Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

for Correction and Detection 18.4 Hadamard Matrices 18.5 Mariner, Hadamard, and Reed–Muller 18.6 Reed–Muller Codes 18.7 Block Designs 18.8 The Rank of Incidence Matrices 18.9 The Main Coding Theory Problem, Bounds 18.10 Update on the Reed–Muller Codes: The Proof of an Old Conjecture 18.11 Problems 18.12 Solutions Chapter 19: Finite Fields, Modular Arithmetic, Linear Algebra, and Number Theory 19.1 Modular Arithmetic 19.2 A Little Linear Algebra 19.3 Applications to RSA 19.4 Primitive Roots for Primes and Diffie–Hellman 19.5 The Extended Euclidean Algorithm 19.6 Proof that the RSA Algorithm Works 19.7 Constructing Finite Fields 19.8 Pollard's p-1 Factoring Algorithm 19.9 Latin Squares 19.10 Computational Complexity, Turing Machines, Quantum Computing 19.11 Problems 19.12 Solutions Note Chapter 20: Introduction to Linear Codes 20.1 Repetition Codes and Parity Checks 20.2 Details of Linear Codes 20.3 Parity Checks, the Syndrome, and Weights 20.4 Hamming Codes, an Inequality 20.5 Perfect Codes, Errors, and the BSC 20.6 Generalizations of Binary Hamming Codes 20.7 The Football Pools Problem, Extended Hamming Codes 20.8 Golay Codes 20.9 McEliece Cryptosystem 20.10 Historical Remarks 20.11 Problems 20.12 Solutions Chapter 21: Cyclic Linear Codes, Shift Registers, and CRC 21.1 Cyclic Linear Codes 21.2 Generators for Cyclic Codes 21.3 The Dual Code 21.4 Linear Feedback Shift Registers and Codes 21.5 Finding the Period of a LFSR 21.6 Cyclic Redundancy Check (CRC) 21.7 Problems 21.8 Solutions Chapter 22: Reed‐Solomon and MDS Codes, and the Main Linear Coding Theory Problem (LCTP) 22.1 Cyclic Linear Codes and Vandermonde 22.2 The Singleton Bound for Linear Codes 22.3 Reed–Solomon Codes 22.4 Reed‐Solomon Codes and the Fourier Transform Approach 22.5 Correcting Burst Errors, Interleaving 22.6 Decoding Reed‐Solomon Codes, Ramanujan, and Berlekamp–Massey 22.7 An Algorithm for Decoding and an Example 22.8 Long MDS Codes and a Partial Solution of a 60 Year‐Old Problem 22.9 Problems 22.10 Solutions Chapter 23: MDS Codes, Secret Sharing, and Invariant Theory 23.1 Some Facts Concerning MDS Codes 23.2 The Case k = 2, Bruck Nets 23.3 Upper Bounds on MDS Codes, Bruck–Ryser 23.4 MDS Codes and Secret Sharing Schemes 23.5 MacWilliams Identities, Invariant Theory 23.6 Codes, Planes, and Blocking Sets 23.7 Long Binary Linear Codes of Minimum Weight at Least 4 23.8 An Inverse Problem and a Basic Question in Linear Algebra Chapter 24: Key Reconciliation, Linear Codes, and New Algorithms 24.1 Symmetric and Public Key Cryptography 24.2 General Background 24.3 The Secret Key and the Reconciliation Algorithm 24.4 Equality of Remnant Keys: The Halting Criterion 24.5 Linear Codes: The Checking Hash Function 24.6 Convergence and Length of Keys 24.7

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