As the first book of its kind, this book is devoted to studying congruences, their problems and applications. The book will be of interest to graduate students and researchers across computer science, electrical engineering, and mathematics.

PrefaceDedication1 Introduction 1.1 Motivation 1.2 Overview of the book 2 The Restricted Congruences Toolbox 62.1 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 code2.11 Gaussian integers, spectral graph theory, characters 3 The GCD-Restricted Linear Congruences 3.1 Introduction 3.2 Linear congruences with3.3 An equivalent form of Theorem 3.2.4 3.4 Some problems 4 Applications in Universal Hashing and Authentication with Secrecy 4.1 Introduction4.2 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 Introduction5.2 Counting surface-kernel epimorphisms from Γ to Zn5.3 A problem 6 Alldiff Congruences, Graph Theoretic Method, and Beyond 6.1 Introduction 6.2 Graph theoretic method6.3 Unweighted alldiff congruences6.4 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 Introduction8.2 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 computing9.2 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 Introduction10.2 Quadratic congruences 10.3 Proof of the conjecture 10.4 A problem Bibliography Index