Restricted Congruences in Computing

Preface

Dedication

1 Introduction
1.1 Motivation
1.2 Overview of the book
2 The Restricted Congruences Toolbox 6
2.1 Ramanujan sums
2.2 Some useful identities
2.3 The discrete Fourier transform
2.4 Universal hashing and its variants
2.5 Multilinear Modular Hashing
2.6 Fuchsian groups and Harvey's theorem
2.7 Counting epimorphisms via homomorphisms
2.8 Generating functions for graph enumeration
2.9 Deletion correcting codes
2.10 Weight enumerator of a code
2.11 Gaussian integers, spectral graph theory, characters
3 The GCD-Restricted Linear Congruences
3.1 Introduction
3.2 Linear congruences with
3.3 An equivalent form of Theorem 3.2.4
3.4 Some problems
4 Applications in Universal Hashing and Authentication with Secrecy
4.1 Introduction
4.2 Generalized Multilinear Modular Hashing
4.3 GRDH
4.4 Applications to authentication with secrecy
4.5 Discussion
5 Applications in String Theory and Quantum Field Theory
5.1 Introduction
5.2 Counting surface-kernel epimorphisms from Γ to Zn
5.3 A problem
6 Alldiff Congruences, Graph Theoretic Method, and Beyond
6.1 Introduction
6.2 Graph theoretic method
6.3 Unweighted alldiff congruences
6.4 More applications and connections
6.5 A problem
7 Alldiff Congruences Meet VT Codes
7.1 Introduction
7.2 Main results
8 Binary Linear Congruence Code
8.1 Introduction
8.2 Weight enumerator of the Binary Linear Congruence Code
8.3 Weight enumerators of the aforementioned codes
9 Applications in Parallel Computing, AI, etc
9.1 Application in parallel computing
9.2 Application in arti□ficial intelligence and computational biology
9.3 Application in the Subset-Sum Problem
10 Quadratic Congruences, Ramanujan Graphs, and the Golomb-Welch Conjecture
10.1 Introduction
10.2 Quadratic congruences
10.3 Proof of the conjecture
10.4 A problem

Bibliography

Index