Abstract
This paper is intended as an overview, presenting several results on the linear complexity of sequences obtained from functions applied to linear shift register sequences. Especially for cryptologic applications it is of course highly desirable that the linear complexity be as large as possible, and not only to get a huge period. The theory reviewed in this paper contains several criteria on how to achieve such goals.
This research was supported in part by the National Swedish Board for Technical Development under grants 81-3323, 83-4364, and 85-3759 at the University of Lund.
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Some Selected References (in chronological order)
L. Fibonacci, Liber Abaci, 1202
E. Galois, “Sur la theorie des nombres”, Bull. Sci. Math. de M. Ferussac, 1830; J. Math. Pures Appl., 1846
L. Kronecker, Werke Bd. 2, pp. 146–149, 1881
D. E. Müller, “Application of Boolean Algebra to Switching Circuit Design and to Error Detection”, IRE Trans. on Electron, Comp., 1954
I. S, Reed, “A Class of Error Correcting Codes and the Decoding Scheme”, IRE Trans. on Electron. Comp., 1954
N. Zierler, “Linear Recurring Sequences”, J. SIAM, 1959; also in W. H. Kautz, Linear Sequential Switching Circuits, Holden-Day, San Francisco, 1965
E. S. Selmer, Linear Recurrence Relations over Finite Fields, Univ. of Bergen, Norway, 1966
B. L. van der Waerden, Algebra I, Springer, Berlin, 1966
S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, 1967
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York. 1968
J. L. Massey, “Shift-Register Synthesis and BCH Decoding”, IEEE Trans on Inform. Th., 1969
E. J. Groth, “Generation of Binary Sequences with Controllable Complexity”, IEEE Trans. on Inform. Th., 1971
N. Zierler and W. H. Mills, “Products of Linear Recurring Sequences”, J. Algebra, 1973
M. P. Ristenbatt et al., “Crack-Resistant Sequences for Data Security”, IEEE Nat. Telecomm. Conf., 1973
P. R. Geffe, “How to Protect Data with Ciphers That Are Really Hard to Break”, Electronics, 1973
P. Nyffeler, Binäre Automaten und ihren linearen Rekursionen, Ph.D. Thesis, Bern, 1975
B. Benjauthrit and I. S. Reed, “Galois Switching Functions and Their Applications”, IEEE Trans. on Comp., 1976
E. L. Key, “An Analysis of the Structure and Complexity of Nonlinear Binary Sequences Generators”, IEEE Trans. on Inform. Th., 1976
K. P. Yiu and R. B. Ward, “A Method for Deciphering a Maximal-Length Sequence”, Proc. IEEE, 1977
T. Herlestam, “On Linearization of Nonlinear Combinations of Linear Shift Register Sequences”, IEEE ISIT, Ithaca, New York, 1977
H. Lüneburg, Galoisfelder, Kreisteilungskörper und Schieberegisterfolgen, Bibliogr. Inst., Zürich, 1979
A. Tucker, Applied Combinatorics, Wiley, New York, 1980
S. M. Jennings, A Special Class of Binary Sequences, Ph.D. Thesis, London, 1980
T. Herlestam, “On Using Prime Polynomials in Crypto Generators”, in Cryptography, Proc. Burg Feuerstein, 1982, ed. by T. Beth, Springer, Berlin, 1983
T. Herlestam, “On the Complexity of Functions of Linear Shift Register Sequences”, IEEE ISIT, Les Arcs, France, 1982
H. Beker and F. Piper, Cipher Systems, Northwood Publ., London, 1982
R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. and Its Appl. Vol. 20, Addison-Wesley, 1983
T. Herlestam, “On the Complexity of Certain Crypto Generators”, in security, IFIP/sec’83, ed. by V. Fåk, North-Holland, 1983
R. Rueppel, New Approaches to Stream Ciphers, Ph.D. Thesis, Zürich, 1984
L. Brynielsson, “On the Linear Complexity of Combined Shift Register Sequences”, Eurocrypt 85, Linz, Austria, 1985
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© 1986 Spnnger-Verlag Berlin Heidelberg
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Herlestam, T. (1986). On Functions of Linear Shift Register Sequences. In: Pichler, F. (eds) Advances in Cryptology — EUROCRYPT’ 85. EUROCRYPT 1985. Lecture Notes in Computer Science, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-39805-8_14
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DOI: https://doi.org/10.1007/3-540-39805-8_14
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