, No cyclotomic family of embedding degree above 32 satisfying (9) was found. For BN, BLS12, BLS24, KSS16, KSS18, we reproduce in Table 9 the results of Guillevic and Singh [23]: BN with a 1022-bit p, BLS12 with a 1150-bit p, KSS16 with a 766-bit prime p, KSS18 with a 638-bit prime p, BLS24 with a 509-bit prime p. We list in Table 10 seed ranges for k ? {14, 15, 20, 21, 27, 28}. We also refer to [18] for alternative curves with ? = 2. We leave to future work a complete study of pairing-friendly curves at the 192-bit security level, ? 37, and with ? = 2 we obtain k ? 10. Curves like Fotiadis-Konstantinou with exactly ? = 2 satisfy, vol.18

D. F. Aranha and C. P. Gouvêa, RELIC is an Efficient LIbrary for Cryptography

C. Arène, T. Lange, M. Naehrig, and C. Ritzenthaler, Faster computation of the Tate pairing, Elliptic Curve Cryptography), vol.131, pp.842-857, 2011.

R. Barbulescu and S. Duquesne, Updating key size estimations for pairings, Journal of Cryptology, vol.32, issue.4, pp.1298-1336, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01534101

R. Barbulescu, N. El-mrabet, and L. Ghammam, A taxonomy of pairings, their security, their complexity, Sept, vol.24, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02129868

R. Barbulescu, P. Gaudry, and T. Kleinjung, The tower number field sieve, ASIACRYPT 2015, Part II, vol.9453, pp.31-55
URL : https://hal.archives-ouvertes.fr/hal-01155635

, , 2015.

P. S. Barreto, B. Lynn, and M. Scott, Constructing elliptic curves with prescribed embedding degrees, SCN 02, vol.2576, pp.257-267, 2003.

S. Bowe, New zk-SNARK elliptic curve construction. Zcash blog, pp.12-381, 2017.

S. Bowe, A. Chiesa, M. Green, I. Miers, P. Mishra et al., Zexe: Enabling decentralized private computation, vol.962, 2018.

F. Brezing and A. Weng, Elliptic curves suitable for pairing based cryptography, Des. Codes Cryptography, vol.37, issue.1, pp.133-141, 2005.

S. Chatterjee, A. Menezes, and F. Rodríguez-henríquez, On instantiating pairingbased protocols with elliptic curves of embedding degree one, IEEE Transactions on Computer, vol.66, issue.6, pp.1061-1070, 2017.

S. Chatterjee, P. Sarkar, and R. Barua, Efficient computation of Tate pairing in projective coordinate over general characteristic fields, ICISC 04, vol.3506, pp.168-181, 2005.

A. Chiesa, L. Chua, and M. Weidner, On cycles of pairing-friendly elliptic curves, SIAM Journal on Applied Algebra and Geometry, vol.3, issue.2, pp.175-192, 2019.

C. Costello, T. Lange, and M. Naehrig, Faster pairing computations on curves with high-degree twists, PKC 2010, vol.6056, pp.224-242, 2010.

C. Costello, K. Lauter, and M. Naehrig, Attractive subfamilies of BLS curves for implementing high-security pairings, DOCRYPT 2011, vol.7107, pp.320-342, 2011.

. Euthereum,

K. Foster, HT90 and "simplest" number fields, Illinois J. Math, vol.55, issue.4, pp.1621-1655, 2011.

G. Fotiadis and E. Konstantinou, TNFS resistant families of pairing-friendly elliptic curves, Theoretical Computer Science, vol.800, pp.73-89, 2019.

G. Fotiadis and C. Martindale, Optimal TNFS-secure pairings on elliptic curves with composite embedding degree, vol.555, 2019.

E. Fouotsa, N. El-mrabet, and A. Pecha, Computing optimal ate pairings on elliptic curves with embedding degree 9, 1187.

D. Freeman, M. Scott, and E. Teske, A taxonomy of pairing-friendly elliptic curves, Journal of Cryptology, vol.23, issue.2, pp.224-280, 2010.

S. Galbraith, Advances in Elliptic Curve Cryptography, pp.183-214, 2005.

A. Guillevic, S. Masson, and E. Thomé, Cocks-Pinch curves of embedding degrees five to eight and optimal ate pairing computation, vol.431, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02305051

A. Guillevic and S. Singh, On the alpha value of polynomials in the tower number field sieve algorithm, vol.885, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02263098

, Elliptic curve generation, 2 edn, ISO: ISO/IEC 15946-5:2017 Information technology -Security techniques -Cryptographic techniques based on elliptic curves -Part, vol.5, 2017.

A. Joux and C. Pierrot, The special number field sieve in Fpn -application to pairingfriendly constructions, PAIRING 2013, vol.8365, pp.45-61, 2014.

T. Kim and R. Barbulescu, Extended tower number field sieve: A new complexity for the medium prime case, CRYPTO 2016, Part I. LNCS, vol.9814, pp.543-571, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01281966

,

T. Kim and J. Jeong, Extended tower number field sieve with application to finite fields of arbitrary composite extension degree, Part I. LNCS, vol.10174, pp.388-408, 2017.

A. K. Lenstra and E. R. Verheul, Selecting cryptographic key sizes, Journal of Cryptology, vol.14, issue.4, pp.255-293, 2001.

A. Menezes, P. Sarkar, and S. Singh, Challenges with assessing the impact of NFS advances on the security of pairing-based cryptography, Mycrypt Conference, vol.10311, pp.83-108, 2016.

P. L. Montgomery, Five, six, and seven-term Karatsuba-like formulae, IEEE Transactions on Computer, vol.54, pp.362-369, 2005.

G. C. Pereira, M. A. Simplício, M. Naehrig, and P. S. Barreto, A family of implementation-friendly BN elliptic curves, Journal of Systems and Software, vol.84, issue.8, pp.1319-1326, 2011.

Y. Sakemi, T. Kobayashi, and T. Saito, Pairing-friendly curves, 2019.

M. Scott and A. Guillevic, A new family of pairing-friendly elliptic curves, pp.43-57, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01875361

B. Smith, Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians, Contemporary mathematics, vol.637, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00874925

R. S. Wahby and D. Boneh, Fast and simple constant-time hashing to the BLS12-381 elliptic curve, IACR TCHES, vol.2019, issue.4, pp.154-179, 2019.

,