A new class of security oriented error correcting robust codes

  • Hila RabiiEmail author
  • Osnat Keren
Part of the following topical collections:
  1. Special Issue on Coding Theory and Applications


Robust codes are codes that can detect any nonzero errore with probability 1 − Q(e) > 0. This property makes them useful in protecting hardware systems from fault injection attacks which cause an arbitrary number of bit flips. This paper presents a new construction of non-linear robust q-ary codes with q = 2m and an error correction capability. The codes are built upon systematic linear codes [n, k, d]q whereas the nk redundant symbols that were originally allocated to increase the minimum distance of the code are modified to provide both correction capability and robustness. The error masking probability of the codes is Q(e) upper bounded by 2/q for odd values of m and by 4/q for even m. Hence, they are more effective in detecting maliciously injected errors and have a higher code rate than codes obtained by concatenation of a linear error correcting code with a security oriented code.


Fault injection attacks Security oriented codes Robust Nonlinear Error correction 



  1. 1.
    Admaty, N., Litsyn, S., Keren, O.: Puncturing, Expurgating and Expanding the Q-Ary BCH Based Robust Codes. In: 2012 IEEE 27Th Convention Of Electrical Electronics Engineers in Israel (IEEEI), pp. 1–5. (2012)
  2. 2.
    Berger, T.P., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear functions over \( \mathbb {F}_{2}^{n}\). IEEE Trans Inf Theory 52(9), 4160–4170 (2006)CrossRefGoogle Scholar
  3. 3.
    Biham, E., Shamir, A.: Differential Fault Analysis of Secret Key Cryptosystems. In: Advances in Cryptology—CRYPTO’97. Springer, pp. 513–525 (1997)Google Scholar
  4. 4.
    Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans Inf Theory 52(3), 1141–1152 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carlet, C., Ding, C.: Highly nonlinear mappings. J Complex 20(2-3), 205–244 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cramer, R., Dodis, Y., Fehr, S., Padró, C., Wichs, D.: Detection of Algebraic Manipulation with Applications to Robust Secret Sharing and Fuzzy Extractors. In: Advances in Cryptology–EUROCRYPT 2008. Springer, pp. 471–488 (2008)Google Scholar
  7. 7.
    Dobbertin, H.: Almost perfect nonlinear power functions on G F(2n): the niho case. Inf Comput 151(1-2), 57–72 (1999)CrossRefGoogle Scholar
  8. 8.
    Dobbertin, H.: Almost Perfect Nonlinear Power Functions on G F(2n): a New Case for N Divisible by 5. In: Finite Fields and Applications. Springer, pp. 113–121 (2001)Google Scholar
  9. 9.
    Dziembowski, S., Pietrzak, K., Wichs, D.: Non-malleable codes. Cryptology ePrint Archive, Report 2009/608. (2009)
  10. 10.
    Engelberg, S., Keren, O.: A comment on the karpovsky–taubin code. IEEE Trans Inf Theory 57(12), 8007–8010 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Etzion, T., Vardy, A.: Perfect binary codes: constructions, properties, and enumeration. IEEE Trans Inf Theory 40(3), 754–763 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gaubatz, G., Sunar, B., Karpovsky, M.: Non-Linear Residue Codes for Robust Public-Key Arithmetic. In: Fault Diagnosis and Tolerance in Cryptography. Springer, pp 173–184 (2006)Google Scholar
  13. 13.
    Gold, R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.) IEEE Trans Inf Theory 14(1), 154–156 (1968)CrossRefGoogle Scholar
  14. 14.
    Karpovsky, M., Taubin, A.: New class of nonlinear systematic error detecting codes. IEEE Trans Inf Theory 50(8), 1818–1819 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Karpovsky, M., Kulikowski, K., Wang, Z.: Robust Error Detection in Communication and Computational Channels. In: SMMSP’2007. 2007 International Workshop On Spectral Methods and Multirate Signal Processing. Citeseer (2007)Google Scholar
  16. 16.
    Kasami, T.: The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes. Inf Control 18(4), 369–394 (1971)CrossRefGoogle Scholar
  17. 17.
    Keren, O., Karpovsky, M.: Relations between the entropy of a source and the error masking probability for security-oriented codes. IEEE Trans Commun 63(1), 206–214 (2015)Google Scholar
  18. 18.
    Keren, O., Levin, I., Stankovic, R.S.: A technique for linearization of logic functions defined by disjoint cubes. i. – theoretical aspects. Autom Remote Control 72 (3), 615–625 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kulikowski, K., Wang, Z., Karpovsky, M.: Comparative Analysis of Fault Attack Resistant Architectures for Private and Public Key Cryptosystems. In: Proceedings of International Workshop Fault-Tolerant Cryptographic Devices, pp. 41–50 (2008)Google Scholar
  20. 20.
    Kulikowski, K., Karpovsky, M., Taubin, A.: Fault Attack Resistant Cryptographic Hardware with Uniform Error Detection. In: Fault Diagnosis and Tolerance in Cryptography. Springer, pp 185–195 (2006)Google Scholar
  21. 21.
    Kyureghyan, G.M., Suder, V.: On inverses of APN exponents. In: 2012 IEEE International Symposium on Information Theory Proceedings (ISIT). IEEE, pp 1207–1211 (2012)Google Scholar
  22. 22.
    Neumeier, Y., Keren, O.: A New Efficiency Criterion for Security Oriented Error Correcting Codes. In: 2014 19Th IEEE European Test Symposium (ETS). IEEE, pp 1–6 (2014)Google Scholar
  23. 23.
    Neumeier, Y., Keren, O.: Robust generalized punctured cubic codes. IEEE Trans Inf Theory 60(5), 2813–2822 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ngo, X.T., Bhasin, S., Danger, J., Guilley, S., Najm, Z.: Linear Complementary Dual Code Improvement to Strengthen Encoded Circuit against Hardware Trojan Horses. In: IEEE International Symposium on Hardware Oriented Security and Trust, HOST 2015, Washington, pp 82–87 (2015)Google Scholar
  25. 25.
    Nyberg, K.: Differentially Uniform Mappings for Cryptography. In: Workshop on the Theory and Application of Of Cryptographic Techniques. Springer, pp 55–64 (1993)Google Scholar
  26. 26.
    Phelps, K.: A combinatorial construction of perfect codes. SIAM J Algebraic Discret Methods 4(3), 398–403 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Phelps, K.T., Levan, M.: Kernels of nonlinear Hamming codes. Des Codes Crypt 6(3), 247–257 (1995)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rabii, H., Keren, O.: A New Construction of Minimum Distance Robust Codes. In: International Castle Meeting on Coding Theory and Applications. Springer, pp 272–282 (2017)Google Scholar
  29. 29.
    Rabii, H., Neumeier, Y., Keren, O.: Low Complexity High Rate Robust Codes. In: Steinbach, B (ed.) Further Improvements in the Boolean Domain, pp. 303–313. Cambridge Scholars Publishing (CSP) (2017)Google Scholar
  30. 30.
    Rabii, H., Neumeier, Y., Keren, O.: High rate robust codes with low implementation complexity. IEEE Transactions on Dependable and Secure Computing, (2018)
  31. 31.
    Tomashevich, V., Neumeier, Y., Kumar, R., Keren, O., Polian, I.: Protecting Cryptographic Hardware against Malicious Attacks by Nonlinear Robust Codes. In: 2014 IEEE International Symposium On Defect and Fault Tolerance in VLSI and Nanotechnology Systems (DFT). IEEE, pp. 40–45 (2014)Google Scholar
  32. 32.
    Vasil’ev, Y.L.: On nongroup close-packed codes. Probl Kibernet 8, 375–378 (1962)Google Scholar
  33. 33.
    Verbauwhede, IM (ed.): Secure integrated circuits and systems. Springer, Berlin (2010)zbMATHGoogle Scholar
  34. 34.
    Wang, Z., Karpovsky, M.: Algebraic Manipulation Detection Codes and Their Applications for Design of Secure Cryptographic Devices. In: On-Line Testing Symposium (IOLTS), 2011 IEEE 17Th International. IEEE, pp. 234–239 (2011)Google Scholar
  35. 35.
    Wang, Z., Karpovsky, M., Joshi, A.: Reliable MLC NAND Flash Memories Based on Nonlinear T-Error-Correcting Codes. In: 2010 IEEE/IFIP International Conference On Dependable Systems and Networks (DSN). IEEE, pp. 41–50 (2010)Google Scholar
  36. 36.
    Wang, Z., Karpovsky, M., Kulikowski, K.: Design of memories with concurrent error detection and correction by nonlinear SEC-DED codes. J Electron Test 26(5), 559–580 (2010)CrossRefGoogle Scholar
  37. 37.
    Wang, Z., Karpovsky, M., Sunar, B.: Multilinear Codes for Robust Error Detection. In: 2009. IOLTS 2009. 15Th IEEE International On-Line Testing Symposium. IEEE, pp 164–169 (2009)Google Scholar
  38. 38.
    Wang, Z., Karpovsky, M., Kulikowski, K.: Replacing Linear Hamming Codes by Robust Nonlinear Codes Results in a Reliability Improvement of Memories. In: 2009. DSN’09. IEEE/IFIP International Conference On Dependable Systems & Networks. IEEE, pp. 514–523 (2009)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of EngineeringBar-Ilan UniversityRamat GanIsrael

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