# American Institute of Mathematical Sciences

February  2014, 8(1): 103-118. doi: 10.3934/amc.2014.8.103

## Heuristics of the Cocks-Pinch method

 1 Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence Cedex, France

Received  April 2013 Revised  December 2013 Published  January 2014

We heuristically analyze the Cocks-Pinch method by using the Bateman-Horn conjecture. Especially, we present the first known heuristic which suggests that any efficient construction of pairing-friendly elliptic curves can efficiently generate such curves over pairing-friendly fields, naturally including the Cocks-Pinch method. Finally, some numerical evidence is given.
Citation: Min Sha. Heuristics of the Cocks-Pinch method. Advances in Mathematics of Communications, 2014, 8 (1) : 103-118. doi: 10.3934/amc.2014.8.103
##### 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). Google Scholar [2] P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order,, in Selected Areas in Cryptography 2005, (2005), 319. doi: 10.1007/11693383_22. Google Scholar [3] P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers,, Math. Comp., 16 (1962), 363. doi: 10.1090/S0025-5718-1962-0148632-7. Google Scholar [4] D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing,, SIAM J. Comput., 32 (2003), 586. doi: 10.1137/S0097539701398521. Google Scholar [5] D. Boneh, E.-J. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts,, in Proc. TCC 2005, (2005), 325. doi: 10.1007/978-3-540-30576-7_18. Google Scholar [6] D. Boneh, B. Lynn and H. Shacham, Short signatures from the Weil pairing,, J. Cryptology, 17 (2004), 297. doi: 10.1007/s00145-004-0314-9. Google Scholar [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. doi: 10.1016/j.jnt.2010.05.001. Google Scholar [8] J. Boxall, Heuristics on pairing-friendly elliptic curves,, J. Math. Cryptol., 6 (2012), 81. Google Scholar [9] C. Cocks and R. G. E. Pinch, Identity-based cryptosystems based on the Weil pairing,, manuscript, (2001). Google Scholar [10] J. Esmonde and M. R. Murty, Problems in Algebraic Number Theory,, Springer-Verlag, (2004). doi: 10.1007/978-3-642-87939-5. Google Scholar [11] S. Finch, G. Martin and P. Sebah, Roots of unity and nullity modulo $n$,, Proc. Amer. Math. Soc., 138 (2010), 2729. doi: 10.1090/S0002-9939-10-10341-4. Google Scholar [12] D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves,, J. Cryptology, 23 (2010), 224. doi: 10.1007/s00145-009-9048-z. Google Scholar [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. doi: 10.2307/2153546. Google Scholar [14] S. Galbraith, Pairings,, in Advances in Elliptic Curve Cryptography (eds. I. Blake, (2005), 183. doi: 10.1017/CBO9780511546570.011. Google Scholar [15] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers,, Oxford University Press, (1979). Google Scholar [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, (2012), 43. Google Scholar [17] A. Joux, A one round protocol for tripartite Diffie-Hellman,, in Algorithmic Number Theory Symposium 2000, (2000), 385. doi: 10.1007/10722028_23. Google Scholar [18] N. Koblitz and A. J. Menezes, Pairing-based cryptography at high security levels,, in Cryptography and Coding, (2005), 13. doi: 10.1007/11586821_2. Google Scholar [19] J. Korevaar and H. Te Riele, Average prime-pair counting formula,, Math. Comp., 79 (2010), 1209. doi: 10.1090/S0025-5718-09-02312-6. Google Scholar [20] F. Luca and I. E. Shparlinski, Elliptic curves with low embedding degree,, J. Cryptology, 19 (2006), 553. doi: 10.1007/s00145-006-0544-0. Google Scholar [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. doi: 10.1109/18.259647. Google Scholar [22] V. Miller, The Weil pairing, and its efficient calculation,, J. Cryptology, 17 (2004), 235. doi: 10.1007/s00145-004-0315-8. Google Scholar [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. Google Scholar [24] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers,, Springer-Verlag, (2004). Google Scholar [25] , PARI/GP, version 2.5.3,, Bordeaux, (2012). Google Scholar [26] K. Paterson, Cryptography from pairings,, in Advances in Elliptic Curve Cryptography (eds. I. Blake, (2005), 215. doi: 10.1017/CBO9780511546570.012. Google Scholar [27] R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing,, in Symp. Crypt. Inf. Secur. 2000, (2000). Google Scholar [28] A. V. Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem,, Math. Comp., 80 (2011), 501. doi: 10.1090/S0025-5718-2010-02373-7. Google Scholar [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. doi: 10.1090/S0025-5718-2011-02543-3. Google Scholar [30] E. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems,, J. Cryptology, 17 (2004), 277. doi: 10.1007/s00145-004-0313-x. Google Scholar

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). Google Scholar [2] P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order,, in Selected Areas in Cryptography 2005, (2005), 319. doi: 10.1007/11693383_22. Google Scholar [3] P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers,, Math. Comp., 16 (1962), 363. doi: 10.1090/S0025-5718-1962-0148632-7. Google Scholar [4] D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing,, SIAM J. Comput., 32 (2003), 586. doi: 10.1137/S0097539701398521. Google Scholar [5] D. Boneh, E.-J. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts,, in Proc. TCC 2005, (2005), 325. doi: 10.1007/978-3-540-30576-7_18. Google Scholar [6] D. Boneh, B. Lynn and H. Shacham, Short signatures from the Weil pairing,, J. Cryptology, 17 (2004), 297. doi: 10.1007/s00145-004-0314-9. Google Scholar [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. doi: 10.1016/j.jnt.2010.05.001. Google Scholar [8] J. Boxall, Heuristics on pairing-friendly elliptic curves,, J. Math. Cryptol., 6 (2012), 81. Google Scholar [9] C. Cocks and R. G. E. Pinch, Identity-based cryptosystems based on the Weil pairing,, manuscript, (2001). Google Scholar [10] J. Esmonde and M. R. Murty, Problems in Algebraic Number Theory,, Springer-Verlag, (2004). doi: 10.1007/978-3-642-87939-5. Google Scholar [11] S. Finch, G. Martin and P. Sebah, Roots of unity and nullity modulo $n$,, Proc. Amer. Math. Soc., 138 (2010), 2729. doi: 10.1090/S0002-9939-10-10341-4. Google Scholar [12] D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves,, J. Cryptology, 23 (2010), 224. doi: 10.1007/s00145-009-9048-z. Google Scholar [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. doi: 10.2307/2153546. Google Scholar [14] S. Galbraith, Pairings,, in Advances in Elliptic Curve Cryptography (eds. I. Blake, (2005), 183. doi: 10.1017/CBO9780511546570.011. Google Scholar [15] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers,, Oxford University Press, (1979). Google Scholar [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, (2012), 43. Google Scholar [17] A. Joux, A one round protocol for tripartite Diffie-Hellman,, in Algorithmic Number Theory Symposium 2000, (2000), 385. doi: 10.1007/10722028_23. Google Scholar [18] N. Koblitz and A. J. Menezes, Pairing-based cryptography at high security levels,, in Cryptography and Coding, (2005), 13. doi: 10.1007/11586821_2. Google Scholar [19] J. Korevaar and H. Te Riele, Average prime-pair counting formula,, Math. Comp., 79 (2010), 1209. doi: 10.1090/S0025-5718-09-02312-6. Google Scholar [20] F. Luca and I. E. Shparlinski, Elliptic curves with low embedding degree,, J. Cryptology, 19 (2006), 553. doi: 10.1007/s00145-006-0544-0. Google Scholar [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. doi: 10.1109/18.259647. Google Scholar [22] V. Miller, The Weil pairing, and its efficient calculation,, J. Cryptology, 17 (2004), 235. doi: 10.1007/s00145-004-0315-8. Google Scholar [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. Google Scholar [24] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers,, Springer-Verlag, (2004). Google Scholar [25] , PARI/GP, version 2.5.3,, Bordeaux, (2012). Google Scholar [26] K. Paterson, Cryptography from pairings,, in Advances in Elliptic Curve Cryptography (eds. I. Blake, (2005), 215. doi: 10.1017/CBO9780511546570.012. Google Scholar [27] R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing,, in Symp. Crypt. Inf. Secur. 2000, (2000). Google Scholar [28] A. V. Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem,, Math. Comp., 80 (2011), 501. doi: 10.1090/S0025-5718-2010-02373-7. Google Scholar [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. doi: 10.1090/S0025-5718-2011-02543-3. Google Scholar [30] E. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems,, J. Cryptology, 17 (2004), 277. doi: 10.1007/s00145-004-0313-x. Google Scholar
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