S. Agrawal, D. Boneh, and X. Boyen, Efficient Lattice (H)IBE in the Standard Model, LNCS, vol.6110, pp.553-572, 2010.
DOI : 10.1007/978-3-642-13190-5_28

M. Alekhnovich, More on Average Case vs Approximation Complexity, computational complexity, vol.64, issue.6, pp.755-786, 2011.
DOI : 10.1016/S0020-0190(97)00190-7

B. Applebaum, Y. Ishai, and E. Kushilevitz, Cryptography with constant input locality, LNCS, vol.4622, pp.92-110, 2007.

S. L. Paulo, R. Barreto, M. A. Misoczki, . Jr, and . Simplicio, One-time signature scheme from syndrome decoding over generic error-correcting codes, Journal of Systems and Software, vol.84, issue.2, pp.198-204, 2011.

E. Berlekamp, R. Mceliece, and H. Van-tilborg, On the inherent intractability of certain coding problems (Corresp.), IEEE Transactions on Information Theory, vol.24, issue.3, pp.384-386, 1978.
DOI : 10.1109/TIT.1978.1055873

L. Bettale, Cryptanalyse algébrique : outils et applications, 2012.

D. Boneh, The Decision Diffie-Hellman problem, International Algorithmic Number Theory Symposium, pp.48-63, 1998.
DOI : 10.1007/BFb0054851

D. Boneh and X. Boyen, Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles, LNCS, vol.3027, pp.223-238, 2004.
DOI : 10.1007/978-3-540-24676-3_14

D. Boneh and X. Boyen, Secure Identity Based Encryption Without Random Oracles, LNCS, vol.3152, pp.443-459, 2004.
DOI : 10.1007/978-3-540-28628-8_27

D. Boneh and M. K. Franklin, Identity-based encryption from the Weil pairing, LNCS, vol.2139, pp.213-229, 2001.

X. Boyen, Lattice Mixing and Vanishing Trapdoors: A Framework for Fully Secure Short Signatures and More, PKC 2010, pp.499-517, 2010.
DOI : 10.1007/978-3-642-13013-7_29

R. Canetti, S. Halevi, and J. Katz, A Forward-Secure Public-Key Encryption Scheme, Journal of Cryptology, vol.20, issue.3, pp.265-294, 2007.
DOI : 10.1007/s00145-006-0442-5

D. Cash, D. Hofheinz, E. Kiltz, and C. Peikert, Bonsai trees, or how to delegate a lattice basis, LNCS, vol.6110, pp.523-552, 2010.

C. Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues, 8th IMA International Conference on Cryptography and Coding, pp.360-363, 2001.
DOI : 10.1007/3-540-45325-3_32

N. Courtois, M. Finiasz, and N. Sendrier, How to Achieve a McEliece-Based Digital Signature Scheme, ASI- ACRYPT 2001, pp.157-174, 2001.
DOI : 10.1007/3-540-45682-1_10
URL : https://hal.archives-ouvertes.fr/inria-00072511

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

T. Elgamal, A public key cryptosystem and a signature scheme based on discrete logarithms, LNCS, vol.84, issue.196, pp.10-18, 1984.

J. Faugère, M. Safey-el-din, and P. Spaenlehauer, Computing loci of rank defects of linear matrices using gröbner bases and applications to cryptology, ISSAC, 2010, Proceedings, pp.257-264, 2010.

J. Faugère, F. Levy-dit, L. Vehel, and . Perret, Cryptanalysis of Minrank Advances in Cryptology -CRYPTO, LNCS, vol.5157, pp.280-296, 2008.

E. Mukhamedovich-gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, vol.21, issue.1, pp.3-16, 1985.

M. Ernst, A. V. Gabidulin, O. V. Paramonov, and . Tretjakov, Ideals over a noncommutative ring and their applications to cryptography, Advances in Cryptology -EUROCRYPT'91, number 547 in LNCS, pp.482-489, 1991.

P. Gaborit, Shorter keys for code based cryptography, Proceedings of the 2005 International Workshop on Coding and Cryptography (WCC 2005), pp.81-91, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00078726

P. Gaborit, A. Hauteville, and J. Tillich, RankSynd a PRNG Based on Rank Metric, Post-Quantum Cryptography 2016, pp.18-28, 2016.
DOI : 10.1007/978-3-319-29360-8_2
URL : https://hal.archives-ouvertes.fr/hal-01289338

P. Gaborit, G. Murat, O. Ruatta, and G. Zémor, Low rank parity check codes and their application to cryptography, Proceedings of the Workshop on Coding and Cryptography WCC'2013, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00913719

P. Gaborit, O. Ruatta, and J. Schrek, On the Complexity of the Rank Syndrome Decoding Problem, IEEE Transactions on Information Theory, vol.62, issue.2, pp.1006-1019, 2016.
DOI : 10.1109/TIT.2015.2511786
URL : https://hal.archives-ouvertes.fr/hal-01280904

P. Gaborit, O. Ruatta, J. Schrek, and G. Zémor, RankSign: An Efficient Signature Algorithm Based on the Rank Metric, Post-Quantum Cryptography 2014, pp.88-107, 2014.
DOI : 10.1007/978-3-319-11659-4_6
URL : https://hal.archives-ouvertes.fr/hal-01261425

P. Gaborit and G. Zémor, On the Hardness of the Decoding and the Minimum Distance Problems for Rank Codes, IEEE Transactions on Information Theory, vol.62, issue.12, pp.7245-7252, 2016.
DOI : 10.1109/TIT.2016.2616127
URL : https://hal.archives-ouvertes.fr/hal-01443111

C. Gentry, Practical Identity-Based Encryption Without Random Oracles, LNCS, vol.32, issue.3, pp.445-464, 2006.
DOI : 10.1007/11426639_7

C. Gentry, C. Peikert, and V. Vaikuntanathan, Trapdoors for hard lattices and new cryptographic constructions, Proceedings of the fourtieth annual ACM symposium on Theory of computing, STOC 08, pp.197-206, 2008.
DOI : 10.1145/1374376.1374407

S. Goldwasser and S. Micali, Probabilistic encryption, Journal of Computer and System Sciences, vol.28, issue.2, pp.270-299, 1984.
DOI : 10.1016/0022-0000(84)90070-9

A. Hauteville, Décodage en métrique rang et attaques sur un système de chiffrement à base de codes LRPC, 2014.

A. Hauteville and J. Tillich, New algorithms for decoding in the rank metric and an attack on the LRPC cryptosystem, 2015 IEEE International Symposium on Information Theory (ISIT), 2015.
DOI : 10.1109/ISIT.2015.7282956
URL : https://hal.archives-ouvertes.fr/hal-01244619

G. Kabatianskii, E. Krouk, and B. J. Smeets, A digital signature scheme based on random error-correcting codes, IMA Int. Conf., volume 1355 of LNCS, pp.161-167, 1997.
DOI : 10.1007/BFb0024461

G. Kabatianskii, E. Krouk, and B. J. Smeets, Error Correcting Coding and Security for Data Networks: Analysis of the Superchannel Concept, 2005.
DOI : 10.1002/0470867574

F. Lévy, -. Vehel, and L. Perret, Algebraic decoding of codes in rank metric, proceedings of YACC06, 2006.

P. Loidreau, Asymptotic behaviour of codes in rank metric over finite fields, Designs, Codes and Cryptography, vol.44, issue.2, pp.105-118, 2014.
DOI : 10.1007/978-3-662-07998-0
URL : https://hal.archives-ouvertes.fr/hal-01097293

R. Misoczki, J. Tillich, N. Sendrier, and P. S. Barreto, MDPC-McEliece: New McEliece variants from Moderate Density Parity-Check codes, 2013 IEEE International Symposium on Information Theory, 2012.
DOI : 10.1109/ISIT.2013.6620590
URL : https://hal.archives-ouvertes.fr/hal-00870929

H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory. Problems of Control and Information Theory, pp.159-166, 1986.

A. Otmani and J. Tillich, An Efficient Attack on All Concrete KKS Proposals, Post-Quantum Cryptography 2011, pp.98-116, 2011.
DOI : 10.1007/BFb0019850
URL : https://hal.archives-ouvertes.fr/hal-00913500

O. Regev, On lattices, learning with errors, random linear codes, and cryptography, 37th ACM STOC, pp.84-93, 2005.

R. Sakai, K. Ohgishi, and M. Kasahara, Cryptosystems based on pairing, SCIS 2000, 2000.

A. Shamir, Identity-Based Cryptosystems and Signature Schemes, LNCS, vol.84, issue.196, pp.47-53, 1984.
DOI : 10.1007/3-540-39568-7_5

D. Silva, F. R. Kschischang, and R. Kötter, Communication Over Finite-Field Matrix Channels, IEEE Transactions on Information Theory, vol.56, issue.3, pp.1296-1305, 2010.
DOI : 10.1109/TIT.2009.2039167

J. Von, Z. Gathen, and J. Gerhard, Modern computer algebra, 2003.

B. Waters, Dual System Encryption: Realizing Fully Secure IBE and HIBE under Simple Assumptions, LNCS, vol.5677, pp.619-636, 2009.
DOI : 10.1007/978-3-642-03356-8_36

R. Brent and . Waters, Efficient identity-based encryption without random oracles, LNCS, vol.3494, pp.114-127, 2005.

A. Discussion and A. , 1 Toward More Advanced Primitives with Rank Metric The possibility of designing an Identity-based encryption opens a new way to design more advanced primitives in Rank Metric. This was the case with Lattice based Cryptography: after the work of Gentry, Peikert, and Vaikuntanathan, more generalized primitives have been designed, especially in multi-receiver encryption such as broadcast encryption, attribute-based encryption and functional encryption. Our framework can fit these primitives as well. We remark that, our IBE encryption, for a public key of an identity H(id), one can sample a lot