Abstract
The use of rank instead of Hamming metric has been proposed to address the main drawback of code-based cryptography: large key sizes. There exist several Key Encapsulation Mechanisms (KEM) and Public Key Encryption (PKE) schemes using rank metric including some submissions to the NIST call for standardization of Post-Quantum Cryptography. In this work, we present an 
PKE scheme based on the McEliece adaptation to rank metric proposed by Loidreau at PQC 2017. This 
PKE scheme based on rank metric does not use a hybrid construction KEM + symmetric encryption. Instead, we take advantage of the bigger message space obtained by the different parameters chosen in rank metric, being able to exchange multiple keys in one ciphertext. Our proposal is designed considering some specific properties of the random error generated during the encryption. We prove our proposal 
-secure in the QROM by using a security notion called disjoint simulatability introduced by Saito et al. in Eurocrypt 2018. Moreover, we provide security bounds by using the semi-oracles introduced by Ambainis et al.
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Notes
- 1.
The idea is that
, once given the public key, is responsible to generate a test instance composed by two messages of its choice, while 
receives a challenge ciphertext generated as a probabilistic function of the test instance, and must output a guess of which of the two messages has been encrypted.
, once given the public key, is responsible to generate a test instance composed by two messages of its choice, while 
receives a challenge ciphertext generated as a probabilistic function of the test instance, and must output a guess of which of the two messages has been encrypted.References
Al Abdouli, A., et al.: Drankula, a McEliece-like rank metric based cryptosystem implementation. In: Proceedings of the 15th International Joint Conference on e-Business and Telecommunications, ICETE 2018, vol. 2, SECRYPT, pp. 230–241 (2018)
Ambainis, A., Hamburg, M., Unruh, D.: Quantum security proofs using semi-classical oracles. Cryptology ePrint Archive, Report 2018/904 (2018). https://eprint.iacr.org/2018/904
Aragon, N., Gaborit, P., Hauteville, A., Tillich, J.: A new algorithm for solving the rank syndrome decoding problem. In: IEEE International Symposium on Information Theory, ISIT, pp. 2421–2425 (2018)
Bellare, M., Desai, A., Pointcheval, D., Rogaway, P.: Relations among notions of security for public-key encryption schemes. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 26–45. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055718
Bernstein, D.J., Buchmann, J., Dahmen, E.: Post Quantum Cryptography, 1st edn. Springer, Heidelberg (2008)
Boneh, D., Dagdelen, Ö., Fischlin, M., Lehmann, A., Schaffner, C., Zhandry, M.: Random oracles in a quantum world. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 41–69. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_3
Coggia, D., Couvreur, A.: On the security of a Loidreau’s rank metric code based encryption scheme. arXiv preprint arXiv:1903.02933 (2019)
Czajkowski, J., Groot Bruinderink, L., Hülsing, A., Schaffner, C., Unruh, D.: Post-quantum security of the sponge construction. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 185–204. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_9
Dolev, D., Dwork, C., Naor, M.: Non-malleable cryptography (extended abstract). In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 542–552 (1991)
Fujisaki, E., Okamoto, T.: Secure integration of asymmetric and symmetric encryption schemes. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 537–554. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48405-1_34
Gabidulin, E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. (English translation of Problemy Peredachi Informatsii) 21(1), 3–16 (1985)
Gabidulin, E.M., Paramonov, A.V., Tretjakov, O.V.: Ideals over a non-commutative ring and their application in cryptology. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 482–489. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-46416-6_41
Gaborit, P., Hauteville, A., Phan, D.H., Tillich, J.-P.: Identity-based encryption from codes with rank metric. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10403, pp. 194–224. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63697-9_7
Gadouleau, M., Yan, Z.: Complexity of decoding Gabidulin codes. In: 42nd Annual Conference on Information Sciences and Systems. CISS 2008, pp. 1081–1085 (2008)
Hofheinz, D., Hövelmanns, K., Kiltz, E.: A modular analysis of the fujisaki-okamoto transformation. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 341–371. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_12
Loidreau, P.: A welch–berlekamp like algorithm for decoding gabidulin codes. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 36–45. Springer, Heidelberg (2006). https://doi.org/10.1007/11779360_4
Loidreau, P.: A new rank metric codes based encryption scheme. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 3–17. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59879-6_1
McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. The Deep Space Network Progress Report, pp. 114–116, January and February 1978
Mosca, M., Stebila, D.: Contributors: Open quantum safe (2017). https://openquantumsafe.org/
NIST: Federal inf. process. stds. (nist fips) - 202 (2015). https://dx.doi.org/10.6028/NIST.FIPS.202
NIST: Submission requirements and evaluation criteria for the post-quantum cryptography standardization process (2016). https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/call-for-proposals-final-dec-2016.pdf
Nojima, R., Imai, H., Kobara, K., Morozov, K.: Semantic security for the McEliece cryptosystem without random oracles. Des. Codes Crypt. 49(1–3), 289–305 (2008)
Overbeck, R.: Structural attacks for public-key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008)
Saito, T., Xagawa, K., Yamakawa, T.: Tightly-secure key-encapsulation mechanism in the quantum random oracle model. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 520–551. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_17
Unruh, D.: Revocable quantum timed-release encryption. J. ACM 62(6), 46:1–46:76 (2015)
Wachter-Zeh, A.: Decoding of block and convolutional codes in rankmetric.Ph.D. thesis, Université Rennes 1 (2013). https://tel.archives-ouvertes.fr/tel-0105674
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Shehhi, H.A., Bellini, E., Borba, F., Caullery, F., Manzano, M., Mateu, V. (2019). An IND-CCA-Secure Code-Based Encryption Scheme Using Rank Metric. In: Buchmann, J., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2019. AFRICACRYPT 2019. Lecture Notes in Computer Science(), vol 11627. Springer, Cham. https://doi.org/10.1007/978-3-030-23696-0_5
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