for computing pairings over ordinary curves, in "Advances in Cryptology - EUROCRYPT 2011,'' (2011), 48-68. |
[2] |
P. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in "Selected Areas in Cryptography - SAC 2005,'' (2006), 319-331. |
[3] |
D. Boneh, B. Lynn, and H. Shacham, Short signatures from the Weil pairing, J. Cryptology, 17 (2004), 297-319.
doi: 10.1007/s00145-004-0314-9. |
[4] |
D. Brown and R. Gallant, The static Diffie-Hellman problem, available online at http://eprint.iacr.org/2004/306 |
[5] |
J. Cheon, Security analysis of the Strong Diffie-Hellman problem, in "Advances in Cryptology - EUROCRYPT 2006,'' (2006), 1-11. |
[6] |
D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two, IEEE Trans. Inform. Theory, 30 (1984), 587-594. |
[7] |
T. ElGamal, A subexponential-time algorithm for computing discrete logarithms over $GF(p^{2})$, IEEE Trans. Inform. Theory, 31 (1985), 473-481. |
[8] |
G. Frey and H. Rück, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comput., 62 (1994), 865-874. |
[9] |
S. Galbraith, Pairings, in "Advances in Elliptic Curve Cryptography'' (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005. |
[10] |
S. Galbraith, C. Ó hÉigeartaigh and C. Sheedy, Simplified pairing computation and security implications, J. Math. Crypt., 1 (2007), 267-281. |
[11] |
S. Galbraith, F. Hess and F. Vercauteren, Aspects of pairing inversion, IEEE Trans. Inform. Theory, 54 (2008), 5719-5728.
doi: 10.1109/TIT.2008.2006431. |
[12] |
S. Galbraith, K. Paterson and N. Smart, Pairings for cryptographers, Discr. Appl. Math., 156 (2008), 3113-3121.
doi: 10.1016/j.dam.2007.12.010. |
[13] |
Z. Hu, M. Xu and Z. Zhou, A generalization of Verheul's theorem for some ordinary curves, in "Inscrypt 2010,'' (2011), 105-114. |
[14] |
N. Koblitz and A. Menezes, Pairing-based cryptography at high security levels, in "Cryptography and Coding: 10th IMA International Conference,'' (2005), 13-36. |
[15] |
A. Lenstra and E. Verheul, The XTR public key system, in "Advances in Cryptology - CRYPTO 2000,'' (2000), 1-19. |
[16] |
A. Menezes, T. Okamoto and S. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Trans. Inform. Theory, 39 (1993), 1639-1646.
doi: 10.1109/18.259647. |
[17] |
A. Menezes and S. Vanstone, ECSTR (XTR): Elliptic curve singular trace representation, in "Rump Session of Crypto 2000''. |
[18] |
D. Mireles Morales, Cheon's algorithm, pairing inversion and the discrete logarithm problem, available online at http://eprint.iacr.org/2008/300 |
[19] |
D. Moody, The Diffie-Hellman problem and generalization of Verheul's theorem, Des. Codes Crypt., 52 (2009), 381-390. |
[20] |
J. Pollard, Monte Carlo methods for index computation mod $p$, Math. Comput., 32 (1978), 918-924. |
[21] |
T. Satoh, On degrees of polynomial interpolations related to elliptic curve cryptography, in "WCC 2005,'' (2006), 155-163. |
[22] |
T. Satoh, On polynomial interpolations related to Verheul homomorphisms, LMS J. Comput. Math., 9 (2006), 135-158. |
[23] |
T. Satoh, Closed formulae for the Weil pairing inversion, Finite Fields Appl., 14 (2008), 743-765.
doi: 10.1016/j.ffa.2007.12.003. |
[24] |
E. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems, J. Cryptology, 17 (2004), 277-296.
doi: 10.1007/s00145-004-0313-x. |
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