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November  2021, 15(4): 677-699. doi: 10.3934/amc.2020089

## Information set decoding in the Lee metric with applications to cryptography

 1 Faculty of Mathematics and Statistics, University of St. Gallen, Bodanstr. 6, St. Gallen, Switzerland 2 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

* Corresponding author: Violetta Weger

Received  May 2019 Revised  February 2020 Published  November 2021 Early access  July 2020

We convert Stern's information set decoding (ISD) algorithm to the ring $\mathbb{Z}/4 \mathbb{Z}$ equipped with the Lee metric. Moreover, we set up the general framework for a McEliece and a Niederreiter cryptosystem over this ring. The complexity of the ISD algorithm determines the minimum key size in these cryptosystems for a given security level. We show that using Lee metric codes can substantially decrease the key size, compared to Hamming metric codes. In the end we explain how our results can be generalized to other Galois rings $\mathbb{Z}/p^s\mathbb{Z}$.

Citation: Anna-Lena Horlemann-Trautmann, Violetta Weger. Information set decoding in the Lee metric with applications to cryptography. Advances in Mathematics of Communications, 2021, 15 (4) : 677-699. doi: 10.3934/amc.2020089
##### References:
 [1] E. F. Assmus and H. F. Mattson, Error-correcting codes: An axiomatic approach, Information and Control, 6 (1963), 315-330.  doi: 10.1016/S0019-9958(63)80010-8.  Google Scholar [2] A. Becker, A. Joux, A. 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 [3] E. Berlekamp, Algebraic Coding Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9407.  Google Scholar [4] D. J. Bernstein, Grover vs. McEliece, In International Workshop on Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, 6061 (2010), 73–80. doi: 10.1007/978-3-642-12929-2_6.  Google Scholar [5] D. J. Bernstein, T. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, In International Workshop on Post-Quantum Cryptography, Springer, (2008), 31–46. doi: 10.1007/978-3-540-88403-3_3.  Google Scholar [6] D. J. Bernstein, T. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, In Annual Cryptology Conference, Springer, (2011), 743–760. doi: 10.1007/978-3-642-22792-9_42.  Google Scholar [7] T. Blackford, Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar [8] I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.  Google Scholar [9] I. F. Blake, Codes over integer residue rings, Information and Control, 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.  Google Scholar [10] E. Byrne, Decoding a class of Lee metric codes over a Galois ring, IEEE Transactions on Information Theory, 48 (2002), 966-975.  doi: 10.1109/18.992804.  Google Scholar [11] A. Canteaut and Hervé Chabanne, A Further Improvement of the Work Factor in an Attempt at Breaking McEliece's Cryptosystem, PhD thesis, INRIA, 1994. Google Scholar [12] A. Canteaut and F. Chabaud, A new algorithm for finding minimum-weight words in a linear code: Application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511, IEEE Transactions on Information Theory, 44 (1998), 367-378.  doi: 10.1109/18.651067.  Google Scholar [13] A. Canteaut and N. Sendrier, Cryptanalysis of the original McEliece cryptosystem, In Advances in Cryptology ASIACRYPT'98 (Beijing), Lecture Notes in Comput. Sci., Springer, Berlin, 1514 (1998), 187–199. doi: 10.1007/3-540-49649-1_16.  Google Scholar [14] F. Chabaud, Asymptotic analysis of probabilistic algorithms for finding short codewords, Eurocode '92 (Udine, 1992), CISM Courses and Lect., Springer, Vienna, 339 (1993), 175–183.  Google Scholar [15] J. T. Coffey and R. M. Goodman, The complexity of information set decoding, IEEE Transactions on Information Theory, 36 (1990), 1031-1037.  doi: 10.1109/18.57202.  Google Scholar [16] I. I. Dumer, Two decoding algorithms for linear codes, Problemy Peredachi Informatsii, 25 (1989), 24-32.   Google Scholar [17] T. Etzion, A. Vardy and E. Yaakobi, Dense error-correcting codes in the Lee metric, In IEEE Information Theory Workshop, (2010), 1–5. Google Scholar [18] M. Finiasz and N. Sendrier, Security bounds for the design of code-based cryptosystems, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, (2009), 88–105. doi: 10.1007/978-3-642-10366-7_6.  Google Scholar [19] E. Gabidulin, A brief survey of metrics in coding theory, Mathematics of Distances and Applications, 66 (2012). Google Scholar [20] M. Greferath, An introduction to ring-linear coding theory, In Gröbner Bases, Coding, and Cryptography, Springer, (2009), 219–238. doi: 10.1007/978-3-540-93806-4_13.  Google Scholar [21] R. A. Hammons, V. P. Kumar, R. A. Calderbank, N. Sloane and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Transactions on Information Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar [22] T. Helleseth and V. Zinoviev, On $\mathbb{Z}_4$-linear Goethals codes and Kloosterman sums, Designs, Codes and Cryptography, 17 (1999), 269-288.  doi: 10.1023/A:1026491513009.  Google Scholar [23] S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, In International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar [24] C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the Ball-Collision algorithm, preprint, arXiv: : 1812.10955, 2018. Google Scholar [25] D. S. Krotov, $\mathbb{Z}_4$-linear Hadamard and extended perfect codes, Electron. Notes Discrete Math., Elsevier Sci. B. V., Amsterdam, 6 (2001), 107-112.   Google Scholar [26] E. A. Kruk, Decoding complexity bound for linear block codes, Problemy Peredachi Informatsii, 25 (1989), 103-107.   Google Scholar [27] P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, In Workshop on the Theory and Application of of Cryptographic Techniques, Springer, 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, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.  doi: 10.1109/18.21270.  Google Scholar [29] A. May, A. Meurer and E. Thomae, Decoding random linear codes in $\mathcal{O} (2^{0.054 n})$, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, 2011 (2011), 107–124. doi: 10.1007/978-3-642-25385-0_6.  Google Scholar [30] A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, In Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer, (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] A. Meurer, A Coding-Theoretic Approach to Cryptanalysis, PhD thesis, Ruhr University Bochum, 2012. Google Scholar [33] A. A. Nechaev, Kerdock code in a cyclic form, Discrete Mathematics and Applications, 1 (1991), 365-384.  doi: 10.1515/dma.1991.1.4.365.  Google Scholar [34] R. Niebuhr, E. Persichetti, Pi erre-Louis Cayrel, S. Bulygin and J. Buchmann, On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, International journal of information and Coding Theory, 4 (2017), 47-78.  doi: 10.1504/IJICOT.2017.081458.  Google Scholar [35] C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, In International Workshop on Post-Quantum Cryptography, Springer, 6061 (2010), 81–94. doi: 10.1007/978-3-642-12929-2_7.  Google Scholar [36] V. S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Transactions on Information Theory, 42 (1996), 1594-1600.  doi: 10.1109/18.532906.  Google Scholar [37] E. Prange, The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), 5-9.  doi: 10.1109/tit.1962.1057777.  Google Scholar [38] R. M. Roth and P. H. Siegel, Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Transactions on Information Theory, 40 (1994), 1083-1096.  doi: 10.1109/18.335966.  Google Scholar [39] C. Satyanarayana, Lee metric codes over integer residue rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 250-254.  doi: 10.1109/TIT.1979.1056017.  Google Scholar [40] P. Shankar, On BCH codes over arbitrary integer rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 480-483.  doi: 10.1109/TIT.1979.1056063.  Google Scholar [41] E. Spiegel, Codes over $\mathbb{Z}_m$, Information and control, 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.  Google Scholar [42] J. Stern, A method for finding codewords of small weight, In International Colloquium on Coding Theory and Applications, Springer, 388 (1989), 106–113. doi: 10.1007/BFb0019850.  Google Scholar [43] I. Tal and R. M. Roth, On list decoding of alternant codes in the Hamming and Lee metrics, In IEEE International Symposium on Information Theory, 2003,364–364. Google Scholar [44] H. Tapia-Recillas, A secret sharing scheme from a chain ring linear code, Congressus Numerantium, 186 (2007), 33-39.   Google Scholar [45] J. van Tilburg, On the McEliece public-key cryptosystem, In Conference on the Theory and Application of Cryptography, Springer, 403 (1990), 119–131. doi: 10.1007/0-387-34799-2_10.  Google Scholar [46] V. Weger, M. Battaglioni, P. Santini, F. Chiaraluce, M. Baldi and E. Persichetti, Information set decoding of Lee-metric codes over finite rings, arXiv preprint, arXiv: 2001.08425, 2020. Google Scholar [47] Y. Wu and C. N. Hadjicostis, Decoding algorithm and architecture for BCH codes under the Lee metric, IEEE Transactions on Communications, 56 (2008), 2050-2059.  doi: 10.1109/TCOMM.2008.041227.  Google Scholar

show all references

##### References:
 [1] E. F. Assmus and H. F. Mattson, Error-correcting codes: An axiomatic approach, Information and Control, 6 (1963), 315-330.  doi: 10.1016/S0019-9958(63)80010-8.  Google Scholar [2] A. Becker, A. Joux, A. 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 [3] E. Berlekamp, Algebraic Coding Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9407.  Google Scholar [4] D. J. Bernstein, Grover vs. McEliece, In International Workshop on Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, 6061 (2010), 73–80. doi: 10.1007/978-3-642-12929-2_6.  Google Scholar [5] D. J. Bernstein, T. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, In International Workshop on Post-Quantum Cryptography, Springer, (2008), 31–46. doi: 10.1007/978-3-540-88403-3_3.  Google Scholar [6] D. J. Bernstein, T. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, In Annual Cryptology Conference, Springer, (2011), 743–760. doi: 10.1007/978-3-642-22792-9_42.  Google Scholar [7] T. Blackford, Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar [8] I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.  Google Scholar [9] I. F. Blake, Codes over integer residue rings, Information and Control, 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.  Google Scholar [10] E. Byrne, Decoding a class of Lee metric codes over a Galois ring, IEEE Transactions on Information Theory, 48 (2002), 966-975.  doi: 10.1109/18.992804.  Google Scholar [11] A. Canteaut and Hervé Chabanne, A Further Improvement of the Work Factor in an Attempt at Breaking McEliece's Cryptosystem, PhD thesis, INRIA, 1994. Google Scholar [12] A. Canteaut and F. Chabaud, A new algorithm for finding minimum-weight words in a linear code: Application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511, IEEE Transactions on Information Theory, 44 (1998), 367-378.  doi: 10.1109/18.651067.  Google Scholar [13] A. Canteaut and N. Sendrier, Cryptanalysis of the original McEliece cryptosystem, In Advances in Cryptology ASIACRYPT'98 (Beijing), Lecture Notes in Comput. Sci., Springer, Berlin, 1514 (1998), 187–199. doi: 10.1007/3-540-49649-1_16.  Google Scholar [14] F. Chabaud, Asymptotic analysis of probabilistic algorithms for finding short codewords, Eurocode '92 (Udine, 1992), CISM Courses and Lect., Springer, Vienna, 339 (1993), 175–183.  Google Scholar [15] J. T. Coffey and R. M. Goodman, The complexity of information set decoding, IEEE Transactions on Information Theory, 36 (1990), 1031-1037.  doi: 10.1109/18.57202.  Google Scholar [16] I. I. Dumer, Two decoding algorithms for linear codes, Problemy Peredachi Informatsii, 25 (1989), 24-32.   Google Scholar [17] T. Etzion, A. Vardy and E. Yaakobi, Dense error-correcting codes in the Lee metric, In IEEE Information Theory Workshop, (2010), 1–5. Google Scholar [18] M. Finiasz and N. Sendrier, Security bounds for the design of code-based cryptosystems, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, (2009), 88–105. doi: 10.1007/978-3-642-10366-7_6.  Google Scholar [19] E. Gabidulin, A brief survey of metrics in coding theory, Mathematics of Distances and Applications, 66 (2012). Google Scholar [20] M. Greferath, An introduction to ring-linear coding theory, In Gröbner Bases, Coding, and Cryptography, Springer, (2009), 219–238. doi: 10.1007/978-3-540-93806-4_13.  Google Scholar [21] R. A. Hammons, V. P. Kumar, R. A. Calderbank, N. Sloane and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Transactions on Information Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar [22] T. Helleseth and V. Zinoviev, On $\mathbb{Z}_4$-linear Goethals codes and Kloosterman sums, Designs, Codes and Cryptography, 17 (1999), 269-288.  doi: 10.1023/A:1026491513009.  Google Scholar [23] S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, In International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar [24] C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the Ball-Collision algorithm, preprint, arXiv: : 1812.10955, 2018. Google Scholar [25] D. S. Krotov, $\mathbb{Z}_4$-linear Hadamard and extended perfect codes, Electron. Notes Discrete Math., Elsevier Sci. B. V., Amsterdam, 6 (2001), 107-112.   Google Scholar [26] E. A. Kruk, Decoding complexity bound for linear block codes, Problemy Peredachi Informatsii, 25 (1989), 103-107.   Google Scholar [27] P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, In Workshop on the Theory and Application of of Cryptographic Techniques, Springer, 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, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.  doi: 10.1109/18.21270.  Google Scholar [29] A. May, A. Meurer and E. Thomae, Decoding random linear codes in $\mathcal{O} (2^{0.054 n})$, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, 2011 (2011), 107–124. doi: 10.1007/978-3-642-25385-0_6.  Google Scholar [30] A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, In Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer, (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] A. Meurer, A Coding-Theoretic Approach to Cryptanalysis, PhD thesis, Ruhr University Bochum, 2012. Google Scholar [33] A. A. Nechaev, Kerdock code in a cyclic form, Discrete Mathematics and Applications, 1 (1991), 365-384.  doi: 10.1515/dma.1991.1.4.365.  Google Scholar [34] R. Niebuhr, E. Persichetti, Pi erre-Louis Cayrel, S. Bulygin and J. Buchmann, On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, International journal of information and Coding Theory, 4 (2017), 47-78.  doi: 10.1504/IJICOT.2017.081458.  Google Scholar [35] C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, In International Workshop on Post-Quantum Cryptography, Springer, 6061 (2010), 81–94. doi: 10.1007/978-3-642-12929-2_7.  Google Scholar [36] V. S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Transactions on Information Theory, 42 (1996), 1594-1600.  doi: 10.1109/18.532906.  Google Scholar [37] E. Prange, The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), 5-9.  doi: 10.1109/tit.1962.1057777.  Google Scholar [38] R. M. Roth and P. H. Siegel, Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Transactions on Information Theory, 40 (1994), 1083-1096.  doi: 10.1109/18.335966.  Google Scholar [39] C. Satyanarayana, Lee metric codes over integer residue rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 250-254.  doi: 10.1109/TIT.1979.1056017.  Google Scholar [40] P. Shankar, On BCH codes over arbitrary integer rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 480-483.  doi: 10.1109/TIT.1979.1056063.  Google Scholar [41] E. Spiegel, Codes over $\mathbb{Z}_m$, Information and control, 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.  Google Scholar [42] J. Stern, A method for finding codewords of small weight, In International Colloquium on Coding Theory and Applications, Springer, 388 (1989), 106–113. doi: 10.1007/BFb0019850.  Google Scholar [43] I. Tal and R. M. Roth, On list decoding of alternant codes in the Hamming and Lee metrics, In IEEE International Symposium on Information Theory, 2003,364–364. Google Scholar [44] H. Tapia-Recillas, A secret sharing scheme from a chain ring linear code, Congressus Numerantium, 186 (2007), 33-39.   Google Scholar [45] J. van Tilburg, On the McEliece public-key cryptosystem, In Conference on the Theory and Application of Cryptography, Springer, 403 (1990), 119–131. doi: 10.1007/0-387-34799-2_10.  Google Scholar [46] V. Weger, M. Battaglioni, P. Santini, F. Chiaraluce, M. Baldi and E. Persichetti, Information set decoding of Lee-metric codes over finite rings, arXiv preprint, arXiv: 2001.08425, 2020. Google Scholar [47] Y. Wu and C. N. Hadjicostis, Decoding algorithm and architecture for BCH codes under the Lee metric, IEEE Transactions on Communications, 56 (2008), 2050-2059.  doi: 10.1109/TCOMM.2008.041227.  Google Scholar
Key sizes and security levels (both in bits) for GV-codes over $\mathbb Z_4$ with $n = 150$ and $d = 81$
 $k_1$ 1 2 3 $\dots$ 18 19 $\dots$ 24 25 26 best $\ell$ 0 0 0 $\dots$ 0 0 $\dots$ 0 1 2 best $v$ 4 4 4 $\dots$ 3 3 $\dots$ 2 2 2 key size 5198 5296 5390 $\dots$ 6110 6160 $\dots$ 6440 6446 6448 security level 31 31 31 $\dots$ 27 27 $\dots$ 28 28 28
 $k_1$ 1 2 3 $\dots$ 18 19 $\dots$ 24 25 26 best $\ell$ 0 0 0 $\dots$ 0 0 $\dots$ 0 1 2 best $v$ 4 4 4 $\dots$ 3 3 $\dots$ 2 2 2 key size 5198 5296 5390 $\dots$ 6110 6160 $\dots$ 6440 6446 6448 security level 31 31 31 $\dots$ 27 27 $\dots$ 28 28 28
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