# American Institute of Mathematical Sciences

February  2016, 10(1): 131-150. doi: 10.3934/amc.2016.10.131

## Complementary dual codes for counter-measures to side-channel attacks

 1 LAGA, UMR 7539, CNRS, University of Paris VIII and University of Paris XIII, Department of Mathematics, 2 rue de la liberte, 93 526 Saint-Denis Cedex, France 2 TELECOM-ParisTech, Crypto Group | Paris-Saclay University | CNRS LTCI, 37/39 rue Dareau, 75 634 Paris Cedex 13, France

Received  December 2014 Revised  September 2015 Published  March 2016

We recall why linear codes with complementary duals (LCD codes) play a role in counter-measures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We recall the known primary construction of such codes with cyclic codes, and investigate other constructions, with expanded Reed-Solomon codes and generalized residue codes, for which we study the idempotents. These constructions do not allow to reach all the desired parameters. We study then those secondary constructions which preserve the LCD property, and we characterize conditions under which codes obtained by direct sum, direct product, puncturing, shortening, extending codes, or obtained by the Plotkin sum, can be LCD.
Citation: Claude Carlet, Sylvain Guilley. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 2016, 10 (1) : 131-150. doi: 10.3934/amc.2016.10.131
##### References:
 [1] S. Barnett, Matrices: Methods and Applications,, Clarendon Press, (1990). Google Scholar [2] K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes,, Des. Codes Crypt., 23 (2001), 11. doi: 10.1023/A:1011203416769. Google Scholar [3] S. Bhasin, J.-L. Danger, S. Guilley and Z. Najm, A low-entropy first-degree secure provable masking scheme for resource-constrained devices,, in Workshop Embedded Syst. Sec., (2013). doi: 10.1145/2527317.2527324. Google Scholar [4] S. Bhasin, J.-L. Danger, S. Guilley, Z. Najm and X. T. Ngo, Linear complementary dual code improvement to strengthen encoded circuit against hardware trojan horses,, in 2015 IEEE Int. Symp. Hardware-Oriented Sec. Trust, (2015), 82. doi: 10.1109/HST.2015.7140242. Google Scholar [5] S. Bhasin, J.-L. Danger, S. Guilley, T. Ngo and L. Sauvage, Hardware trojan horses in cryptographic IP cores,, in FDTC, (2013), 15. Google Scholar [6] S. Bhasin, J.-L. Danger, X. T. Ngo, S. Guilley and Z. Najm, Encoding the state of integrated circuits: a proactive and reactive protection against hardware trojans horses,, in Proc. 9th Workshop Embedded Syst. Sec., (2014). doi: 10.1145/2668322.2668329. Google Scholar [7] A. Bojilov, A. J. van Zanten and S. M. Dodunekov, Minimal distances in generalized residue codes,, in Proc. 12th Int. Workshop Algebr. Combin. Coding Theory, (2010). Google Scholar [8] J. Bringer, C. Carlet, H. Chabanne, S. Guilley and H. Maghrebi, Orthogonal direct sum masking - a smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks,, in WISTP, (2014), 40. Google Scholar [9] C. Carlet, Boolean functions for cryptography and error correcting codes,, in Chapter of the Monography Boolean Models and Methods (eds. Y. Crama and P. Hammer), (2010), 257. Google Scholar [10] C. Carlet, A. Daif, J.-L. Danger, S. Guilley, Z. Najm, X. T. Ngo and C. Tavernier, Optimized linear complementary codes implementation for hardware trojan prevention,, in 22nd Europ. Conf. Circuit Theory Des., (2015). Google Scholar [11] C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations,, IEEE Trans. Inf. Theory, 58 (2012), 6000. doi: 10.1109/TIT.2012.2200651. Google Scholar [12] B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length $2l^mp^n$,, preprint, (). Google Scholar [13] J. Etesami, F. Hu and W. Henkel, LCD codes and iterative decoding by projections, a first step towards an intuitive description of iterative decoding,, in GLOBECOM, (2011), 1. Google Scholar [14] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at , (). Google Scholar [15] V. Grosso, F.-X. Standaert and E. Prouff, Low entropy masking schemes, revisited,, in CARDIS (eds. A. Francillon and P. Rohatgi), (2013), 33. Google Scholar [16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003). doi: 10.1017/CBO9780511807077. Google Scholar [17] W. B. V. Kandasamy, F. Smarandache, R. Sujatha and R. S. R. Durai, Erasure Techniques in MRD Codes,, Infinite Study, (2012). Google Scholar [18] S. Ling and C. Xing, Polyadic codes revisited,, IEEE Trans. Inf. Theory, 50 (2004), 200. doi: 10.1109/TIT.2003.821986. Google Scholar [19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, Elsevier, (1977). Google Scholar [20] J. L. Massey, Linear codes with complementary duals,, Discrete Math., 106/107 (1992), 337. doi: 10.1016/0012-365X(92)90563-U. Google Scholar [21] G. L. Mullen and D. Panario, Handbook of Finite Fields,, Chapman and Hall/CRC, (2013). doi: 10.1201/b15006. Google Scholar [22] E. Prouff, M. Rivain and R. Bevan, Statistical Analysis of Second Order Differential Power Analysis,, IEEE Trans. Computers, 58 (2009), 799. doi: 10.1109/TC.2009.15. Google Scholar [23] N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound,, Discrete Math., 285 (2004), 345. doi: 10.1016/j.disc.2004.05.005. Google Scholar [24] A. Sharma, G. K. Bakshi and M. Raka, Polyadic codes of prime power length,, Finite Fields Appl., 13 (2007), 1071. doi: 10.1016/j.ffa.2006.12.006. Google Scholar [25] J. H. van Lint and F. J. MacWilliams, Generalized quadratic residue codes,, IEEE Trans. Inf. Theory, 24 (1978), 730. doi: 10.1109/TIT.1978.1055965. Google Scholar [26] H. N. Ward, Quadratic residue codes and divisibility,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 827. Google Scholar [27] X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual,, Discrete Math., 126 (1994), 391. doi: 10.1016/0012-365X(94)90283-6. Google Scholar

show all references

##### References:
 [1] S. Barnett, Matrices: Methods and Applications,, Clarendon Press, (1990). Google Scholar [2] K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes,, Des. Codes Crypt., 23 (2001), 11. doi: 10.1023/A:1011203416769. Google Scholar [3] S. Bhasin, J.-L. Danger, S. Guilley and Z. Najm, A low-entropy first-degree secure provable masking scheme for resource-constrained devices,, in Workshop Embedded Syst. Sec., (2013). doi: 10.1145/2527317.2527324. Google Scholar [4] S. Bhasin, J.-L. Danger, S. Guilley, Z. Najm and X. T. Ngo, Linear complementary dual code improvement to strengthen encoded circuit against hardware trojan horses,, in 2015 IEEE Int. Symp. Hardware-Oriented Sec. Trust, (2015), 82. doi: 10.1109/HST.2015.7140242. Google Scholar [5] S. Bhasin, J.-L. Danger, S. Guilley, T. Ngo and L. Sauvage, Hardware trojan horses in cryptographic IP cores,, in FDTC, (2013), 15. Google Scholar [6] S. Bhasin, J.-L. Danger, X. T. Ngo, S. Guilley and Z. Najm, Encoding the state of integrated circuits: a proactive and reactive protection against hardware trojans horses,, in Proc. 9th Workshop Embedded Syst. Sec., (2014). doi: 10.1145/2668322.2668329. Google Scholar [7] A. Bojilov, A. J. van Zanten and S. M. Dodunekov, Minimal distances in generalized residue codes,, in Proc. 12th Int. Workshop Algebr. Combin. Coding Theory, (2010). Google Scholar [8] J. Bringer, C. Carlet, H. Chabanne, S. Guilley and H. Maghrebi, Orthogonal direct sum masking - a smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks,, in WISTP, (2014), 40. Google Scholar [9] C. Carlet, Boolean functions for cryptography and error correcting codes,, in Chapter of the Monography Boolean Models and Methods (eds. Y. Crama and P. Hammer), (2010), 257. Google Scholar [10] C. Carlet, A. Daif, J.-L. Danger, S. Guilley, Z. Najm, X. T. Ngo and C. Tavernier, Optimized linear complementary codes implementation for hardware trojan prevention,, in 22nd Europ. Conf. Circuit Theory Des., (2015). Google Scholar [11] C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations,, IEEE Trans. Inf. Theory, 58 (2012), 6000. doi: 10.1109/TIT.2012.2200651. Google Scholar [12] B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length $2l^mp^n$,, preprint, (). Google Scholar [13] J. Etesami, F. Hu and W. Henkel, LCD codes and iterative decoding by projections, a first step towards an intuitive description of iterative decoding,, in GLOBECOM, (2011), 1. Google Scholar [14] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at , (). Google Scholar [15] V. Grosso, F.-X. Standaert and E. Prouff, Low entropy masking schemes, revisited,, in CARDIS (eds. A. Francillon and P. Rohatgi), (2013), 33. Google Scholar [16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003). doi: 10.1017/CBO9780511807077. Google Scholar [17] W. B. V. Kandasamy, F. Smarandache, R. Sujatha and R. S. R. Durai, Erasure Techniques in MRD Codes,, Infinite Study, (2012). Google Scholar [18] S. Ling and C. Xing, Polyadic codes revisited,, IEEE Trans. Inf. Theory, 50 (2004), 200. doi: 10.1109/TIT.2003.821986. Google Scholar [19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, Elsevier, (1977). Google Scholar [20] J. L. Massey, Linear codes with complementary duals,, Discrete Math., 106/107 (1992), 337. doi: 10.1016/0012-365X(92)90563-U. Google Scholar [21] G. L. Mullen and D. Panario, Handbook of Finite Fields,, Chapman and Hall/CRC, (2013). doi: 10.1201/b15006. Google Scholar [22] E. Prouff, M. Rivain and R. Bevan, Statistical Analysis of Second Order Differential Power Analysis,, IEEE Trans. Computers, 58 (2009), 799. doi: 10.1109/TC.2009.15. Google Scholar [23] N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound,, Discrete Math., 285 (2004), 345. doi: 10.1016/j.disc.2004.05.005. Google Scholar [24] A. Sharma, G. K. Bakshi and M. Raka, Polyadic codes of prime power length,, Finite Fields Appl., 13 (2007), 1071. doi: 10.1016/j.ffa.2006.12.006. Google Scholar [25] J. H. van Lint and F. J. MacWilliams, Generalized quadratic residue codes,, IEEE Trans. Inf. Theory, 24 (1978), 730. doi: 10.1109/TIT.1978.1055965. Google Scholar [26] H. N. Ward, Quadratic residue codes and divisibility,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 827. Google Scholar [27] X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual,, Discrete Math., 126 (1994), 391. doi: 10.1016/0012-365X(94)90283-6. Google Scholar
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