Lower bounds for the linear complexity of sequences over residue rings 1st edition by Zong duo Dai, Thomas Beth Dieter Gollmann ISBN 3540535874 9783540535874

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ISBN 10:       3540535874
ISBN 13: 978-3540535874
Author:         Zong-duo Dai, Thomas Beth Dieter Gollmann 

Linear feedback shift registers over the ring Z ₂ ^e can be implemented efficiently on standard microprocessors. The most significant bits of the elements of a sequence in Z ^e∞₂ constitute a binary pseudo-random sequence. We derive lower bounds for the linear complexity over F ₂ of these binary sequences.

Lower bounds for the linear complexity of sequences over residue rings 1st Table of contents:

Chapter 1: Introduction

• Overview of Linear Complexity
• Importance of Studying Sequences Over Residue Rings
• Applications in Cryptography, Coding Theory, and Pseudorandom Number Generation
• Goals and Structure of the Work

Chapter 2: Preliminaries

• Definition of Sequences and Residue Rings
• Algebraic Structures of Residue Rings
• Linear Complexity of Sequences
• Linear Recurrence Relations and Feedback Polynomials
• Notations and Terminology

Chapter 3: Linear Complexity of Sequences

• Definition and Key Properties of Linear Complexity
• Methods for Computing Linear Complexity
• Relationships Between Linear Complexity and Other Properties of Sequences
• Bounds and Estimates in Linear Complexity Theory

Chapter 4: Residue Rings and Their Structure

• Properties of Residue Rings: Z_n and Generalizations
• Modulo Operations and Their Impact on Sequence Behavior
• Representation of Sequences Over Residue Rings
• Basic Operations in Residue Rings and Their Role in Linear Complexity

Chapter 5: Lower Bounds for Linear Complexity

• Lower Bound Techniques for Linear Complexity
• Analytical Methods and Approaches
• Known Results and Classical Bounds
• Tightening Lower Bounds for Special Classes of Sequences

Chapter 6: Methods for Constructing Sequences with High Linear Complexity

• Construction Methods for Pseudorandom Sequences
• Methods for Maximizing Linear Complexity
• Example Sequences Over Residue Rings
• Application of Bounds in Sequence Construction

Chapter 7: Asymptotic Behavior of Linear Complexity

• Asymptotic Analysis for Sequences Over Residue Rings
• Limits of Linear Complexity in Infinite Sequences
• Comparison with Sequences Over Other Algebraic Structures
• Stability of Lower Bounds in Asymptotic Settings

Chapter 8: Applications of Linear Complexity Bounds

• Cryptographic Applications: Stream Ciphers and Security
• Coding Theory: Error Detection and Correction
• Pseudorandom Number Generation
• Other Areas of Practical Interest

Chapter 9: Computational Techniques and Algorithms

• Algorithms for Computing Linear Complexity
• Efficient Techniques for Sequence Analysis
• Numerical Methods for Lower Bound Estimation
• Computational Complexity Considerations

Chapter 10: Open Problems and Future Directions

• Limitations of Current Techniques and Bounds
• Unsolved Problems in Linear Complexity Theory
• Potential Areas for Further Research
• New Techniques and Approaches for Improved Bounds

Chapter 11: Conclusion

• Summary of Key Results
• Implications of Lower Bounds in Practical Applications
• Final Remarks on the Future of Sequence Analysis

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