On the Cryptanalysis of Nonlinear Sequences [Invited Paper]

  • Solomon W. Golomb
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1746)


A nonlinear boolean function f(x1; x2; : : : ; xk) of k binary variables may be used in two basically different ways to generate a non-linear binary sequence, internally or externally. Internally, f may be part of the feedback computation of a nonlinear feedback shift register. Externally, f may be applied to the output bit stream of another sequence generator (e.g. a linear shift register) to introduce nonlinearity, or greater nonlinearity. A third approach is to use f to obtain a nonlinear combination of k linear sequences. The vulnerability of systems using f in any of these ways to cryptanalysis depends on the multidimensional correlations of f with the modulo 2 sums of the subsets of its variables. This principle was published by the present author in [1] in 1959, and included as Chapter 8 in his book [2] in 1967. It was subsequently rediscovered and republished in 1988 in [3], on the basis of which it is sometimes known as the Xiao-Massey algorithm. Some practical aspects of the use of this principle in code construction as well as code breaking, and for other types of signal design, are discussed.


Boolean Function Binary Sequence Truth Table Shift Register Linear Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Golomb, S.W.: On the Classification of Boolean Functions. Transactions of the International Symposium on Circuit and Information Theory: IRE Transactions on Circuit Theory. CT-6 (1959) 176–186; IRE Transactions on Information Theory. IT-5 (1959) 176-186.CrossRefGoogle Scholar
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    Golomb, S.W.: Shift Register Sequences. Holden-Day, Inc., San Francisco (1967).zbMATHGoogle Scholar
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    Xiao, G.-Z., Massey, J.L.: A spectral characterization of correlation-immune combining functions. IEEE Trans. on Information Theory, IT-34,no. 3 (1988) 569–571.CrossRefMathSciNetGoogle Scholar
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    Slepian, D.: On the number of symmetry types of boolean functions of n variables, Can. J. Math. 5,no. 2 (1953) 185–193.zbMATHMathSciNetGoogle Scholar
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    Golomb, S.W., ed.: Digital Communications with Space Applications. Prentice-Hall, Englewood Cliffs, NJ (1964).zbMATHGoogle Scholar
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    Siegenthaler, T., Decrypting a Class of Stream Ciphers Using Ciphertext Only. IEEE Trans. on Computers, C-34 (1985) 81–85.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Solomon W. Golomb
    • 1
  1. 1.Communication Sciences Institute University of Southern CaliforniaLos AngelesUSA

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