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Error-correcting codes and cryptography

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Abstract

In this paper, we give and explain some illustrative examples of research topics where error-correcting codes overlap with cryptography. In some of these examples, error-correcting codes employed in the implementation of secure cryptographic protocols. In the others, the codes are used in attacks against cryptographic schemes. Throughout this paper, we show the interrelation between error-correcting codes and cryptography, as well as point out the common features and the differences between these two fields.

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References

  1. Shannon C.E. (1948). A mathematical theory of communication (parts 1 and 2). Bell Syst. Tech. J. 27: 379–423

    MathSciNet  MATH  Google Scholar 

  2. Shannon C.E. (1949). Communication theory of secrecy systems. Bell Syst. Tech. J. 28: 656–715

    MathSciNet  Google Scholar 

  3. Beth T. and Sagerer G. (1981). CODEC—a program system for the interactive development of error-correcting coders/decoders. Elektronische Informations verarbeitung und Kybernetik 17(2/3): 145–147

    MATH  Google Scholar 

  4. Beth, T.: Codes, groups and invariants, AAECC 1984, pp.71–78 (1984)

  5. Beth, T.: Algebraic and Dymbolic computation in digital signal processing, coding and cryptography. In: European Conference on Computer Algebra. vol. 1, pp. 93–101 (1985)

  6. Beth T. (1996). Designs, codes and crypts—a puzzle altogether. Des. Codes Cryptogr. 8(1–2): 91–101

    Article  MATH  MathSciNet  Google Scholar 

  7. Beth T., Charnes C., Grassl M., Alber G., Delgado A. and Mussinger M. (2003). A new class of designs which protect against quantum jumps. Des. Codes Cryptogr. 29(1–3): 51–70

    Article  MATH  MathSciNet  Google Scholar 

  8. Shamir A. (1979). How to share a secret. Commun. ACM 22: 612–613

    Article  MATH  MathSciNet  Google Scholar 

  9. Blakley, G.:Safeguarding cryptographic keys. In: Proceedings of AFIPS National Computer Conference. vol. 48, pp. 313–317 (1979)

  10. Karnin E., Green J. and Hellman M. (1983). On secret sharing systems. IEEE Trans. Inf. Theory IT-29(1): 35–41

    Article  Google Scholar 

  11. Massey, J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp. 276–279, (1993)

  12. McEliece R.J. and Sarwate D.V. (1979). On sharing secrets and Reed-Solomon codes. Comm. ACM 22(11): 612–613

    Article  MathSciNet  Google Scholar 

  13. Brickell E.F. (1990). Some ideal secret sharing schemes, in advances in cryptology—Eurocrypt’89. Lect. Notes Comput. Sci. 434: 468–475

    MathSciNet  Google Scholar 

  14. Cramer, R., Daza, V., Gracia I., Urrorz, J.J., Leander, G., Martíi-Farré J., Padró, C.: On codes, matroids and secure multi-party computation from linear secret sharing schemes, Lecture Notes in Computer Science, vol. 3621, Crypto 2005, pp. 327–343 (2005)

  15. Ding C., Kohel D.R. and Ling S. (2000). Secret-sharing with a class of ternary codes. Theor. Comput. Sci. 246: 285–298

    Article  MATH  MathSciNet  Google Scholar 

  16. Gilbert E.N., MacWilliams F.J. and Sloane N.J.A. (1974). Codes which detect deception. Bell Syst. Tech. J. 53(3): 405–424

    MathSciNet  Google Scholar 

  17. Pei, D.: Authentication Codes and Combinatorial Designs, Discrete Mathematics and its Applications, Chapman and Hall/CRC, (2006)

  18. Berlekamp E.R., McEliece R.J. and van Tilborg H.C.A. (1978). On the inherent intractability of certain coding problems. IEEE Trans. Inf. Theory IT-24(3): 384–386

    Article  MathSciNet  Google Scholar 

  19. McEliece R.J. (1978). A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep. 42(44): 114–116

    Google Scholar 

  20. Kobara, K., Imai, H.: Semantically secure McEliece public-key cryptosystems-conversions for McEliece PKC-. In: PKC’2001, Lecture Notes in Computer Science, Cheju Island, Korea, Springer, Heidelberg (2001)

  21. Niederreiter, H.: Error-correcting Codes and Cryptography, Public-Key Cryptography and Computational Number Theory. In: Alster, K., Urbanowicz, J., Williams, H.C. (eds.) pp. 209–219, W. de Gruyter, Berlin, 2001

  22. Loidreau, P.: How to reduce public-key size in McEliece like PKCs? In: CLC 2006, Workshop on Codes and Lattices in Cryptography, pp. 57–60, (2006)

  23. Kobara, K., Imai, H.: Personalized-public-key cryptosystem (P 2 KC)—application where public-key size of Niederreiter PKC can be reduced-, CLC 2006. In: Workshop on Codes and Lattices in Cryptography, pp. 61–68, 2006

  24. Davida, G.I., Frankel, Y., Matt, B.J.: On enabling secure applications through off-line biometric identification. In: Proc. IEEE Symposium and Privacy, pp. 148–157, 1998

  25. Martinian, E., Yekhanin S., Yadidia, J.S.: Secure biometrics via syndromes, Mitsubishi electric research laboratories. Technical Reports (2005)

  26. Gallager R.G. (1963). Low Density Parity Check Codes. MIT Press, Cambridge

    Google Scholar 

  27. Richardson, T.J., Shokrollahi, M.A., Urbanke, R.L.: Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans. Inf. Theory 47(2), (2001)

  28. Menezes A.J., Vanstone S.A. and Oorschot P.C. (1997). Handbook of Applied Cryptography. CRC Press, Boca Raton

    MATH  Google Scholar 

  29. Mihaljević, M.J., Fossorier, M.P.C., Imai H.: A general formulation of algebraic and fast correlation attacks based on dedicated sample decimation, AAECC2006. Lecture Notes Computer Science, vol. 3857. pp. 203–214, Feb. (2006)

  30. Mihaljević M.J., Fossorier M.P.C. and Imai H. (2002). Fast correlation attack algorithm with list decoding and an application, FSE 2001. Lect. Notes Comput. Sci. 2355: 196–210

    Google Scholar 

  31. Fossorier Marc P.C., Mihaljević M.J. and Imai H. (1999). Critical noise for convergence of iterative probabilistic decoding with belief propagation in cryptographic applications, applied algebra, algebraic algorithms and error correcting codes—AAECC 13. Lect. Notes Comput. Sci. 1719: 282–293

    Google Scholar 

  32. Johansson T. and Jonsson F. (1999). Improved fast correlation attacks on stream ciphers via convolutional codes, advanced in cryptology—EUROCRYPTO’99. Lect. Notes Comput. Sci. 1592: 347–362

    Google Scholar 

  33. Johansson T. and Jonsson F. (1999). Fast correlation attacks based on turbo code techniques, advances in cryptology—CRYPTO’99. Lect. Notes Comput. Sci. 1666: 181–197

    Article  Google Scholar 

  34. Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption, EUROCRYPTf88. Springer, Heidelberg, pp. 419–453, (1988)

  35. Grassl M., Geiselmann W. and Beth T. (1999). Quantum Reed-Solomon codes, AAECC 13. Lect. Notes Comput. Sci. 1719: 231–244

    MathSciNet  Google Scholar 

  36. Calderbank A.R. and Shor P.W. (1996). Good quantum error correcting codes exist. Phys. Rev. A 54: 1098–1105

    Article  Google Scholar 

  37. Steane A.M. (1996). Multiple particle interference and quantum error correction. Proc. Roy. Soc. London A 452: 2551–2577

    MATH  MathSciNet  Google Scholar 

  38. Shor P.W. and Preskill J. (2000). Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85: 441–444

    Article  Google Scholar 

  39. Lo H.-K. and Chau H.F. (1999). Unconditional security of quantum key distribution over arbitrarily long distances. Science 283(5410): 2050–2056

    Article  Google Scholar 

  40. MacKay D., Mitchison G. and McFadden P. (2004). Sparse graph codes for quantum error-correction, quant-ph/0304161. IEEE Trans. Inf. Theory 50(10): 2315–2330

    Article  MathSciNet  Google Scholar 

  41. Hagiwara, M., Imai, H.: A simple construction of quantum quasi-cyclic LDPC codes. In: Proceeding of HISC 2007

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Imai, H., Hagiwara, M. Error-correcting codes and cryptography. AAECC 19, 213–228 (2008). https://doi.org/10.1007/s00200-008-0074-0

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