Decoding a Perturbed Sequence Generated by an LFSR
Given a sequence of bits produced by a linear feedback shift register (LFSR), the Berlekamp-Massey algorithm finds a register of minimal length able to generate the sequence. The situation is different when the sequence is perturbed; for instance, when it is sent through a transmission channel. LFSRs can be described as autonomous systems. A perturbed sequence of bits generated by an LFSR can be interpreted as a codeword in the binary linear code generated by the corresponding observability matrix. The problem of finding the original sequence can then be stated as the decoding problem, “given the received codeword, find the information transmitted”. We propose two decoding algorithms, one based on a brute force attack and the other one based on the representation technique of the syndromes introduced by Becker, Joux, May, and Meurer (2012).
KeywordsLFSR Correlation attack Keystream sequence Companion matrix Autonomous system Syndrome decoding Decoding representation technique
The first author was supported by FAPESP with number of process 2015/07246-0. The second author was partially supported by grants MIMECO MTM2015-68805-REDT and MTM2015-69138-REDT. The third author was partially supported by grants MINECO MTM2013-40960-P and MTM2015-68805-REDT.
- 2.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). doi: 10.1007/978-3-642-29011-4_31 CrossRefGoogle Scholar
- 5.Geffe, P.: How to protect data with ciphers that are really hard to break. Electronics 46(1), 99–101 (1973)Google Scholar
- 15.Meier, W., Staffelbach, O.: Fast correlation attacks on stream ciphers. In: Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegmüller, G., Stoer, J., Wirth, N., Günther, C.G. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 301–314. Springer, Heidelberg (1988). doi: 10.1007/3-540-45961-8_28 Google Scholar