On the matrix feedback shift register synthesis for matrix sequences

矩阵序列的矩阵反馈移位寄存器综合问题

Abstract

In this paper, a generalization of the linear feedback shift register synthesis problem is presented for synthesizing minimum-length matrix feedback shift registers (MFSRs for short) to generate prescribed matrix sequences and so a new complexity measure, that is, matrix complexity, is introduced. This problem is closely related to the minimal partial realization in linear systems and so can be solved through any minimal partial realization algorithm. All minimum-length MFSRs capable of generating a given matrix sequence with finite length are characterized and a necessary and sufficient condition for the uniqueness issue is obtained. Furthermore, the asymptotic behavior of the matrix complexity profile of random vector sequences is determined.

摘要

创新点

  1. (1)

    提出并解决了矩阵序列的矩阵反馈移位寄存器综合问题;

  2. (2)

    利用对偶格的性质, 对于有限长的矩阵序列, 给出了所有生成该序列的最短的矩阵反馈移位寄存器及其唯一的充要条件。

  3. (3)

    提出了矩阵序列的矩阵复杂度并给出了向量序列的矩阵复杂度轮廓的渐进性质。

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References

  1. 1

    Dawson E, Simpson L. Analysis and design issues for synchronous stream ciphers. In: Niederreiter H, ed. Coding Theory and Cryptology. Singapore: World Scientific, 2002. 49–90

    Google Scholar 

  2. 2

    Ekdahl P, Johansson T. A new version of the stream ciphers SNOW. In: Proceedings of 9th Annual International Workshop on Selected Areas in Cryptography, Newfoundland, 2002. 47–61

    Google Scholar 

  3. 3

    Hawkes P, Rose G G. Exploiting multiples of the connection polynomial in word-oriented stream ciphers. In: Proceedings of 6th International Conference on the Theory and Application of Cryptology and Information Security, Kyoto, 2000. 303–316

    Google Scholar 

  4. 4

    Niederreiter H. Factorization of polynomials and some linear algebra problems over finite fields. Linear Alg Appl, 1993, 192: 301–328

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Tsaban B, Vishne U. Efficient linear feedback shift registers with maximal period. Finite Fields Appl, 2002, 8: 256–267

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Zeng G, Han W, He K. High efficiency feedback shift register: σ-LFSR. Cryptology ePrint Archive, Report 2007/114, 2007

    Google Scholar 

  7. 7

    Zeng G, He K, Han W. A trinomial type of s-LFSR oriented toward software implementation. Sci China Ser-F: Inf Sci, 2007, 50: 359–372

    MathSciNet  MATH  Google Scholar 

  8. 8

    Zeng G, Yang Y, Han W, et al. Word oriented cascade jump σ-LFSR. In: Proceedings of 18th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Tarragona, 2009. 127–136

    Google Scholar 

  9. 9

    Berlekamp E R. Algebraic Coding Theory. New York: McGraw-Hill, 1968

    MATH  Google Scholar 

  10. 10

    Massey J L. Shift-register synthesis and BCH decoding. IEEE Trans Inform Theory, 1969, 15: 122–127

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Dai Z D, Wang K P, Ye D F. m-Continued fraction expansions of multi-Laurent series (in Chinese). Adv Math, 2004, 33: 246–248

    Google Scholar 

  12. 12

    Dai Z D, Wang K P, Ye D F. Multi-continued fraction algorithm on multi-formal Laurent series. Acta Arithmet, 2006, 122: 1–16

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Dai Z D, Yang J H. Multi-continued fraction algorithm and generalized B-M algorithm over Fq. Finite Fields Appl, 2006, 12: 379–402

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Ding C S. Proof of Massey’s conjectured algorithm. In: Proceedings of Workshop on the Theory and Application of Cryptographic Techniques, Davos, 1988. 345–349

    Google Scholar 

  15. 15

    Feng G L, Tzeng K K. A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Trans Inform Theory, 1991, 37: 1274–1287

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Wang L P, Zhu Y F, Pei D Y. On the lattice basis reduction multisequence synthesis algorithm. IEEE Trans Inform Theory, 2004, 50: 2905–2910

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Kaltofen F, Yuhasz G. On the matrix Berlekamp-Massey algorithm. ACM Trans Algorithm, 2013, 9: 33

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Kaltofen F, Yuhasz G. A fraction free matrix Berlekamp/Massey algorithm. Linear Alg Appl, 2013, 439: 2515–2526

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Antoulas A C. On recursiveness and related topics in linear systems. IEEE Trans Automat Control, 1985, 31: 1121–1135

    MathSciNet  Article  MATH  Google Scholar 

  20. 20

    Dickinson B W, Morf M, Kailath D. A minimal realization algorithm for matrix sequences. IEEE Trans Automat Control, 1974, 19: 31–38

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Gragg W B, Lindquist A. On the partial realization problem. Linear Alg Appl, 1983, 50: 277–319

    MathSciNet  Article  MATH  Google Scholar 

  22. 22

    Kuijper M. An algorithm for constructing a minimal partial realization in the multivariable case. Syst Contr Lett, 1997, 31: 225–233

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    van Barel M, Bultheel M A. A generalized minimal partial realization problem. Linear Alg Appl, 1997, 254: 527–551

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Wang L P. A lattice-based minimal partial realization algorithm. In: Proceedings of 5th International Conference on Sequences and Their Applications, Lexington, 2008. 278–289

    Google Scholar 

  25. 25

    Wang L P. A lattice-based minimal partial realization algorithm for matrix sequences of varying length. Cryptogr Commun, 2011, 3: 29–42

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Wang L P. Lagrange interpolation polynomials and generalized Reed-Solomon codes over rings of matrices. In: Proceedings of IEEE International Symposium on Information Theory, Cambridge, 2012. 3098–3100

    Google Scholar 

  27. 27

    Quintin G, Barbier M, Chabot C. On generalized Reed-Solomon codes over commutative and noncommutative rings. IEEE Trans Inform Theory, 2013, 59: 5882–5897

    MathSciNet  Article  Google Scholar 

  28. 28

    Dai Z D, Imamura K, Yang J H. Asymptotic behavior of normalized linear complexity of multi-sequences. In: Proceeding of 3rd International Conference on Sequences and Their Applications, Seoul, 2004. 126–142

    Google Scholar 

  29. 29

    Niederreiter H, Wang L P. Proof of a conjecture on the joint linear complexity profile of multisequences. In: Proceeding of 6th International Conference on Cryptology in India, Bangalore, 2005. 13–22

    Google Scholar 

  30. 30

    Niederreiter H, Wang L P. The asymptotic behavior of the joint linear complexity profile of multisequences. Monatsh Math, 2007, 150: 141–155

    MathSciNet  Article  MATH  Google Scholar 

  31. 31

    Niederreiter H, Vielhaber M, Wang L P. Improved results on the probabilistic theory of the joint linear complexity of multisequences. Sci China Inf Sci, 2012, 55: 165–170

    MathSciNet  Article  MATH  Google Scholar 

  32. 32

    Wang L P, Niederreiter H. Enumeration results on the joint linear complexity of multisequences. Finite Fields Appl, 2006, 12: 613–637

    MathSciNet  Article  MATH  Google Scholar 

  33. 33

    Mahler K. An analogue to Minkowski’s geometry of numbers in a field of series. Ann Math, 1941, 42: 488–522

    MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Couture R, L’Ecuyer P. Lattice computations for random numbers. Math Comput, 2000, 69: 757–765

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Liping Wang.

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Wang, L., Zeng, G. On the matrix feedback shift register synthesis for matrix sequences. Sci. China Inf. Sci. 59, 32107 (2016). https://doi.org/10.1007/s11432-015-5302-1

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Keywords

  • Berlekamp-Massey algorithm
  • minimal partial realization
  • multisequences
  • σ-LFSR

关键词

  • Berlekamp-Massey 算法
  • 极小部分实现
  • 多重序列
  • σ-LFSR