Abstract
This paper studies the task of inferring a binary vector s given noisy observations of the binary vector t=As modulo 2, where A is an M×N binary matrix. This task arises in correlation attack on a class of stream ciphers and in the decoding of error correcting codes.
The unknown binary vector is replaced by a real vector of probabilities that are optimized by variational free energy minimization. The derived algorithms converge in computational time of order between w A and Nw A , where w A is the number of 1s in the matrix A, but convergence to the correct solution is not guaranteed.
Applied to error correcting codes based on sparse matrices A, these algorithms give a system with empirical performance comparable to that of BCH and Reed-Muller codes.
Applied to the inference of the state of a linear feedback shift register given the noisy output sequence, the algorithms offer a principled version of Meier and Staffelbach's (1989) algorithm B, thereby resolving the open problem posed at the end of their paper. The algorithms presented here appear to give superior performance.
Supported by the Royal Society Smithson Research Fellowship
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References
Aiyer, S. V. B., Niranjan, M. and Fallside, F. (1990). A theoretical investigation into the performance of the Hopfield model, IEEE Trans. on Neural Networks 1(2): 204–215.
Anderson, R. J. (1995). Searching for the optimum correlation attack, in B. Preneel (ed.), Fast Software Encryption Lecture Notes in Computer Science, Springer-Verlag, pp. 137–143 (these proceedings).
Berlekamp, E. R. (1968). Algebraic Coding Theory, McGraw-Hill, New York.
Berlekamp, E. R., McEliece, R. J. and van Tilborg, H. C. A. (1978). On the intractability of certain coding problems, IEEE Transactions on Information Theory 24(3): 384–386.
Blake, A. and Zisserman, A. (1987). Visual Reconstruction, MIT Press, Cambridge Mass.
Durbin, R. and Willshaw, D. (1987). An analogue approach to the travelling salesman problem using an elastic net method, Nature 326: 689–91.
Feynman, R. P. (1972). Statistical Mechanics, W. A. Benjamin, Inc.
Gallager, R. G. (1963). Low density parity check codes, number 21 in Research monograph series, MIT Press, Cambridge, Mass.
Gee, A. H. and Prager, R. W. (1994). Polyhedral combinatorics and neural networks, Neural Computation 6: 161–180.
Hopfield, J. J. and Tank, D. W. (1985). Neural computation of decisions in optimization problems, Biological Cybernetics 52: 1–25.
MacKay, D. J. C. and Neal, R. M. (1995). Error correcting codes using free energy minimization, in preparation.
McEliece, R. J. (1978). A public-key cryptosystem based on algebraic coding theory, Technical Report DSN 42-44, JPL, Pasadena.
Meier, W. and Staffelbach, O. (1989). Fast correlation attacks on certain stream ciphers, J. Cryptology 1: 159–176.
Mihaljević, M. J. and Golić, J. D. (1992). A fast iterative algorithm for a shift register initial state reconstruction given the noisy output sequence, Advances in Cryptology — AUSCRYPT'90, Vol. 453, Springer-Verlag, pp. 165–175.
Mihaljević, M. J. and Golić, J. D. (1993). Convergence of a Bayesian iterative errorcorrection procedure on a noisy shift register sequence, Advances in Cryptology — EUROCRYPT 92, Vol. 658, Springer-Verlag, pp. 124–137.
Peterson, C. and Soderberg, B. (1989). A new method for mapping optimization problems onto neural networks, Int. Journal Neural Systems.
Peterson, W. W. and Weldon, Jr., E. J. (1972). Error-Correcting Codes, 2nd edn, MIT Press, Cambridge, Massachusetts.
Van den Bout, D. E. and Miller, III, T. K. (1989). Improving the performance of the Hopfield-Tank neural network through normalization and annealing, Biological Cybernetics 62: 129–139.
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MacKay, D.J.C. (1995). A free energy minimization framework for inference problems in modulo 2 arithmetic. In: Preneel, B. (eds) Fast Software Encryption. FSE 1994. Lecture Notes in Computer Science, vol 1008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60590-8_15
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DOI: https://doi.org/10.1007/3-540-60590-8_15
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