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16-08-2022 | Original Paper

Two modifications for Loidreau’s code-based cryptosystem

Authors: Wenshuo Guo, Fang-Wei Fu

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2024

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Abstract

This paper presents two modifications for Loidreau’s cryptosystem, a rank metric-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break this cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in Modification I. In Modification II, we choose a random matrix of low column rank to mix with the secret matrix. Our analysis shows that these two modifications can both resist the existing structural attacks. Furthermore, these modifications have a much more compact representation of public keys compared to Classic McEliece, which has been selected into the fourth round of the NIST-PQC project.

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Literature
1.
go back to reference Albrecht, M.R., Bernstein, D.J., Chou, T., Cid, C., Gilcher, J., Lange, T., Maram, V., Maurich, I., Misoczki, R., Niederhagen, R., Paterson, K., Persichetti, E., Peters, C., Schwabe, P., Sendrier, N., Szefer, J., Tjhai, C.J., Tomlinson, M., Wang, W.: Classic McEliece: conservative code-based cryptography. https://classic.mceliece.org/nist/mceliece-20201010.pdf. Accessed October 10 (2020) Albrecht, M.R., Bernstein, D.J., Chou, T., Cid, C., Gilcher, J., Lange, T., Maram, V., Maurich, I., Misoczki, R., Niederhagen, R., Paterson, K., Persichetti, E., Peters, C., Schwabe, P., Sendrier, N., Szefer, J., Tjhai, C.J., Tomlinson, M., Wang, W.: Classic McEliece: conservative code-based cryptography. https://​classic.​mceliece.​org/​nist/​mceliece-20201010.​pdf. Accessed October 10 (2020)
3.
go back to reference Aragon, N., Gaborit, P., Hauteville, A., Tillich, J.-P.: A new algorithm for solving the rank syndrome decoding problem. In: Proceedings of ISIT 2018, pp. 2421–2425. IEEE (2018) Aragon, N., Gaborit, P., Hauteville, A., Tillich, J.-P.: A new algorithm for solving the rank syndrome decoding problem. In: Proceedings of ISIT 2018, pp. 2421–2425. IEEE (2018)
4.
go back to reference Augot, D., Finiasz, M.: A public key encryption scheme based on the polynomial reconstrution problem. In: Biham, E. (ed.): Proceedings of EUROCRYPT 2003, LNCS, vol. 2656, pp. 229–240. Springer (2003) Augot, D., Finiasz, M.: A public key encryption scheme based on the polynomial reconstrution problem. In: Biham, E. (ed.): Proceedings of EUROCRYPT 2003, LNCS, vol. 2656, pp. 229–240. Springer (2003)
5.
go back to reference Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Enhanced public key security for the McEliece cryptosystem. J. Cryptol. 29(1), 1–27 (2016)MathSciNetCrossRef Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Enhanced public key security for the McEliece cryptosystem. J. Cryptol. 29(1), 1–27 (2016)MathSciNetCrossRef
6.
go back to reference Bardet, M., Briaud, P., Bros, M., Gaborit, P., Neiger, V., Ruatta, O., Tillich, J.-P.: An algebraic attack on rank metric code-based cryptosystems. In: Proceedings of EUROCRYPT 2020, LNCS, vol. 12107, pp. 64–93. Springer (2020) Bardet, M., Briaud, P., Bros, M., Gaborit, P., Neiger, V., Ruatta, O., Tillich, J.-P.: An algebraic attack on rank metric code-based cryptosystems. In: Proceedings of EUROCRYPT 2020, LNCS, vol. 12107, pp. 64–93. Springer (2020)
7.
go back to reference Bardet, M., Bros, M., Cabarcas, D., Gaborit, P., Perlner, R., Smith-Tone, D., Tillich, J.-P., Verbel, J.: Improvements of algebraic attacks for solving the rank decoding and MinRank problems. In: Proceedings of ASIACRYPT 2020, LNCS, vol. 12491, pp. 507–536. Springer (2020) Bardet, M., Bros, M., Cabarcas, D., Gaborit, P., Perlner, R., Smith-Tone, D., Tillich, J.-P., Verbel, J.: Improvements of algebraic attacks for solving the rank decoding and MinRank problems. In: Proceedings of ASIACRYPT 2020, LNCS, vol. 12491, pp. 507–536. Springer (2020)
8.
go back to reference Berlekamp, E.R., McEliece, R.J., van Tilborg, H.: On the inherent intractability of certain coding problems. IEEE Trans. Inf. Theory 24(3), 384–386 (1978)MathSciNetCrossRef Berlekamp, E.R., McEliece, R.J., van Tilborg, H.: On the inherent intractability of certain coding problems. IEEE Trans. Inf. Theory 24(3), 384–386 (1978)MathSciNetCrossRef
9.
go back to reference Bombar, M., Couvreur, A.: Decoding supercodes of Gabidulin codes and applications to cryptanalysis. In: Proceedings of PQCrypto 2021, LNCS, vol. 12841, pp. 3–22. Springer (2021) Bombar, M., Couvreur, A.: Decoding supercodes of Gabidulin codes and applications to cryptanalysis. In: Proceedings of PQCrypto 2021, LNCS, vol. 12841, pp. 3–22. Springer (2021)
10.
go back to reference Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997)MathSciNetCrossRef Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997)MathSciNetCrossRef
11.
go back to reference Canteaut, A., Sendrier, N.: Cryptanalysis of the original McEliece cryptosystem. In: Ohta, K., Pei, D. (eds.): Proceedings of ASIACRYPT 1998, LNCS, vol. 1514, pp. 187–199. Springer (2000) Canteaut, A., Sendrier, N.: Cryptanalysis of the original McEliece cryptosystem. In: Ohta, K., Pei, D. (eds.): Proceedings of ASIACRYPT 1998, LNCS, vol. 1514, pp. 187–199. Springer (2000)
12.
go back to reference Chabaud, F., Stern, J.: The cryptographic security of the syndrome decoding problem for rank distance codes. In: Proceedings of ASIACRYPT 1996, pp. 368–381. Springer (1996) Chabaud, F., Stern, J.: The cryptographic security of the syndrome decoding problem for rank distance codes. In: Proceedings of ASIACRYPT 1996, pp. 368–381. Springer (1996)
13.
go back to reference Coggia, D., Couvreur, A.: On the security of a Loidreau rank metric code based encryption scheme. Des. Codes Cryptogr. 88(9), 1941–1957 (2020)MathSciNetCrossRef Coggia, D., Couvreur, A.: On the security of a Loidreau rank metric code based encryption scheme. Des. Codes Cryptogr. 88(9), 1941–1957 (2020)MathSciNetCrossRef
14.
go back to reference Couvreur, A., Gaborit, P., Otmani, A., Tillich, J.-P.: Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes. Des. Codes Cryptogr. 73(2), 641–666 (2014)MathSciNetCrossRef Couvreur, A., Gaborit, P., Otmani, A., Tillich, J.-P.: Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes. Des. Codes Cryptogr. 73(2), 641–666 (2014)MathSciNetCrossRef
15.
go back to reference Couvreur, A., Otmani, A., Tillich, J.-P., Gauthier-Umaña.: A polynomial-time attack on the BBCRS cryptosystem. In: Proceedings of PKC 2015, LNCS, vol. 9020, pp. 175–193. Springer (2015) Couvreur, A., Otmani, A., Tillich, J.-P., Gauthier-Umaña.: A polynomial-time attack on the BBCRS cryptosystem. In: Proceedings of PKC 2015, LNCS, vol. 9020, pp. 175–193. Springer (2015)
16.
go back to reference Faugère, J.-C., Otmani, A., Perret, L., de Portzamparc, F., Tillich, J.-P.: Structural cryptanalysis of McEliece schemes with compact keys. Des. Codes Cryptogr. 79(1), 87–112 (2016)MathSciNetCrossRef Faugère, J.-C., Otmani, A., Perret, L., de Portzamparc, F., Tillich, J.-P.: Structural cryptanalysis of McEliece schemes with compact keys. Des. Codes Cryptogr. 79(1), 87–112 (2016)MathSciNetCrossRef
17.
go back to reference Faure, C., Loidreau, P.: A new public-key cryptosystem based on the problem of reconstructing \(p\)-polynomials. In: Ytrehus, \(\varnothing\). (ed.): Proceedings of WCC 2005, LNCS, vol. 3969, pp. 304–315. Springer (2005) Faure, C., Loidreau, P.: A new public-key cryptosystem based on the problem of reconstructing \(p\)-polynomials. In: Ytrehus, \(\varnothing\). (ed.): Proceedings of WCC 2005, LNCS, vol. 3969, pp. 304–315. Springer (2005)
18.
go back to reference Gabidulin, E.M.: Theory of codes with maximum rank distance. Prob. Peredachi Inf. 21(1), 3–16 (1985)MathSciNet Gabidulin, E.M.: Theory of codes with maximum rank distance. Prob. Peredachi Inf. 21(1), 3–16 (1985)MathSciNet
19.
go back to reference Gabidulin, E.M.: Attacks and counter-attacks on the GPT public key cryptosystem. Des. Codes Cryptogr. 48(2), 171–177 (2008)MathSciNetCrossRef Gabidulin, E.M.: Attacks and counter-attacks on the GPT public key cryptosystem. Des. Codes Cryptogr. 48(2), 171–177 (2008)MathSciNetCrossRef
20.
go back to reference Gabidulin, E.M., Ourivski, A.V., Honary, B., Ammar, B.: Reducible rank codes and their applications to cryptography. IEEE Trans. Inf. Theory 49(12), 3289–3293 (2003)MathSciNetCrossRef Gabidulin, E.M., Ourivski, A.V., Honary, B., Ammar, B.: Reducible rank codes and their applications to cryptography. IEEE Trans. Inf. Theory 49(12), 3289–3293 (2003)MathSciNetCrossRef
21.
go back to reference 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.): Proceedings of EUROCRYPT 1991, LNCS, vol. 547, pp. 482–489. Springer (1991) 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.): Proceedings of EUROCRYPT 1991, LNCS, vol. 547, pp. 482–489. Springer (1991)
22.
go back to reference Gabidulin, E.M., Rashwan, H., Honary, B.: On improving security of GPT cryptosystems. In: Proceedings of ISIT 2009, pp. 1110–1114. IEEE (2009) Gabidulin, E.M., Rashwan, H., Honary, B.: On improving security of GPT cryptosystems. In: Proceedings of ISIT 2009, pp. 1110–1114. IEEE (2009)
23.
go back to reference Gaborit, P., Otmani, A., Kalachi, H.T.: Polynomial-time key recovery attack on the Faure–Loidreau scheme based on Gabidulin codes. Des. Codes Cryptogr. 86(7), 1391–1403 (2018)MathSciNetCrossRef Gaborit, P., Otmani, A., Kalachi, H.T.: Polynomial-time key recovery attack on the Faure–Loidreau scheme based on Gabidulin codes. Des. Codes Cryptogr. 86(7), 1391–1403 (2018)MathSciNetCrossRef
24.
go back to reference Gaborit, P., Ruatta, O., Schrek, J.: On the complexity of the rank syndrome decoding problem. IEEE Trans. Inf. Theory 62(2), 1006–1019 (2016)MathSciNetCrossRef Gaborit, P., Ruatta, O., Schrek, J.: On the complexity of the rank syndrome decoding problem. IEEE Trans. Inf. Theory 62(2), 1006–1019 (2016)MathSciNetCrossRef
25.
go back to reference Gaborit, P., Zémor, G.: On the hardness of the decoding and the minimum distance problems for rank codes. IEEE Trans. Inf. Theory 62(12), 7245–7252 (2016)MathSciNetCrossRef Gaborit, P., Zémor, G.: On the hardness of the decoding and the minimum distance problems for rank codes. IEEE Trans. Inf. Theory 62(12), 7245–7252 (2016)MathSciNetCrossRef
26.
go back to reference Ghatak, A.: Extending Coggia–Couvreur attack on Loidreau’s rank-metric cryptosystem. Des. Codes Cryptogr. 90(1), 215–238 (2022)MathSciNetCrossRef Ghatak, A.: Extending Coggia–Couvreur attack on Loidreau’s rank-metric cryptosystem. Des. Codes Cryptogr. 90(1), 215–238 (2022)MathSciNetCrossRef
27.
go back to reference Gibson, K.: The security of the Gabidulin public key cryptosystem. In: Proceedings of EUROCRYPT 1996, LNCS, vol. 1070, pp. 212–223. Springer (1996) Gibson, K.: The security of the Gabidulin public key cryptosystem. In: Proceedings of EUROCRYPT 1996, LNCS, vol. 1070, pp. 212–223. Springer (1996)
28.
go back to reference Horlemann-Trautmann, A.-L., Marshall, K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. Adv. Math. Commun. 11(3), 533–548 (2017)MathSciNetCrossRef Horlemann-Trautmann, A.-L., Marshall, K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. Adv. Math. Commun. 11(3), 533–548 (2017)MathSciNetCrossRef
29.
go back to reference Horlemann-Trautmann, A.-L., Marshall, K., Rosenthal, J.: Considerations for rank-based cryptosystems. In: Proceedings of ISIT 2016, pp. 2544–2548. IEEE (2016) Horlemann-Trautmann, A.-L., Marshall, K., Rosenthal, J.: Considerations for rank-based cryptosystems. In: Proceedings of ISIT 2016, pp. 2544–2548. IEEE (2016)
30.
go back to reference Horlemann-Trautmann, A.-L., Marshall, K., Rosenthal, J.: Extension of Overbeck’s attack for Gabidulin-based cryptosystems. Des. Codes Cryptogr. 86(2), 319–340 (2018)MathSciNetCrossRef Horlemann-Trautmann, A.-L., Marshall, K., Rosenthal, J.: Extension of Overbeck’s attack for Gabidulin-based cryptosystems. Des. Codes Cryptogr. 86(2), 319–340 (2018)MathSciNetCrossRef
31.
go back to reference Kobara, K., Imai, H.: Semantically secure McEliece public-key cryptosystems-conversions for McEliece PKC. Kim, K. (ed.): Proceedings of PKC 2001, LNCS, vol. 1992, pp. 19–35. Springer (2001) Kobara, K., Imai, H.: Semantically secure McEliece public-key cryptosystems-conversions for McEliece PKC. Kim, K. (ed.): Proceedings of PKC 2001, LNCS, vol. 1992, pp. 19–35. Springer (2001)
32.
33.
go back to reference Lee, P.J., Brickell, E.F.: An observation on the security of McEliece’s public-key cryptosystem. In: Guenther, C.G. (ed.): Proceedings of EUROCRYPT 1988, LNCS, vol. 330, pp. 275–280. Springer (1988) Lee, P.J., Brickell, E.F.: An observation on the security of McEliece’s public-key cryptosystem. In: Guenther, C.G. (ed.): Proceedings of EUROCRYPT 1988, LNCS, vol. 330, pp. 275–280. Springer (1988)
34.
go back to reference Li, Y.X., Deng, R.H., Wang, X.M.: On the equivalence of McEliece’s and Niederreiter’s public-key cryptosystems. IEEE Trans. Inf. Theory 40(1), 271–273 (1994)MathSciNetCrossRef Li, Y.X., Deng, R.H., Wang, X.M.: On the equivalence of McEliece’s and Niederreiter’s public-key cryptosystems. IEEE Trans. Inf. Theory 40(1), 271–273 (1994)MathSciNetCrossRef
35.
go back to reference Ling, S., Xing, C.: Coding Theory: A First Course. Cambridge University Press, Cambridge (2004)CrossRef Ling, S., Xing, C.: Coding Theory: A First Course. Cambridge University Press, Cambridge (2004)CrossRef
36.
go back to reference Loidreau, P.: A Welch-Berlekamp like algorithm for decoding Gabidulin codes. In: Ytrehus, \(\varnothing\). (ed.): Proceedings of WCC 2005, LNCS, vol. 3969, pp. 36–45. Springer (2005) Loidreau, P.: A Welch-Berlekamp like algorithm for decoding Gabidulin codes. In: Ytrehus, \(\varnothing\). (ed.): Proceedings of WCC 2005, LNCS, vol. 3969, pp. 36–45. Springer (2005)
37.
go back to reference Loidreau, P.: Designing a rank metric based McEliece cryptosystem. In: Sendrier, N. (ed.): Proceedings of PQCrypto 2010, LNCS, vol. 6061, pp. 142–152. Springer (2010) Loidreau, P.: Designing a rank metric based McEliece cryptosystem. In: Sendrier, N. (ed.): Proceedings of PQCrypto 2010, LNCS, vol. 6061, pp. 142–152. Springer (2010)
38.
go back to reference Loidreau, P.: A new rank metric codes based encryption scheme. In: Lange, T., Takagi, T. (Eds.): Proceedings of PQCrypto 2017, LNCS, vol. 10346, pp. 3–17. Springer (2017) Loidreau, P.: A new rank metric codes based encryption scheme. In: Lange, T., Takagi, T. (Eds.): Proceedings of PQCrypto 2017, LNCS, vol. 10346, pp. 3–17. Springer (2017)
40.
go back to reference Loidreau, P., Sendrier, N.: Weak keys in the McEliece public-key cryptosystem. IEEE Trans. Inf. Theory 47(3), 1207–1211 (2001)MathSciNetCrossRef Loidreau, P., Sendrier, N.: Weak keys in the McEliece public-key cryptosystem. IEEE Trans. Inf. Theory 47(3), 1207–1211 (2001)MathSciNetCrossRef
41.
go back to reference McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Jet Propuls. Lab. DSN Progr. Rep. 42–44, 114–116 (1978) McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Jet Propuls. Lab. DSN Progr. Rep. 42–44, 114–116 (1978)
43.
go back to reference Niederreiter, H.: Knapsack-type cryptosystems and algebraic coding theory. Prob. Control Inf. Theory 15(2), 159–166 (1986)MathSciNet Niederreiter, H.: Knapsack-type cryptosystems and algebraic coding theory. Prob. Control Inf. Theory 15(2), 159–166 (1986)MathSciNet
44.
go back to reference Otmani, A., Kalachi, H.T., Ndjeya, S.: Improved cryptanalysis of rank metric schemes based on Gabidulin codes. Des. Codes Cryptogr. 86(9), 1983–1996 (2018)MathSciNetCrossRef Otmani, A., Kalachi, H.T., Ndjeya, S.: Improved cryptanalysis of rank metric schemes based on Gabidulin codes. Des. Codes Cryptogr. 86(9), 1983–1996 (2018)MathSciNetCrossRef
45.
go back to reference Ourivski, A.V., Johansson, T.: New technique for decoding codes in the rank metric and its cryptography applications. Probl. Inf. Transm. 38(3), 237–246 (2002)MathSciNetCrossRef Ourivski, A.V., Johansson, T.: New technique for decoding codes in the rank metric and its cryptography applications. Probl. Inf. Transm. 38(3), 237–246 (2002)MathSciNetCrossRef
46.
go back to reference Overbeck, R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008)MathSciNetCrossRef Overbeck, R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008)MathSciNetCrossRef
47.
go back to reference Rashwan, H., Gabidulin, E.M., Honary, B.: A smart approach for GPT cryptosystem based on rank codes. In: Proceedings of ISIT 2010, pp. 2463–2467. IEEE (2010) Rashwan, H., Gabidulin, E.M., Honary, B.: A smart approach for GPT cryptosystem based on rank codes. In: Proceedings of ISIT 2010, pp. 2463–2467. IEEE (2010)
48.
go back to reference Rashwan, H., Gabidulin, E.M., Honary, B.: Security of the GPT cryptosystem and its applications to cryptography. Secur. Commun. Netw. 4(8), 937–946 (2011)CrossRef Rashwan, H., Gabidulin, E.M., Honary, B.: Security of the GPT cryptosystem and its applications to cryptography. Secur. Commun. Netw. 4(8), 937–946 (2011)CrossRef
49.
go back to reference Renner, J., Puchinger, S., Wachter-Zeh, A.: LIGA: a cryptosystem based on the hardness of rank-metric list and interleaved decoding. Des. Codes Cryptogr. 89(6), 1279–1319 (2021)MathSciNetCrossRef Renner, J., Puchinger, S., Wachter-Zeh, A.: LIGA: a cryptosystem based on the hardness of rank-metric list and interleaved decoding. Des. Codes Cryptogr. 89(6), 1279–1319 (2021)MathSciNetCrossRef
50.
go back to reference Richter, G., Plass, S.: Error and erasure decoding of rank-codes with a modified Berlekamp–Massey algorithm. ITG FACHBERICHT, pp. 203–210 (2004) Richter, G., Plass, S.: Error and erasure decoding of rank-codes with a modified Berlekamp–Massey algorithm. ITG FACHBERICHT, pp. 203–210 (2004)
51.
go back to reference Sidelnikov, V.M., Shestakov, S.O.: On insecurity of cryptosystems based on generalized Reed–Solomon codes. Discret. Math. Appl. 2(4), 439–444 (1992) Sidelnikov, V.M., Shestakov, S.O.: On insecurity of cryptosystems based on generalized Reed–Solomon codes. Discret. Math. Appl. 2(4), 439–444 (1992)
Metadata
Title
Two modifications for Loidreau’s code-based cryptosystem
Authors
Wenshuo Guo
Fang-Wei Fu
Publication date
16-08-2022
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 5/2024
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-022-00577-0

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