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August  2013, 7(3): 267-278. doi: 10.3934/amc.2013.7.267

The classification of complementary information set codes of lengths $14$ and $16$

Finley Freibert 1,

1. 

Department of Mathematics, Ohio Dominican University, 1216 Sunbury Road, Columbus, OH 43219, United States

Received  June 2012 Published  July 2013

In the paper ``A new class of codes for Boolean masking of cryptographic computations,'' Carlet, Gaborit, Kim, and Solé defined a new class of rate one-half binary codes called complementary information set (or CIS) codes. The authors then classified all CIS codes of length less than or equal to 12. CIS codes have relations to classical Coding Theory as they are a generali-zation of self-dual codes. As stated in the paper, CIS codes also have important practical applications as they may improve the cost of masking cryptographic algorithms against side channel attacks. In this paper, we give a complete classification result for length 14 CIS codes using an equivalence relation on $GL(n,\mathbb{F}_2)$. We also give a new classification for all binary $[16,8,3]$ and $[16,8,4]$ codes. We then complete the classification for length 16 CIS codes and give additional classifications for optimal CIS codes of lengths 20 and 26.
Keywords: CIS codes, computerized search, graph isomorphism., classification, formally self-dual codes.
Mathematics Subject Classification: 94B05, 11T7.
Citation: Finley Freibert. The classification of complementary information set codes of lengths $14$ and $16$. Advances in Mathematics of Communications, 2013, 7 (3) : 267-278. doi: 10.3934/amc.2013.7.267
References:
[1]

K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes, Des. Codes Crypt., 23 (2001), 11-21. doi: 10.1023/A:1011203416769.  Google Scholar

[2]

K. Betsumiya and M. Harada, Classification of formally self-dual even codes of lengths up to 16, Des. Codes Crypt., 23 (2001), 325-332. doi: 10.1023/A:1011223128089.  Google Scholar

[3]

K. Betsumiya, M. Harada and A. Munemasa, A complete classification of doubly-even self-dual codes of length 40,, preprint, ().   Google Scholar

[4]

J. Cannon and C. Playoust, "An Introduction to Magma,'' University of Sydney, Sydney, Australia, 1994. Google Scholar

[5]

C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations,, preprint, ().  doi: 10.1109/TIT.2012.2200651.  Google Scholar

[6]

I. A. Faradzev, Constructive enumeration of combinatorial objects, in "Problemes Combinatoires et Theorie des Graphes Colloque Internat,'' CNRS, Paris, (1978), 131-135.  Google Scholar

[7]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths 20 and 22, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.  Google Scholar

[8]

F. Freibert, A classification of binary [16,8,4] codes; A classification of [14,7] CIS codes, available online at http://finleyfreibert.wordpress.com/mathematics-research/, 2012. Google Scholar

[9]

T. A. Gulliver and P. R. J. Östergard, Binary optimal linear rate 1/2 codes, Discrete Math., 283 (2004), 255-261. doi: 10.1016/j.disc.2003.10.027.  Google Scholar

[10]

S. Han, H. Lee and Y. Lee, Binary formally self-dual odd codes, Des. Codes Crypt., 61 (2010), 141-150. doi: 10.1007/s10623-010-9444-2.  Google Scholar

[11]

W. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.  Google Scholar

[12]

P. Kaski and P. R. J. Östergard, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006.  Google Scholar

[13]

H. Maghrebi, C. Carlet, S. Guilley and J.-L. Danger, Optimal first-order masking with linear and non-linear bijections, in "Progress in Cryptology - AFRICACRYPT 2012,'' (2012), 360-377. doi: 10.1007/978-3-642-31410-0_22.  Google Scholar

[14]

H. Maghrebi, S. Guilley, C. Carlet and J.-L. Danger, Classification of high-order Boolean masking Schemes and improvements of their efficiency, available online at http://eprint.iacr.org/2011/520.pdf, 2011. Google Scholar

[15]

H. Maghrebi, S. Guilley and J.-L. Danger, Leakage squeezing countermeasure against high-order attacks, in "Information Security Theory and Practice,'' Springer, Berlin, (2011), 208-223. doi: 10.1007/978-3-642-21040-2_14.  Google Scholar

[16]

B. D. McKay, Nauty user's guide (version 2.4), available online at http://cs.anu.edu.au/~bdm/nauty/nug.pdf, 2009. Google Scholar

[17]

P. R. J. Östergard, Classifying subspaces of hamming spaces, Des. Codes Crypt., 27 (2000), 297-305. doi: 10.1023/A:1019903407222.  Google Scholar

[18]

V. Pless, A classification of self-orthogonal codes over $GF(2)$, Discrete Math., 3 (1972), 215-228. doi: 10.1016/0012-365X(72)90034-9.  Google Scholar

[19]

R. C. Read, Every one a winner; or, how to avoid isomorphism search when cataloguing combinatorial configurations, Ann. Discrete Math., 2 (1978), 107-120. doi: 10.1016/S0167-5060(08)70325-X.  Google Scholar

[20]

M. Rivain and E. Prouff, Provably secure higher-order masking of AES, in "Cryptographic Hardware and Embedded Systems, CHES 2010,'' Springer, Berlin, (2010), 413-427. doi: 10.1007/978-3-642-15031-9_28.  Google Scholar

[21]

H. G. Schaathun, On higher weights and code existence, in "Cryptography and Coding,'' Springer, Berlin, (2009), 56-64. doi: 10.1007/978-3-642-10868-6_4.  Google Scholar

[22]

J. Simonis, A description of the $[16,7,6]$ codes, in "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (1991), 25-35. doi: 10.1007/3-540-54195-0_36.  Google Scholar

show all references

References:
[1]

K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes, Des. Codes Crypt., 23 (2001), 11-21. doi: 10.1023/A:1011203416769.  Google Scholar

[2]

K. Betsumiya and M. Harada, Classification of formally self-dual even codes of lengths up to 16, Des. Codes Crypt., 23 (2001), 325-332. doi: 10.1023/A:1011223128089.  Google Scholar

[3]

K. Betsumiya, M. Harada and A. Munemasa, A complete classification of doubly-even self-dual codes of length 40,, preprint, ().   Google Scholar

[4]

J. Cannon and C. Playoust, "An Introduction to Magma,'' University of Sydney, Sydney, Australia, 1994. Google Scholar

[5]

C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations,, preprint, ().  doi: 10.1109/TIT.2012.2200651.  Google Scholar

[6]

I. A. Faradzev, Constructive enumeration of combinatorial objects, in "Problemes Combinatoires et Theorie des Graphes Colloque Internat,'' CNRS, Paris, (1978), 131-135.  Google Scholar

[7]

J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths 20 and 22, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.  Google Scholar

[8]

F. Freibert, A classification of binary [16,8,4] codes; A classification of [14,7] CIS codes, available online at http://finleyfreibert.wordpress.com/mathematics-research/, 2012. Google Scholar

[9]

T. A. Gulliver and P. R. J. Östergard, Binary optimal linear rate 1/2 codes, Discrete Math., 283 (2004), 255-261. doi: 10.1016/j.disc.2003.10.027.  Google Scholar

[10]

S. Han, H. Lee and Y. Lee, Binary formally self-dual odd codes, Des. Codes Crypt., 61 (2010), 141-150. doi: 10.1007/s10623-010-9444-2.  Google Scholar

[11]

W. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.  Google Scholar

[12]

P. Kaski and P. R. J. Östergard, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006.  Google Scholar

[13]

H. Maghrebi, C. Carlet, S. Guilley and J.-L. Danger, Optimal first-order masking with linear and non-linear bijections, in "Progress in Cryptology - AFRICACRYPT 2012,'' (2012), 360-377. doi: 10.1007/978-3-642-31410-0_22.  Google Scholar

[14]

H. Maghrebi, S. Guilley, C. Carlet and J.-L. Danger, Classification of high-order Boolean masking Schemes and improvements of their efficiency, available online at http://eprint.iacr.org/2011/520.pdf, 2011. Google Scholar

[15]

H. Maghrebi, S. Guilley and J.-L. Danger, Leakage squeezing countermeasure against high-order attacks, in "Information Security Theory and Practice,'' Springer, Berlin, (2011), 208-223. doi: 10.1007/978-3-642-21040-2_14.  Google Scholar

[16]

B. D. McKay, Nauty user's guide (version 2.4), available online at http://cs.anu.edu.au/~bdm/nauty/nug.pdf, 2009. Google Scholar

[17]

P. R. J. Östergard, Classifying subspaces of hamming spaces, Des. Codes Crypt., 27 (2000), 297-305. doi: 10.1023/A:1019903407222.  Google Scholar

[18]

V. Pless, A classification of self-orthogonal codes over $GF(2)$, Discrete Math., 3 (1972), 215-228. doi: 10.1016/0012-365X(72)90034-9.  Google Scholar

[19]

R. C. Read, Every one a winner; or, how to avoid isomorphism search when cataloguing combinatorial configurations, Ann. Discrete Math., 2 (1978), 107-120. doi: 10.1016/S0167-5060(08)70325-X.  Google Scholar

[20]

M. Rivain and E. Prouff, Provably secure higher-order masking of AES, in "Cryptographic Hardware and Embedded Systems, CHES 2010,'' Springer, Berlin, (2010), 413-427. doi: 10.1007/978-3-642-15031-9_28.  Google Scholar

[21]

H. G. Schaathun, On higher weights and code existence, in "Cryptography and Coding,'' Springer, Berlin, (2009), 56-64. doi: 10.1007/978-3-642-10868-6_4.  Google Scholar

[22]

J. Simonis, A description of the $[16,7,6]$ codes, in "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (1991), 25-35. doi: 10.1007/3-540-54195-0_36.  Google Scholar

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