Feedback shift registers, 2-adic Span, and combiners with memory

Journal of Cryptology

Volumn 10, Issue 2, 1997, Pages 111-147

  Klapper, Andrew ^a  

^a UNIVERSITY OF KENTUCKY   (United States)

Author keywords

2 adic numbers; Arithmetic code; Binary sequence; Combiner with memory; Cryptanalysis; Linear span; Sequence; Shift register; Stream cipher

Indexed keywords

EID: 0001760011     PISSN: 09332790     EISSN: None     Source Type: Journal    
DOI: 10.1007/s001459900024     Document Type: Article

References (48)

1. [1 ] E. Bach, Efficient prediction of Marsaglia-Zaman random number generators, Draft, University of Wisconsin, 1993.
2. J. T. Barrows, Jr., A new method for constructing multiple error correcting linear residue codes, Repor R-277, Coordinated Science Laboratory, University of Illinois, Urbana, 1966.
3. L. Blum, M. Blum, and M. Shub, A simple unpredictable pseudo-random number generator, SIA J. Comput., vol. 15,1986, pp. 364-383.
4. A. Blumer and J. Blumer, Linear size finite automata for the set of all subwords of a word: An outline o results, Bull. European Assoc. Theoret. Comput. Sci., vol. 21, 1983, pp. 68-77.
5. E. Bombieri, Personal communication.
6. A. Chan and R. Games, On the quadratic span of de Bruijn sequences, IEEE Trans. Inform. Theory vol. 36, 1990, pp. 822-829.
7. U. Cheng, On the continued fraction and Berlekamp's algorithm, IEEE Trans. Inform. Theory, vol. 30 1984, pp. 541-544
8. H. Cohen.A Coursein Computational Algebraic Number Theory, Springer- Verlag, New York, 1993.
9. Z. D. Dai and K. C. Zeng, Continued fractions and the Berlekamp-Massey algorithm. Advances i Cryptology-AUSCRYPT '90. Lecture Notes in Computer Science, vol. 453. Springer-Verlag, Berlin 1990
10. C. Ding, Stream Ciphers and Number Theory, to appear.
11. H. D. Ebbinghaus et al.. Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York 1990.
12. C. F. Gauss, Disquisitiones Arithmeticae, 1801; reprinted in English translation by Yale University Press New Haven, CT, 1966.
13. S. Golomb, Shift Register Sequences, Aegean Park Press, Laguna Hills, CA, 1982.
14. R. T. Gregory and E. V. Krishnamurthy, Methods and Applications of Error-Free Computation, SpringerVerlag, New York, 1984.
15. G. H. Hardy and J. E. Littlewood, Some problems of "Partitio Numcrorum"; III: On the expression of number as a sum of primes, Acta Mathematica, vol. 44, 1922, pp. 1-70.
16. G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford 1979
17. C. Hooley, On Artin's conjecture, J. Reine Angew. Math., vol. 22, 1967, pp. 209-220.
18. K. Ireland and M. Rosen, A Classical Introduction to Modem Number Theory, Springer-Verlag, Ne York, 1990
19. C. J. A. Jansen, Information theory of shift registers, In: Proceedings of the Tenth Symposium on Information Theory in the Benelux (A. M. Barbe, ed.), Werkgemeenschap voor Inf.- & Communicatietheorie Enschede, 1989, pp. 153-160.
20. C. J. A. Jansen and D. E. Boekee, The shortest feedback shift register that can generate a given sequence In: Advances in Cryptology-CRYPTO '89 (G. Brassard, ed.). Lecture Notes in Computer Science vol. 435, Springer-Verlag, Berlin, 1990, pp. 90-99.
21. C. J. A. Jansen and D. E. Boekee, On the significance of the directed acyclic word graph in cryptology In: Advances in Cryplology-AUSCRYPT '90. Lecture Notes in Computer Science, vol. 453, SpringerVerlag, Berlin, 1990, pp. 318-326.
22. A. Klapper, Feedback with carry shift registers over finite fields. Fast Sofnvare Encryption, Secon International Workshop. Lecture Notes in Computer Science, vol. 1008, Springer-Verlag, Berlin, 1995 pp. 170-178.
23. A. Klapper and M. Goresky, 2-adic shift registers. Fast Sofuvare Encryption. Lecture Notes in Compute Science, vol. 809, Springer-Verlag, Berlin, 1994, pp. 174-178.
24. A. Klapper and M. Goresky, Feedback registers based on ramified extensions of the 2-adic numbers Advances in Cryptology-Eurocrypt 1994, Perugia, Italy. Lecture Notes in Computer Science, vol. 950 Springer-Verlag, Berlin, 1995, pp. 215-222.
25. A. Klapper and M. Goresky, Large period nearly deBruijn FCSR sequences, Advances in Cryptology Eurocrypt 1995. Lecture Notes in Computer Science, vol. 921, Springer-Verlag, Berlin, 1995, pp. 263 273.
26. A. Klapper and M. Goresky, Cryptanalysis based on 2-adic rational approximation, Advances i Cryptology-CRYPTO '95. Springer Lecture Notes in Computer Science, vol. 963, Springer-Verlag Berlin, 1995, pp. 262-273.
27. A. Klapper and M. Goresky, Arithmetic cross-correlation of FCSR sequences. University of Kentucky Technical Report, no. 262-96, 1996.
28. D. Knuth, The Art of Computer Programming, vol. 2, Seininumerical Algorithms, Addison-Wesley Reading, M A, 1981.
29. N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta Functions, Graduate Texts in Mathematics vol. 58, Springer-Verlag, New York, 1984.
30. E. V. Krishnamurthy and R. T. Gregory, Mapping integers and Hensel codes onto Farey fractions, BIT vol. 23, 1983, pp. 9-20.
31. A. Lempel, M. Cohn, and W. Eastman, A class of balanced binary sequences with optimal autocorrelatio properties, IEEE Trans. Inform. Theory, vol. IT-23, 1977, pp. 38-42
32. K. Mahler, On a geometrical representation of p-adic numbers, Ann. of Math., vol. 41, 1940, pp. 8-56.
33. D. Mandelbaum, Arithmetic codes with large distance, IEEE Trans. Inform. Theory, vol. IT-13, 1967 pp. 237-242.
34. D. Mandelbaum, An approach to an arithmetic analog of Berlekamp's algorithm, IEEE Trans. Inform Theory, vol. IT-30, 1984, pp. 758-762.
35. G. Marsaglia, The mathematics of random number generators. The unreasonable Effectiveness of Numbe Theory, American Mathematical Society, Providence, RI, 1992, pp. 73-90.
36. G. Marsaglia and A. Zaman, A new class of random number generators, Ann.Appl. Probab., vol. 1, 1991 pp. 462-480.
37. J. Massey and R. Rueppel, Method of, and apparatus for, transforming a digital data sequence into a encoded form, U.S. Patent No. 4,797,922, 1989.
38. W. Meier and O. Staffelbach, Correlation properties of combiners with memory in stream ciphers Advances in Cryptology-EUROCRYPT '90. Workshop on the Theory and Application of Cryptographi Techniques Proceedings, Springer-Verlag, Berlin, 1991, pp. 204-213.
39. W. Meier and O. Staffelbach, Correlation properties of combiners with memory in stream ciphers J. Cryptology vol. 5, 1992, pp. 67-86
40. W. H. Mills, Continued fractions and linear recurrences, Malh. Comput., vol. 29, 1975, pp. 173-180.
41. W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, 2nd edn., MIT Press, Cambridge, MA 1972
42. J. Pollard, The fast Fourier transform in a finite field, Math. Comput., vol. 25, 1971, pp. 365-374.
43. T. R. N. Rao, Error Coding For Arithmetic Processors, Academic Press, New York, 1974.
44. R. Rueppel, Analysis and Design of Stream Ciphers, Springer-4 Verlag, New York, 1986.
45. B. Schneier, Applied Cry-ptography.Witey, New York, 1996.
46. A. Schönhage and V. Strassen, Schnelle Multipl ikation Grosser Zahlen, Computing, vol. 7,1971, pp. 281 292.
47. B. M. M. de Weger, Approximation lattices of p-adic numbers, J. Number Theory, vol. 24, 1986, pp. 70 88.
48. L. R. Welch and R. A. Scholtz, Continued fractions and Berlekamp's algorithm, IEEE Trans. Inform Theory, vol. 25, 1979 pp. 19-27.