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Sets of zero-difference balanced functions and their applications
Heuristics of the Cocks-Pinch method
1. | Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence Cedex, France |
References:
[1] |
R. Avanzi, H. Cohen, C. Doche, G. Frey, T. Lange, K. Nguyen and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, 2005. |
[2] |
P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in Selected Areas in Cryptography 2005, Springer-Verlag, 2006, 319-331.
doi: 10.1007/11693383_22. |
[3] |
P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp., 16 (1962), 363-367.
doi: 10.1090/S0025-5718-1962-0148632-7. |
[4] |
D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing, SIAM J. Comput., 32 (2003), 586-615.
doi: 10.1137/S0097539701398521. |
[5] |
D. Boneh, E.-J. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts, in Proc. TCC 2005, Springer-Verlag, 2005, 325-341.
doi: 10.1007/978-3-540-30576-7_18. |
[6] |
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. |
[7] |
D. Boneh, K. Rubin and A. Silverberg, Finding composite order ordinary elliptic curves using the Cocks-Pinch method, J. Number Theory, 131 (2011), 832-841.
doi: 10.1016/j.jnt.2010.05.001. |
[8] |
J. Boxall, Heuristics on pairing-friendly elliptic curves, J. Math. Cryptol., 6 (2012), 81-104. |
[9] |
C. Cocks and R. G. E. Pinch, Identity-based cryptosystems based on the Weil pairing, manuscript, 2001. |
[10] |
J. Esmonde and M. R. Murty, Problems in Algebraic Number Theory, Springer-Verlag, 2004.
doi: 10.1007/978-3-642-87939-5. |
[11] |
S. Finch, G. Martin and P. Sebah, Roots of unity and nullity modulo $n$, Proc. Amer. Math. Soc., 138 (2010), 2729-2743.
doi: 10.1090/S0002-9939-10-10341-4. |
[12] |
D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology, 23 (2010), 224-280.
doi: 10.1007/s00145-009-9048-z. |
[13] |
G. Frey and H. Rück, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp., 62 (1994), 865-874.
doi: 10.2307/2153546. |
[14] |
S. Galbraith, Pairings, in Advances in Elliptic Curve Cryptography (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005, 183-213.
doi: 10.1017/CBO9780511546570.011. |
[15] |
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1979. |
[16] |
T. Hayashi, T. Shimoyama, N. Shinohara and T. Takagi, Breaking pairing-based cryptosystems using $\eta_T$ pairing over $GF(3^{97})$, in Asiacrypt 2012, Springer-Verlag, 2012, 43-60. |
[17] |
A. Joux, A one round protocol for tripartite Diffie-Hellman, in Algorithmic Number Theory Symposium 2000, Springer-Verlag, 2000, 385-393.
doi: 10.1007/10722028_23. |
[18] |
N. Koblitz and A. J. Menezes, Pairing-based cryptography at high security levels, in Cryptography and Coding, Springer-Verlag, 2005, 13-36.
doi: 10.1007/11586821_2. |
[19] |
J. Korevaar and H. Te Riele, Average prime-pair counting formula, Math. Comp., 79 (2010), 1209-1229.
doi: 10.1090/S0025-5718-09-02312-6. |
[20] |
F. Luca and I. E. Shparlinski, Elliptic curves with low embedding degree, J. Cryptology, 19 (2006), 553-562.
doi: 10.1007/s00145-006-0544-0. |
[21] |
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. |
[22] |
V. Miller, The Weil pairing, and its efficient calculation, J. Cryptology, 17 (2004) 235-261.
doi: 10.1007/s00145-004-0315-8. |
[23] |
A. Miyaji, M. Nakabayashi and S. Takano, New explicit conditions of elliptic curve traces for FR-reduction, IEICE Trans. Fundam., E84-A (2001), 1234-1243. |
[24] |
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, 2004. |
[25] | |
[26] |
K. Paterson, Cryptography from pairings, in Advances in Elliptic Curve Cryptography (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005, 215-251.
doi: 10.1017/CBO9780511546570.012. |
[27] |
R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing, in Symp. Crypt. Inf. Secur. 2000, Okinawa, Japan, 2000. |
[28] |
A. V. Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem, Math. Comp., 80 (2011), 501-538.
doi: 10.1090/S0025-5718-2010-02373-7. |
[29] |
J. J. Urroz, F. Luca and I. E. Shparlinski, On the number of isogeny classes and pairing-friendly elliptic curves and statistics for MNT curves, Math. Comp., 81 (2012), 1093-1110.
doi: 10.1090/S0025-5718-2011-02543-3. |
[30] |
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] |
R. Avanzi, H. Cohen, C. Doche, G. Frey, T. Lange, K. Nguyen and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, 2005. |
[2] |
P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in Selected Areas in Cryptography 2005, Springer-Verlag, 2006, 319-331.
doi: 10.1007/11693383_22. |
[3] |
P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp., 16 (1962), 363-367.
doi: 10.1090/S0025-5718-1962-0148632-7. |
[4] |
D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing, SIAM J. Comput., 32 (2003), 586-615.
doi: 10.1137/S0097539701398521. |
[5] |
D. Boneh, E.-J. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts, in Proc. TCC 2005, Springer-Verlag, 2005, 325-341.
doi: 10.1007/978-3-540-30576-7_18. |
[6] |
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. |
[7] |
D. Boneh, K. Rubin and A. Silverberg, Finding composite order ordinary elliptic curves using the Cocks-Pinch method, J. Number Theory, 131 (2011), 832-841.
doi: 10.1016/j.jnt.2010.05.001. |
[8] |
J. Boxall, Heuristics on pairing-friendly elliptic curves, J. Math. Cryptol., 6 (2012), 81-104. |
[9] |
C. Cocks and R. G. E. Pinch, Identity-based cryptosystems based on the Weil pairing, manuscript, 2001. |
[10] |
J. Esmonde and M. R. Murty, Problems in Algebraic Number Theory, Springer-Verlag, 2004.
doi: 10.1007/978-3-642-87939-5. |
[11] |
S. Finch, G. Martin and P. Sebah, Roots of unity and nullity modulo $n$, Proc. Amer. Math. Soc., 138 (2010), 2729-2743.
doi: 10.1090/S0002-9939-10-10341-4. |
[12] |
D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology, 23 (2010), 224-280.
doi: 10.1007/s00145-009-9048-z. |
[13] |
G. Frey and H. Rück, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp., 62 (1994), 865-874.
doi: 10.2307/2153546. |
[14] |
S. Galbraith, Pairings, in Advances in Elliptic Curve Cryptography (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005, 183-213.
doi: 10.1017/CBO9780511546570.011. |
[15] |
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1979. |
[16] |
T. Hayashi, T. Shimoyama, N. Shinohara and T. Takagi, Breaking pairing-based cryptosystems using $\eta_T$ pairing over $GF(3^{97})$, in Asiacrypt 2012, Springer-Verlag, 2012, 43-60. |
[17] |
A. Joux, A one round protocol for tripartite Diffie-Hellman, in Algorithmic Number Theory Symposium 2000, Springer-Verlag, 2000, 385-393.
doi: 10.1007/10722028_23. |
[18] |
N. Koblitz and A. J. Menezes, Pairing-based cryptography at high security levels, in Cryptography and Coding, Springer-Verlag, 2005, 13-36.
doi: 10.1007/11586821_2. |
[19] |
J. Korevaar and H. Te Riele, Average prime-pair counting formula, Math. Comp., 79 (2010), 1209-1229.
doi: 10.1090/S0025-5718-09-02312-6. |
[20] |
F. Luca and I. E. Shparlinski, Elliptic curves with low embedding degree, J. Cryptology, 19 (2006), 553-562.
doi: 10.1007/s00145-006-0544-0. |
[21] |
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. |
[22] |
V. Miller, The Weil pairing, and its efficient calculation, J. Cryptology, 17 (2004) 235-261.
doi: 10.1007/s00145-004-0315-8. |
[23] |
A. Miyaji, M. Nakabayashi and S. Takano, New explicit conditions of elliptic curve traces for FR-reduction, IEICE Trans. Fundam., E84-A (2001), 1234-1243. |
[24] |
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, 2004. |
[25] | |
[26] |
K. Paterson, Cryptography from pairings, in Advances in Elliptic Curve Cryptography (eds. I. Blake, G. Seroussi and N. Smart), Cambridge University Press, 2005, 215-251.
doi: 10.1017/CBO9780511546570.012. |
[27] |
R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing, in Symp. Crypt. Inf. Secur. 2000, Okinawa, Japan, 2000. |
[28] |
A. V. Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem, Math. Comp., 80 (2011), 501-538.
doi: 10.1090/S0025-5718-2010-02373-7. |
[29] |
J. J. Urroz, F. Luca and I. E. Shparlinski, On the number of isogeny classes and pairing-friendly elliptic curves and statistics for MNT curves, Math. Comp., 81 (2012), 1093-1110.
doi: 10.1090/S0025-5718-2011-02543-3. |
[30] |
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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