doi: 10.3934/amc.2020053

Encryption scheme based on expanded Reed-Solomon codes

Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

Received  June 2019 Revised  October 2019 Published  January 2020

We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly $ 45 \% $ compared to the classic McEliece cryptosystem proposed by Bernstein et al.

Citation: Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020053
References:
[1]

C. Aguilar-MelchorO. BlazyJ.-C. DeneuvilleP. Gaborit and G. Zémor, Efficient encryption from random quasi-cyclic codes, IEEE Transactions on Information Theory, 64 (2018), 3927-3943.  doi: 10.1109/TIT.2018.2804444.  Google Scholar

[2]

M. Albrecht, C. Cid, K. G. Paterson, C. J. Tjhai and M. Tomlinson, NTS-KEM, 2018. Google Scholar

[3]

N. Aragon, P. S. L. M. Barreto, S. Bettaieb, Lo. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, S. Gueron, T. Guneysu, C. A. Melchor, R. Misoczki, E. Persichetti, N. Sendrier, J.-P. Tillich and G. Zémor, Bike: Bit Flipping Key Encapsulation, 2017. Google Scholar

[4]

M. BaldiA. BarenghiF. ChiaraluceG. Pelosi and P. Santini, LEDAkem: A post-quantum key encapsulation mechanism based on QC-LDPC codes, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Cham, 10786 (2018), 3-24.   Google Scholar

[5]

M. Baldi, M. Bianchi, F. Chiaraluce, J. Rosenthal and D. Schipani, A variant of the McEliece cryptosystem with increased public key security, Proceedings of the Seventh International Workshop on Coding and Cryptography (WCC) 2011, (2011), 173–182. Google Scholar

[6]

M. Baldi, M. Bianchi, F. Chiaraluce, J. Rosenthal and D. Schipani, Method and Apparatus for Public-Key Cryptography Based on Error Correcting Codes, 2015, US Patent 9,191,199. Google Scholar

[7]

M. Baldi, M. Bodrato and F. Chiaraluce, A new analysis of the McEliece cryptosystem based on QC-LDPC codes, International Conference on Security and Cryptography for Networks, Springer Berlin Heidelberg, (2008), 246–262. Google Scholar

[8]

M. Baldi, F. Chiaraluce, J. Rosenthal, P. Santini and D. Schipani, On the security of generalized Reed-Solomon code-based cryptosystems, IET Information Security, (2019). Google Scholar

[9]

A. BeckerA. JouxA. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How $1+ 1 = 0$ improves information set decoding, Advances in Cryptology—EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520-536.  doi: 10.1007/978-3-642-29011-4_31.  Google Scholar

[10]

T. P. Berger and P. Loidreau, How to mask the structure of codes for a cryptographic use, Des. Codes Cryptogr., 35 (2005), 63-79.  doi: 10.1007/s10623-003-6151-2.  Google Scholar

[11]

D. J. BernsteinT. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 5299 (2008), 31-46.  doi: 10.1007/978-3-540-88403-3_3.  Google Scholar

[12]

D. J. BernsteinT. Lange and C. Peters, Wild McEliece, Selected Areas in Cryptography, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6544 (2011), 143-158.  doi: 10.1007/978-3-642-19574-7_10.  Google Scholar

[13]

D. J. BernsteinT. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, Advances in Cryptology—CRYPTO 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6841 (2011), 743-760.  doi: 10.1007/978-3-642-22792-9_42.  Google Scholar

[14]

J. BolkemaH. Gluesing-LuerssenC. A. KelleyK. E. LauterB. Malmskog and J. Rosenthal, Variations of the McEliece cryptosystem, Algebraic Geometry for Coding Theory and Cryptography, Assoc. Women Math. Ser., Springer, Cham, 9 (2017), 129-150.   Google Scholar

[15]

A. CouvreurP. GaboritV. Gauthier-UmañaA. Otmani and J.-P. Tillich, Distinguisher-based attacks on public-key cryptosystems using reed-solomon codes, Designs, Codes and Cryptography, 73 (2014), 641-666.  doi: 10.1007/s10623-014-9967-z.  Google Scholar

[16]

A. CouvreurI. Márquez-Corbella and R. Pellikaan, Cryptanalysis of McEliece cryptosystem based on algebraic geometry codes and their subcodes, IEEE Trans. Inform. Theory, 63 (2017), 5404-5418.  doi: 10.1109/TIT.2017.2712636.  Google Scholar

[17]

A. CouvreurA. Otmani and J.-P. Tillich, Polynomial time attack on wild McEliece over quadratic extensions, IEEE Transactions on Information Theory, 63 (2017), 404-427.  doi: 10.1109/TIT.2016.2574841.  Google Scholar

[18]

A. Couvreur, A. Otmani, J.-P. Tillich and V. Gauthier-Umaña, A polynomial-time attack on the BBCRS scheme, Public-key Cryptography-PKC 2015, Lecture Notes in Comput. Sci., Springer, Heidelberg, 9020 (2015), 175–193. doi: 10.1007/978-3-662-46447-2_8.  Google Scholar

[19]

J.-C. FaugèreV. Gauthier-UmañaA. OtmaniL. Perret and J.-P. Tillich, A distinguisher for high-rate McEliece cryptosystems, IEEE Transactions on Information Theory, 59 (2013), 6830-6844.  doi: 10.1109/TIT.2013.2272036.  Google Scholar

[20]

V. Gauthier-Umaña, A. Otmani and J.-P. Tillich, A distinguisher-based attack on a variant of McEliece's cryptosystem based on reed-solomon codes, Preprint, (2012), arXiv: 1204.6459. Google Scholar

[21]

C. T. GueyeJ. B. Klamti and S. Hirose, Generalization of BJMM-ISD using May-Ozerov nearest neighbor algorithm over an arbitrary finite field $\mathbb{F}_q$, Codes, Cryptology and Information Security, Lecture Notes in Comput. Sci., Springer, Cham, 10194 (2017), 96-109.   Google Scholar

[22]

S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar

[23]

C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the ball-collision algorithm, Preprint, (2018), arXiv: 1812.10955. Google Scholar

[24]

H. Janwa and O. Moreno, McEliece public key cryptosystems using algebraic-geometric codes, Designs, Codes and Cryptography, 8 (1996), 293-307.  doi: 10.1023/A:1027351723034.  Google Scholar

[25]

K. Khathuria, J. Rosenthal and V. Weger, Weight two masking of the reed-solomon structure in conjugation with list decoding, Proceedings of 23rd International Symposium on Mathematical Theory of Networks and Systems, Hong Kong University of Science and Technology, Hong Kong, (2018), 309–314. Google Scholar

[26]

G. Landais and J.-P. Tillich, An efficient attack of a McEliece cryptosystem variant based on convolutional codes, International Workshop on Post-Quantum Cryptography, Springer, (2013), 102–117. Google Scholar

[27]

P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, Advances in Cryptology—EUROCRYPT '88 (Davos, 1988), Lecture Notes in Comput. Sci., Springer, Berlin, 330 (1988), 275-280.  doi: 10.1007/3-540-45961-8_25.  Google Scholar

[28]

J. S. Leon, A probabilistic algorithm for computing minimum weights of large error-correcting codes. Coding techniques and coding theory, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.  doi: 10.1109/18.21270.  Google Scholar

[29]

C. Löndahl and T. Johansson, A new version of McEliece PKC based on convolutional codes, International Conference on Information and Communications Security, Springer, (2012), 461–470. Google Scholar

[30]

A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, Advances in Cryptology—EUROCRYPT 2015. Part Ⅰ, Lecture Notes in Comput. Sci., Springer, Heidelberg, 9056 (2015), 203-228.  doi: 10.1007/978-3-662-46800-5_9.  Google Scholar

[31]

R. J. McEliece, A Public-Key Cryptosystem Based on Algebraic Coding Theory, Technical report, DSN Progress report, Jet Propulsion Laboratory, Pasadena, 1978. Google Scholar

[32]

C. A. Melchor, N. Aragon, M. Bardet, S. Bettaieb, L. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, A. Hauteville, A. Otmani, O. Ruatta, J.-P. Tillich and G. Zémor, ROLLO-Rank-Ouroboros, LAKE & LOCKER, 2018. Google Scholar

[33]

L. Minder and A. Shokrollahi, Cryptanalysis of the Sidelnikov cryptosystem, Advances in Cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci., Springer, Berlin, 4515 (2007), 347-360.  doi: 10.1007/978-3-540-72540-4_20.  Google Scholar

[34]

R. Misoczki, J.-P. Tillich, N. Sendrier and P. S. L. M. Barreto, MDPC-McEliece: New McEliece variants from moderate density parity-check codes, 2013 IEEE International Symposium on Information Theory, (2013), 2069–2073. doi: 10.1109/ISIT.2013.6620590.  Google Scholar

[35]

R. NiebuhrE. PersichettiP.-L. CayrelS. Bulygin and J. Buchmann, On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, Int. J. Inf. Coding Theory, 4 (2017), 47-78.  doi: 10.1504/IJICOT.2017.081458.  Google Scholar

[36]

H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform., 15 (1986), 159-166.   Google Scholar

[37]

A. OtmaniJ.-P. Tillich and L. Dallot, Cryptanalysis of two McEliece cryptosystems based on quasi-cyclic codes, Mathematics in Computer Science, 3 (2010), 129-140.  doi: 10.1007/s11786-009-0015-8.  Google Scholar

[38]

C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 6061 (2010), 81–94, https://bitbucket.org/cbcrypto/isdfq/src/master/. doi: 10.1007/978-3-642-12929-2_7.  Google Scholar

[39]

E. Prange, The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), S5–S9. doi: 10.1109/tit.1962.1057777.  Google Scholar

[40]

P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), IEEE Comput. Soc. Press, Los Alamitos, CA, (1994), 124–134. doi: 10.1109/SFCS.1994.365700.  Google Scholar

[41]

V. M. Sidelnikov, A public key cryptosystem based on Reed-Muller binary codes, Discrete Math. Appl., 4 (1994), 191-207.  doi: 10.1515/dma.1994.4.3.191.  Google Scholar

[42]

V. M. Sidelnikov and S. O. Shestakov, On an encoding system constructed on the basis of generalized Reed-Solomon codes, Diskret. Mat., 4 (1992), 57-63.  doi: 10.1515/dma.1992.2.4.439.  Google Scholar

[43]

V. M. Sidelnikov and S. O. Shestakov, On insecurity of cryptosystems based on generalized Reed-Solomon codes, Discrete Mathematics and Applications, 2 (1992), 439-444.   Google Scholar

[44]

J. Stern, A method for finding codewords of small weight, Coding Theory and Applications, Lecture Notes in Comput. Sci., Springer, New Yorkpages, 388 (1989), 106-113.  doi: 10.1007/BFb0019850.  Google Scholar

[45]

A. Vardy and Y. Be'ery, Bit-level soft-decision decoding of Reed-Solomon codes, IEEE Transactions on Communications, 39 (1991), 440-444.   Google Scholar

[46]

C. Wieschebrink, Cryptanalysis of the Niederreiter public key scheme based on GRS subcodes, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 6061 (2010), 61-72.  doi: 10.1007/978-3-642-12929-2_5.  Google Scholar

[47]

Y. Q. Wu, On expanded cyclic and Reed-Solomon codes, IEEE Transactions on Information Theory, 57 (2011), 601-620.  doi: 10.1109/TIT.2010.2095150.  Google Scholar

show all references

References:
[1]

C. Aguilar-MelchorO. BlazyJ.-C. DeneuvilleP. Gaborit and G. Zémor, Efficient encryption from random quasi-cyclic codes, IEEE Transactions on Information Theory, 64 (2018), 3927-3943.  doi: 10.1109/TIT.2018.2804444.  Google Scholar

[2]

M. Albrecht, C. Cid, K. G. Paterson, C. J. Tjhai and M. Tomlinson, NTS-KEM, 2018. Google Scholar

[3]

N. Aragon, P. S. L. M. Barreto, S. Bettaieb, Lo. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, S. Gueron, T. Guneysu, C. A. Melchor, R. Misoczki, E. Persichetti, N. Sendrier, J.-P. Tillich and G. Zémor, Bike: Bit Flipping Key Encapsulation, 2017. Google Scholar

[4]

M. BaldiA. BarenghiF. ChiaraluceG. Pelosi and P. Santini, LEDAkem: A post-quantum key encapsulation mechanism based on QC-LDPC codes, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Cham, 10786 (2018), 3-24.   Google Scholar

[5]

M. Baldi, M. Bianchi, F. Chiaraluce, J. Rosenthal and D. Schipani, A variant of the McEliece cryptosystem with increased public key security, Proceedings of the Seventh International Workshop on Coding and Cryptography (WCC) 2011, (2011), 173–182. Google Scholar

[6]

M. Baldi, M. Bianchi, F. Chiaraluce, J. Rosenthal and D. Schipani, Method and Apparatus for Public-Key Cryptography Based on Error Correcting Codes, 2015, US Patent 9,191,199. Google Scholar

[7]

M. Baldi, M. Bodrato and F. Chiaraluce, A new analysis of the McEliece cryptosystem based on QC-LDPC codes, International Conference on Security and Cryptography for Networks, Springer Berlin Heidelberg, (2008), 246–262. Google Scholar

[8]

M. Baldi, F. Chiaraluce, J. Rosenthal, P. Santini and D. Schipani, On the security of generalized Reed-Solomon code-based cryptosystems, IET Information Security, (2019). Google Scholar

[9]

A. BeckerA. JouxA. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How $1+ 1 = 0$ improves information set decoding, Advances in Cryptology—EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520-536.  doi: 10.1007/978-3-642-29011-4_31.  Google Scholar

[10]

T. P. Berger and P. Loidreau, How to mask the structure of codes for a cryptographic use, Des. Codes Cryptogr., 35 (2005), 63-79.  doi: 10.1007/s10623-003-6151-2.  Google Scholar

[11]

D. J. BernsteinT. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 5299 (2008), 31-46.  doi: 10.1007/978-3-540-88403-3_3.  Google Scholar

[12]

D. J. BernsteinT. Lange and C. Peters, Wild McEliece, Selected Areas in Cryptography, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6544 (2011), 143-158.  doi: 10.1007/978-3-642-19574-7_10.  Google Scholar

[13]

D. J. BernsteinT. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, Advances in Cryptology—CRYPTO 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6841 (2011), 743-760.  doi: 10.1007/978-3-642-22792-9_42.  Google Scholar

[14]

J. BolkemaH. Gluesing-LuerssenC. A. KelleyK. E. LauterB. Malmskog and J. Rosenthal, Variations of the McEliece cryptosystem, Algebraic Geometry for Coding Theory and Cryptography, Assoc. Women Math. Ser., Springer, Cham, 9 (2017), 129-150.   Google Scholar

[15]

A. CouvreurP. GaboritV. Gauthier-UmañaA. Otmani and J.-P. Tillich, Distinguisher-based attacks on public-key cryptosystems using reed-solomon codes, Designs, Codes and Cryptography, 73 (2014), 641-666.  doi: 10.1007/s10623-014-9967-z.  Google Scholar

[16]

A. CouvreurI. Márquez-Corbella and R. Pellikaan, Cryptanalysis of McEliece cryptosystem based on algebraic geometry codes and their subcodes, IEEE Trans. Inform. Theory, 63 (2017), 5404-5418.  doi: 10.1109/TIT.2017.2712636.  Google Scholar

[17]

A. CouvreurA. Otmani and J.-P. Tillich, Polynomial time attack on wild McEliece over quadratic extensions, IEEE Transactions on Information Theory, 63 (2017), 404-427.  doi: 10.1109/TIT.2016.2574841.  Google Scholar

[18]

A. Couvreur, A. Otmani, J.-P. Tillich and V. Gauthier-Umaña, A polynomial-time attack on the BBCRS scheme, Public-key Cryptography-PKC 2015, Lecture Notes in Comput. Sci., Springer, Heidelberg, 9020 (2015), 175–193. doi: 10.1007/978-3-662-46447-2_8.  Google Scholar

[19]

J.-C. FaugèreV. Gauthier-UmañaA. OtmaniL. Perret and J.-P. Tillich, A distinguisher for high-rate McEliece cryptosystems, IEEE Transactions on Information Theory, 59 (2013), 6830-6844.  doi: 10.1109/TIT.2013.2272036.  Google Scholar

[20]

V. Gauthier-Umaña, A. Otmani and J.-P. Tillich, A distinguisher-based attack on a variant of McEliece's cryptosystem based on reed-solomon codes, Preprint, (2012), arXiv: 1204.6459. Google Scholar

[21]

C. T. GueyeJ. B. Klamti and S. Hirose, Generalization of BJMM-ISD using May-Ozerov nearest neighbor algorithm over an arbitrary finite field $\mathbb{F}_q$, Codes, Cryptology and Information Security, Lecture Notes in Comput. Sci., Springer, Cham, 10194 (2017), 96-109.   Google Scholar

[22]

S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar

[23]

C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the ball-collision algorithm, Preprint, (2018), arXiv: 1812.10955. Google Scholar

[24]

H. Janwa and O. Moreno, McEliece public key cryptosystems using algebraic-geometric codes, Designs, Codes and Cryptography, 8 (1996), 293-307.  doi: 10.1023/A:1027351723034.  Google Scholar

[25]

K. Khathuria, J. Rosenthal and V. Weger, Weight two masking of the reed-solomon structure in conjugation with list decoding, Proceedings of 23rd International Symposium on Mathematical Theory of Networks and Systems, Hong Kong University of Science and Technology, Hong Kong, (2018), 309–314. Google Scholar

[26]

G. Landais and J.-P. Tillich, An efficient attack of a McEliece cryptosystem variant based on convolutional codes, International Workshop on Post-Quantum Cryptography, Springer, (2013), 102–117. Google Scholar

[27]

P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, Advances in Cryptology—EUROCRYPT '88 (Davos, 1988), Lecture Notes in Comput. Sci., Springer, Berlin, 330 (1988), 275-280.  doi: 10.1007/3-540-45961-8_25.  Google Scholar

[28]

J. S. Leon, A probabilistic algorithm for computing minimum weights of large error-correcting codes. Coding techniques and coding theory, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.  doi: 10.1109/18.21270.  Google Scholar

[29]

C. Löndahl and T. Johansson, A new version of McEliece PKC based on convolutional codes, International Conference on Information and Communications Security, Springer, (2012), 461–470. Google Scholar

[30]

A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, Advances in Cryptology—EUROCRYPT 2015. Part Ⅰ, Lecture Notes in Comput. Sci., Springer, Heidelberg, 9056 (2015), 203-228.  doi: 10.1007/978-3-662-46800-5_9.  Google Scholar

[31]

R. J. McEliece, A Public-Key Cryptosystem Based on Algebraic Coding Theory, Technical report, DSN Progress report, Jet Propulsion Laboratory, Pasadena, 1978. Google Scholar

[32]

C. A. Melchor, N. Aragon, M. Bardet, S. Bettaieb, L. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, A. Hauteville, A. Otmani, O. Ruatta, J.-P. Tillich and G. Zémor, ROLLO-Rank-Ouroboros, LAKE & LOCKER, 2018. Google Scholar

[33]

L. Minder and A. Shokrollahi, Cryptanalysis of the Sidelnikov cryptosystem, Advances in Cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci., Springer, Berlin, 4515 (2007), 347-360.  doi: 10.1007/978-3-540-72540-4_20.  Google Scholar

[34]

R. Misoczki, J.-P. Tillich, N. Sendrier and P. S. L. M. Barreto, MDPC-McEliece: New McEliece variants from moderate density parity-check codes, 2013 IEEE International Symposium on Information Theory, (2013), 2069–2073. doi: 10.1109/ISIT.2013.6620590.  Google Scholar

[35]

R. NiebuhrE. PersichettiP.-L. CayrelS. Bulygin and J. Buchmann, On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, Int. J. Inf. Coding Theory, 4 (2017), 47-78.  doi: 10.1504/IJICOT.2017.081458.  Google Scholar

[36]

H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform., 15 (1986), 159-166.   Google Scholar

[37]

A. OtmaniJ.-P. Tillich and L. Dallot, Cryptanalysis of two McEliece cryptosystems based on quasi-cyclic codes, Mathematics in Computer Science, 3 (2010), 129-140.  doi: 10.1007/s11786-009-0015-8.  Google Scholar

[38]

C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 6061 (2010), 81–94, https://bitbucket.org/cbcrypto/isdfq/src/master/. doi: 10.1007/978-3-642-12929-2_7.  Google Scholar

[39]

E. Prange, The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), S5–S9. doi: 10.1109/tit.1962.1057777.  Google Scholar

[40]

P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), IEEE Comput. Soc. Press, Los Alamitos, CA, (1994), 124–134. doi: 10.1109/SFCS.1994.365700.  Google Scholar

[41]

V. M. Sidelnikov, A public key cryptosystem based on Reed-Muller binary codes, Discrete Math. Appl., 4 (1994), 191-207.  doi: 10.1515/dma.1994.4.3.191.  Google Scholar

[42]

V. M. Sidelnikov and S. O. Shestakov, On an encoding system constructed on the basis of generalized Reed-Solomon codes, Diskret. Mat., 4 (1992), 57-63.  doi: 10.1515/dma.1992.2.4.439.  Google Scholar

[43]

V. M. Sidelnikov and S. O. Shestakov, On insecurity of cryptosystems based on generalized Reed-Solomon codes, Discrete Mathematics and Applications, 2 (1992), 439-444.   Google Scholar

[44]

J. Stern, A method for finding codewords of small weight, Coding Theory and Applications, Lecture Notes in Comput. Sci., Springer, New Yorkpages, 388 (1989), 106-113.  doi: 10.1007/BFb0019850.  Google Scholar

[45]

A. Vardy and Y. Be'ery, Bit-level soft-decision decoding of Reed-Solomon codes, IEEE Transactions on Communications, 39 (1991), 440-444.   Google Scholar

[46]

C. Wieschebrink, Cryptanalysis of the Niederreiter public key scheme based on GRS subcodes, Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 6061 (2010), 61-72.  doi: 10.1007/978-3-642-12929-2_5.  Google Scholar

[47]

Y. Q. Wu, On expanded cyclic and Reed-Solomon codes, IEEE Transactions on Information Theory, 57 (2011), 601-620.  doi: 10.1109/TIT.2010.2095150.  Google Scholar

Table 1.  Comparing key sizes of the proposed cryptosystem with $ m = 3 $ and $ \lambda = 2 $ reaching a $ 256 $-bit security level against the modified ISD algorithm
Rate $ q $ $ n $ $ k $ $ t $ Key Size (bits)
0.60 13 1382 829 277 6783627
0.65 13 1270 825 223 5952804
0.70 13 1207 844 182 5339456
0.75 13 1192 894 149 4929077
0.80 13 1230 984 123 4702652
0.82 13 1258 1031 114 4624198
0.85 13 1340 1139 101 4634545
0.87 13 1420 1235 93 4692805
0.90 13 1602 1441 81 4863276
Rate $ q $ $ n $ $ k $ $ t $ Key Size (bits)
0.60 13 1382 829 277 6783627
0.65 13 1270 825 223 5952804
0.70 13 1207 844 182 5339456
0.75 13 1192 894 149 4929077
0.80 13 1230 984 123 4702652
0.82 13 1258 1031 114 4624198
0.85 13 1340 1139 101 4634545
0.87 13 1420 1235 93 4692805
0.90 13 1602 1441 81 4863276
Table 2.  Comparing key sizes of the proposed cryptosystem with $ m = 4 $ and $ \lambda = 2 $ reaching a $ 256 $-bit security level against the modified ISD algorithm
Rate $ q $ $ n $ $ k $ $ t $ Key Size (bits)
0.65 7 2360 1534 413 13134108
0.70 7 1945 1361 292 10191102
0.75 7 1738 1303 218 8480009
0.80 7 1662 1329 167 7448878
0.85 7 1700 1445 128 6815134
0.87 7 1770 1539 116 6785893
0.89 7 1872 1666 103 6754721
0.91 7 2024 1841 92 6814326
Rate $ q $ $ n $ $ k $ $ t $ Key Size (bits)
0.65 7 2360 1534 413 13134108
0.70 7 1945 1361 292 10191102
0.75 7 1738 1303 218 8480009
0.80 7 1662 1329 167 7448878
0.85 7 1700 1445 128 6815134
0.87 7 1770 1539 116 6785893
0.89 7 1872 1666 103 6754721
0.91 7 2024 1841 92 6814326
Table 3.  Comparing the key sizes of the proposed parameters against different cryptosystems
$ q $ $ m $ $ n $ $ k $ Key Size (in bits)
Proposed system Type Ⅰs 13 3 1258 1031 4624198
Type Ⅱ 7 4 1872 1666 6754721
classical McEliece 2 13 6960 5413 8373911
BBCRS based schemes $ w=1.708 $ and $ z=1 $ 1423 1 1422 786 5113520
$ w=1.2 $ and $ z=10 $ 1163 1 1162 928 2274160
$ w=2 $ and $ z=0 $ 1993 1 1992 1593 6966714
$ q $ $ m $ $ n $ $ k $ Key Size (in bits)
Proposed system Type Ⅰs 13 3 1258 1031 4624198
Type Ⅱ 7 4 1872 1666 6754721
classical McEliece 2 13 6960 5413 8373911
BBCRS based schemes $ w=1.708 $ and $ z=1 $ 1423 1 1422 786 5113520
$ w=1.2 $ and $ z=10 $ 1163 1 1162 928 2274160
$ w=2 $ and $ z=0 $ 1993 1 1992 1593 6966714
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