Using LDGM Codes and Sparse Syndromes to Achieve Digital Signatures

  • Marco Baldi
  • Marco Bianchi
  • Franco Chiaraluce
  • Joachim Rosenthal
  • Davide Schipani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7932)


In this paper, we address the problem of achieving efficient code-based digital signatures with small public keys. The solution we propose exploits sparse syndromes and randomly designed low-density generator matrix codes. Based on our evaluations, the proposed scheme is able to outperform existing solutions, permitting to achieve considerable security levels with very small public keys.


Code-based digital signatures LDGM codes sparse syndromes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baldi, M., Chiaraluce, F.: Cryptanalysis of a new instance of McEliece cryptosystem based on QC-LDPC codes. In: Proc. IEEE International Symposium on Information Theory (ISIT 2007), Nice, France, pp. 2591–2595 (June 2007)Google Scholar
  2. 2.
    Baldi, M., Chiaraluce, F., Garello, R., Mininni, F.: Quasi-cyclic low-density parity-check codes in the McEliece cryptosystem. In: Proc. IEEE International Conference on Communications (ICC 2007), Glasgow, Scotland, pp. 951–956 (June 2007)Google Scholar
  3. 3.
    Baldi, M., Bodrato, M., Chiaraluce, F.: A new analysis of the McEliece cryptosystem based on QC-LDPC codes. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 246–262. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Baldi, M., Bambozzi, F., Chiaraluce, F.: On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes. IEEE Trans. on Information Theory 57(9), 6052–6067 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Enhanced public key security for the McEliece cryptosystem (2011),
  6. 6.
    M. Baldi, M. Bianchi, and F. Chiaraluce. “Security and complexity of the McEliece cryptosystem based on QC-LDPC codes. IET Information Security (in press),
  7. 7.
    Baldi, M., Bianchi, M., Chiaraluce, F.: Optimization of the parity-check matrix density in QC-LDPC code-based McEliece cryptosystems. To be presented at the IEEE International Conference on Communications (ICC 2013) - Workshop on Information Security over Noisy and Lossy Communication Systems, Budapest, Hungary (June 2013)Google Scholar
  8. 8.
    Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in 2n/20: How 1 + 1 = 0 improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Bernstein, D.J., Lange, T., Peters, C.: Attacking and defending the mcEliece cryptosystem. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 31–46. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Bernstein, D.J., Lange, T., Peters, C.: Smaller decoding exponents: ball-collision decoding. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 743–760. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Chabaud, F., Stern, J.: The cryptographic security of the syndrome decoding problem for rank distance codes. In: Kim, K.-c., Matsumoto, T. (eds.) ASIACRYPT 1996. LNCS, vol. 1163, pp. 368–381. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  12. 12.
    Cheng, J.F., McEliece, R.J.: Some high-rate near capacity codecs for the Gaussian channel. In: Proc. 34th Allerton Conference on Communications, Control and Computing, Allerton, IL (October 1996)Google Scholar
  13. 13.
    Courtois, N.T., Finiasz, M., Sendrier, N.: How to achieve a McEliece-based digital signature scheme. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 157–174. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Finiasz, M., Sendrier, N.: Security bounds for the design of code-based cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 88–105. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Finiasz, M.: Parallel-CFS strengthening the CFS McEliece-based signature scheme. In: Proc. PQCrypto, Darmstadt, Germany, pp. 61–72, May 25-28 (2010)Google Scholar
  16. 16.
    Garcia-Frias, J., Zhong, W.: Approaching Shannon performance by iterative decoding of linear codes with low-density generator matrix. IEEE Commun. Lett. 7(6), 266–268 (2003)CrossRefGoogle Scholar
  17. 17.
    González-López, M., Vázquez-Araújo, F.J., Castedo, L., Garcia-Frias, J.: Serially-concatenated low-density generator matrix (SCLDGM) codes for transmission over AWGN and Rayleigh fading channels. IEEE Trans. Wireless Commun. 6(8), 2753–2758 (2007)CrossRefGoogle Scholar
  18. 18.
    Kabatianskii, G., Krouk, E., Smeets, B.: A digital signature scheme based on random error correcting codes. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 161–167. Springer, Heidelberg (1997)Google Scholar
  19. 19.
    Lim, C.H., Lee, P.J.: On the length of hash-values for digital signature schemes. In: Proc. CISC 1995, Seoul, Korea, November 1995, pp. 29–31 (1995)Google Scholar
  20. 20.
    May, A., Meurer, A., Thomae, E.: Decoding random linear codes in \(\tilde{\mathcal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Progress Report, pp. 114–116 (1978)Google Scholar
  22. 22.
    Minder, L., Sinclair, A.: The extended k-tree algorithm. Journal of Cryptology 25(2), 349–382 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Misoczki, R., Tillich, J.-P., Sendrier, N., Barreto, P.S.L.M.: MDPC-McEliece: New McEliece variants from moderate density parity-check codes (2012),
  24. 24.
    Monico, C., Rosenthal, J., Shokrollahi, A.: Using low density parity check codes in the McEliece cryptosystem. In: Proc. IEEE International Symposium on Information Theory (ISIT 2000), Sorrento, Italy, p. 215 (June 2000)Google Scholar
  25. 25.
    Niebuhr, R., Cayrel, P.-L., Buchmann, J.: Improving the efficiency of Generalized Birthday Attacks against certain structured cryptosystems. In: Proc. WCC 2011, Paris, France, April 11-15 (2011)Google Scholar
  26. 26.
    Otmani, A., Tillich, J.-P.: An efficient attack on all concrete KKS proposals. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 98–116. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  27. 27.
    Peters, C.: Information-set decoding for linear codes over F q. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 81–94. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  28. 28.
    Sendrier, N.: Decoding one out of many. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 51–67. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Stern, J.: A method for finding codewords of small weight. In: Wolfmann, J., Cohen, G. (eds.) Coding Theory and Applications 1988. LNCS, vol. 388, pp. 106–113. Springer, Heidelberg (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marco Baldi
    • 1
  • Marco Bianchi
    • 1
  • Franco Chiaraluce
    • 1
  • Joachim Rosenthal
    • 2
  • Davide Schipani
    • 3
  1. 1.Universitá Politecnica delle MarcheAnconaItaly
  2. 2.University of ZurichZurichSwitzerland
  3. 3.Nottingham Trent UniversityNottinghamUK

Personalised recommendations