A New Analysis of the McEliece Cryptosystem Based on QC-LDPC Codes
We improve our proposal of a new variant of the McEliece cryptosystem based on QC-LDPC codes. The original McEliece cryptosystem, based on Goppa codes, is still unbroken up to now, but has two major drawbacks: long key and low transmission rate. Our variant is based on QC-LDPC codes and is able to overcome such drawbacks, while avoiding the known attacks. Recently, however, a new attack has been discovered that can recover the private key with limited complexity. We show that such attack can be avoided by changing the form of some constituent matrices, without altering the remaining system parameters. We also propose another variant that exhibits an overall increased security level. We analyze the complexity of the encryption and decryption stages by adopting efficient algorithms for processing large circulant matrices. The Toom-Cook algorithm and the short Winograd convolution are considered, that give a significant speed-up in the cryptosystem operations.
KeywordsMcEliece cryptosystem QC-LDPC codes Cryptanalysis Toom-Cook Winograd
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- 1.McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Progress Report, 114–116 (1978)Google Scholar
- 3.Lee, P., Brickell, E.: An observation on the security of McEliece’s public-key cryptosystem. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 275–280. Springer, Heidelberg (1988)Google Scholar
- 7.Riek, J.: Observations on the application of error correcting codes to public key encryption. In: Proc. IEEE International Carnahan Conference on Security Technology. Crime Countermeasures, Lexington, KY, USA, October 1990, pp. 15–18 (1990)Google Scholar
- 9.Baldi, M., Chiaraluce, F.: Cryptanalysis of a new instance of McEliece cryptosystem based on QC-LDPC codes. In: Proc. IEEE ISIT 2007, Nice, France, June 2007, pp. 2591–2595 (2007)Google Scholar
- 10.Monico, C., Rosenthal, J., Shokrollahi, A.: Using low density parity check codes in the McEliece cryptosystem. In: Proc. IEEE ISIT 2000, Sorrento, Italy, June 2000, p. 215 (2000)Google Scholar
- 11.Otmani, A., Tillich, J.P., Dallot, L.: Cryptanalysis of two McEliece cryptosystems based on quasi-cyclic codes. In: Proc. First International Conference on Symbolic Computation and Cryptography (SCC 2008), Beijing, China (April 2008)Google Scholar
- 12.Gaborit, P.: Shorter keys for code based cryptography. In: Proc. Int. Workshop on Coding and Cryptography WCC, Bergen, Norway, March 2005, pp. 81–90 (2005)Google Scholar
- 14.Neal, R.M.: Faster encoding for low-density parity check codes using sparse matrix methods (1999), http://www.cs.toronto.edu/~radford/ftp/ima-part1.pdf.
- 16.Baldi, M., Chiaraluce, F.: LDPC Codes in the McEliece Cryptosystem (September 2007), http://arxiv.org/abs/0710.0142
- 17.Karatsuba, A.A., Ofman, Y.: Multiplication of multidigit numbers on automata. Soviet Physics Doklady 7, 595–596 (1963)Google Scholar
- 18.Toom, A.L.: The complexity of a scheme of functional elements realizing the multiplication of integers. Soviet Mathematics Doklady 3, 714–716 (1963)Google Scholar
- 19.Cook, S.A.: On the minimum computation time of functions. PhD thesis, Dept. of Mathematics, Harvard University (1966)Google Scholar
- 20.Bodrato, M., Zanoni, A.: Integer and polynomial multiplication: Towards optimal Toom-Cook matrices. In: Brown, C.W. (ed.) Proceedings of the ISSAC 2007 Conference, July 2007, pp. 17–24. ACM Press, New York (2007)Google Scholar
- 28.Silverman, J.H.: High-speed multiplication of (truncated) polynomials. Technical Report 10, NTRU CryptoLab (January 1999)Google Scholar