G. Adj, Discrete logarithms in the cryptographically-interesting field GF(3 6 * 509 Invited talk slides available at http://ecc2016.yasar.edu.tr/slides/ecc2016-gora.pdf. 2. , Logaritmo discreto en campos finitos de característica pequeña: atacando la criptogrfía basada en emparejamientos de tipo 1, Elliptic Curve Cryptography Conference (ECC) Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, 2016.

G. Adj, I. Canales-martínez, N. Cruz-cortés, A. Menezes, T. Oliveira et al., Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields, Cryptology ePrint Archive, 2016.

G. Adj, A. Menezes, T. Oliveira, and F. , Computing discrete logarithms in F 3 6·137 and F 3 6·163 using Magma Arithmetic of Finite Fields, Weakness of F 3 6509 for discrete logarithm cryptography, PAIRING 2013 (Zhenfu Cao and Fangguo Zhang, pp.3-22, 2014.

L. Adleman, The function field sieve Algorithmic Number Theory (ANTS-I), LNCS, vol.877, pp.141-154, 1994.

M. Leonard, . Adleman, A. Ming-deh, and . Huang, Function field sieve method for discrete logarithms over finite fields, Information and Computation, vol.151, issue.12, pp.5-16, 1999.

R. Barbulescu, Algorithmes de logarithmes discrets dans les corps finis, thèse de doctorat

R. Barbulescu, An appendix for a recent paper of Kim, Cryptology ePrint Archive, 1076.

R. Barbulescu, C. Bouvier, J. Detrey, P. Gaudry, H. Jeljeli et al., Discrete logarithm in GF(2 809 ) with FFS, PKC LNCS, vol.2014, issue.8383, pp.221-238, 2014.
DOI : 10.1007/978-3-642-54631-0_13

URL : https://hal.archives-ouvertes.fr/hal-00818124

R. Barbulescu and P. Gaudry, Aurore Guillevic, and François Morain, Improving NFS for the discrete logarithm problem in non-prime finite fields, LNCS, vol.9056, pp.129-155, 2015.

R. Barbulescu, P. Gaudry, A. Joux, and E. Thomé, A Heuristic Quasi-Polynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic, LNCS, vol.8441, pp.1-16, 2014.
DOI : 10.1007/978-3-642-55220-5_1

URL : https://hal.archives-ouvertes.fr/hal-00835446

R. Barbulescu, P. Gaudry, and T. Kleinjung, The Tower Number Field Sieve, LNCS, vol.9453, pp.31-55, 2015.
DOI : 10.1007/978-3-662-48800-3_2

URL : https://hal.archives-ouvertes.fr/hal-01155635

I. F. Blake, R. Fuji-hara, R. C. Mullin, and S. A. Vanstone, Computing Logarithms in Finite Fields of Characteristic Two, SIAM Journal on Algebraic Discrete Methods, vol.5, issue.2, pp.276-285, 1984.
DOI : 10.1137/0605029

I. F. Blake, R. C. Mullin, and S. A. Vanstone, Computing logarithms in GF(2 n ), CRYPTO'84, LNCS, vol.196, pp.73-82, 1984.
DOI : 10.1007/3-540-39568-7_8

A. Commeine and I. Semaev, An Algorithm to Solve the Discrete Logarithm Problem with the Number Field Sieve, LNCS, vol.3958, pp.174-190, 2006.
DOI : 10.1007/11745853_12

D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two, IEEE Transactions on Information Theory, vol.30, issue.4, pp.587-594, 1984.
DOI : 10.1109/TIT.1984.1056941

D. Coppersmith, A. M. Odlyzko, and R. Schroeppel, Discrete logarithms inGF(p), Algorithmica, vol.13, issue.1-4, pp.1-15, 1986.
DOI : 10.6028/jres.045.026

W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, vol.22, issue.6, pp.644-654, 1976.
DOI : 10.1109/TIT.1976.1055638

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.9720

M. Drmota and D. Panario, A rigorous proof of the Waterloo algorithm for the discrete logarithm problem, Designs, Codes and Cryptography, vol.26, issue.1/3, pp.229-241, 2002.
DOI : 10.1023/A:1016521712726

J. Dumas and C. Pernet, Handbook of finite fields, ch. Computational linear algebra over finite fields, pp.520-535, 2013.

P. Flajolet, X. Gourdon, and D. Panario, The Complete Analysis of a Polynomial Factorization Algorithm over Finite Fields, Journal of Algorithms, vol.40, issue.1, pp.37-81, 2001.
DOI : 10.1006/jagm.2001.1158

URL : https://hal.archives-ouvertes.fr/inria-00073319

J. Fried, P. Gaudry, N. Heninger, and E. Thomé, A Kilobit Hidden SNFS Discrete Logarithm Computation, LNCS, vol.33, issue.5, pp.202-231, 2017.
DOI : 10.1007/3-540-68697-5_8

URL : https://hal.archives-ouvertes.fr/hal-01376934

M. Daniel and . Gordon, Discrete logarithms in GF(p) using the number field sieve, SIAM Journal on Discrete Mathematics, vol.6, issue.1, pp.124-138, 1993.

R. Granger, P. Jovanovic, B. Mennink, and S. Neves, Improved masking for tweakable blockciphers with applications to authenticated encryption, Cryptology ePrint Archive Improved masking for tweakable blockciphers with applications to authenticated encryption, EUROCRYPT 2016, Part I (Marc Fischlin and Jean-Sébastien Coron, LNCS, vol.999, issue.9665, pp.263-293, 2015.

R. Granger, T. Kleinjung, and J. Zumbrägel, Breaking '128-bit secure' supersingular binary curves -(or how to solve discrete logarithms in F 2 4·1223 and F 2 12·367 ), CRYPTO 2014 Breaking '128-bit secure' supersingular binary curves (or how to solve discrete logarithms, Cryptology ePrint Archive, pp.126-145, 2014.
DOI : 10.1007/978-3-662-44381-1_8

URL : https://infoscience.epfl.ch/record/215153/files/Neuchatel_short.pdf

L. Grémy, A. Guillevic, F. Morain, and E. Thomé, Computing discrete logarithms in GF, Selected areas in cryptography conference, 2017.

A. Guillevic, Computing Individual Discrete Logarithms Faster in $${{\mathrm{GF}}}(p^n)$$ with the NFS-DL Algorithm, LNCS, vol.9452, pp.149-173, 2015.
DOI : 10.1007/978-3-662-48797-6_7

A. Joux and R. Lercier, The function field sieve is quite special Algorithmic 35. , Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the Gaussian integer method, LNCS, vol.2369, issue.242, pp.431-445, 2002.
DOI : 10.1007/3-540-45455-1_34

A. Joux, R. Lercier, N. Smart, and F. Vercauteren, The Number Field Sieve in the Medium Prime Case, LNCS, vol.4117, pp.326-344, 2006.
DOI : 10.1007/11818175_19

URL : https://hal.archives-ouvertes.fr/hal-01102034

A. Joux and C. Pierrot, Improving the polynomial time precomputation of frobenius representation discrete logarithm algorithms -simplified setting for small characteristic finite fields, ASIACRYPT 2014, Part I (Palash Sarkar and Tetsu Iwata The special number field sieve in F p n -application to pairing-friendly constructions, LNCS LNCS, vol.8873, issue.8365, pp.378-397, 2014.

A. Joux and C. Pierrot, Discrete logarithm record in characteristic 3, GF(3 5·479 ) a 3796-bit field, Number Theory list, item 004745, 2014.

M. Kalkbrener, An upper bound on the number of monomials in determinants of sparse matrices with symbolic entries, Mathematica Pannonica, vol.8, pp.73-82, 1997.

T. Kim, Extended tower number field sieve: A new complexity for medium prime case, Cryptology ePrint Archive, 1027.
DOI : 10.1007/978-3-662-53018-4_20

T. Kim and R. Barbulescu, Extended tower number field sieve: A new complexity for the medium prime case, CRYPTO 2016, LNCS, vol.9814, pp.543-571, 2016.

T. Kim and J. Jeong, Extended Tower Number Field Sieve with Application to Finite Fields of Arbitrary Composite Extension Degree, Proceedings, Part I (Serge Fehr, pp.388-408, 2017.
DOI : 10.1007/978-3-642-34931-7_24

T. Kleinjung, Discrete logarithms in GF(2 1279 ), Number Theory list, item 004751, 2014.

T. Kleinjung, C. Diem, A. K. Lenstra, C. Priplata, and C. Stahlke, Computation of a 768-Bit Prime Field Discrete Logarithm, Coron and Nielsen [21], pp.185-201
DOI : 10.1109/TIT.1986.1057137

A. Brian, A. M. Lamacchia, and . Odlyzko, Computation of discrete logarithms in prime fields, Des. Codes Cryptography, vol.1, issue.1, pp.47-62, 1991.

A. K. Lenstra, . Jr, H. W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen, vol.32, issue.4, pp.515-534, 1982.
DOI : 10.1007/BF01457454

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.310.318

H. W. Lenstra, J. Jr, C. Pila, and . Pomerance, A Hyperelliptic Smoothness Test. I, A hyperelliptic smoothness test. II, Proc. London Math, pp.397-408, 1993.
DOI : 10.1098/rsta.1993.0138

URL : https://openaccess.leidenuniv.nl/bitstream/handle/1887/3840/346_118.pdf?sequence=1

D. Matyukhin, Effective version of the number field sieve for discrete logarithms in the field GF(p k ) (in Russian, Trudy po Discretnoi Matematike 9, pp.121-151, 2006.

M. Andrew and . Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, LNCS, vol.209, issue.84, pp.224-314, 1985.

C. Pomerance, Analysis and comparison of some integer factoring algorithms, Computational methods in number theory, Mathematical Centre Tracts, pp.89-139, 1982.

F. Rodríguez-henríquez, Another initial splitting in small characteristic finite fields, Personal communication, 2015.

P. Sarkar and S. Singh, A generalisation of the conjugation method for polynomial selection for the extended tower number field sieve algorithm, Cryptology ePrint Archive New complexity trade-offs for the (multiple) number field sieve algorithm in non-prime fields, EUROCRYPT 2016 Tower number field sieve variant of a recent polynomial selection method, LNCS LNCS, vol.10031401, issue.9665, pp.37-62, 2016.

O. Schirokauer, Discrete Logarithms and Local Units, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.345, issue.1676, pp.409-423, 1993.
DOI : 10.1098/rsta.1993.0139

M. Van, D. , and D. P. Woodruff, Asymptotically optimal communication for torusbased cryptography, LNCS, vol.3152, pp.157-178, 2004.

Y. Zhu, J. Zhuang, C. Lv, and D. Lin, Improvements on the individual logarithm step in extended tower number field sieve, Cryptology ePrint Archive, 2016.

I. Nancy and ?. Est, 615 rue du jardin botanique, CS 20101, 54603 Villers-l` es-Nancy Cedex, France E-mail address: aurore.guillevic@inria.fr URL: https://members