R. Balasubramanian and N. Koblitz, The improbability that an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm, J. Cryptology, vol.11, pp.141-145, 1998.

P. S. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, Selected Areas in Cryptography (SAC 2005), vol.3897, pp.319-331, 2006.

P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp, vol.16, pp.363-367, 1962.

K. Belabas, , 2010.

D. Boneh, E. Goh, and K. Nissim, Evaluating 2-DNF formulas on ciphertexts, Theory of Cryptography (TCC 2005), vol.3897, pp.325-341, 2005.
DOI : 10.1007/978-3-540-30576-7_18

URL : https://link.springer.com/content/pdf/10.1007%2F978-3-540-30576-7_18.pdf

D. Boneh, K. Rubin, and A. Silverberg, Finding composite order ordinary elliptic curves using the Cocks-Pinch method, J. Number Theory, vol.131, pp.832-841, 2011.

F. Brezing and A. Weng, Elliptic curves suitable for pairing based cryptography, Des. Codes Cryptogr, vol.37, pp.133-141, 2005.
DOI : 10.1007/s10623-004-3808-4

C. Cocks and R. G. Pinch, ID-based cryptosystems based on the Weil pairing, 2001.

K. Conrad, . Hardy-littlewood, and . Constants, Mathematical Properties of Sequences and Other Combinatorical Structures, pp.133-154, 2003.

H. Davenport and A. Schinzel, A note on certain arithmetical constants, Illinois J. Math, vol.10, pp.181-185, 1966.
DOI : 10.1215/ijm/1256055100

URL : https://doi.org/10.1215/ijm/1256055100

M. Deuring, Die Typen der Multiplikatorenringe elliptische Funktionenkörper, Abh. Math. Sem. Univ. Hamburg, vol.14, pp.197-272, 1941.
DOI : 10.1007/bf02940746

A. Enge and A. V. Sutherland, Class invariants by the CRT method, Algorithmic Number Theory, vol.6197, pp.142-156, 2010.
DOI : 10.1007/978-3-642-14518-6_14

URL : https://hal.archives-ouvertes.fr/inria-00448729

D. Freeman, Converting pairing-based cryptosystems from composite order groups to prime order groups, Advances in Cryptology, vol.6110, pp.44-61, 2010.
DOI : 10.1007/978-3-642-13190-5_3

URL : https://link.springer.com/content/pdf/10.1007%2F978-3-642-13190-5_3.pdf

D. Freeman, M. Scott, and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology, vol.23, pp.224-280, 2010.

S. D. Galbraith, J. F. Mckee, and P. C. Valença, Ordinary abelian varieties having small embedding degree, Finite Fields Appl, vol.13, pp.800-814, 2007.
DOI : 10.1016/j.ffa.2007.02.003

URL : https://doi.org/10.1016/j.ffa.2007.02.003

D. Goldfeld and J. Hoffstein, Eisenstein series of 1 2 -integral weight and the mean value of Dirichlet L-series, Invent. Math, vol.80, pp.185-208, 1985.

T. Granlund, GMP multiprecision arithmetic library

A. Joux, A one-round protocol for tripartite Diffie-Hellman, vol.1838, pp.385-394, 2000.

J. Korevaar and H. Te-riele, Average prime-pair counting formula, Math. Comp, vol.79, pp.1209-1229, 2010.
DOI : 10.1090/s0025-5718-09-02312-6

URL : https://www.ams.org/mcom/2010-79-270/S0025-5718-09-02312-6/S0025-5718-09-02312-6.pdf

F. Luca and I. Shparlinski, Elliptic curves of low embedding degree, J. Cryptology, vol.19, pp.553-562, 2006.

A. Miyaji, M. Nakabayashi, and S. Takano, New explicit conditions of elliptic curve traces for FR-reduction, IEICE Trans. Fundamentals E84-A, pp.1234-1243, 2001.

W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2004.

J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math, vol.2, pp.134-144, 1966.

J. J. Urroz, F. Luca, and I. Shparlinski, On the number of isogeny classes and pairingfriendly elliptic curves and statistics for MNT curves, Math. Comp, vol.81, pp.1093-1110, 2012.

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math, vol.83, 1982.

W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér, vol.2, issue.4, pp.521-560, 1969.

, Author information, 2011.

J. Boxall, 14032 Caen cedex, Laboratoire de Mathématiques Nicolas Oresme, vol.5186