Targeting high-school students transitioning to proof-based mathematics, this textbook focuses on developing skills in writing and understanding mathematical proofs. It explores commonly used proof techniques across various mathematical fields, particularly number theory, combinatorics, and analysis. Beyond mechanics, it introduces engaging mathematical concepts, enriching students' understanding while honing their proof-writing abilities.
Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text.
This text studies cryptography from its earliest roots (2000 years ago), to the main cryptosystems used today for secure online communication. Many topics covered - originate from number theory, combinatorics, probability, and abstract algebra. The book is partitioned into three sections; the first part discusses classical ciphers and their cryptanalysis. The second - focuses on modern public key cryptosystems: Diffie-Hellman, ElGamal, RSA, and elliptic curve cryptography. This section analyzes attacks against these systems as well as attacks on the underlying math problems such as various clever factorization algorithms. Specialized topics such as zero knowledge proofs, cryptographic voting, coding theory, and new research are covered in the final section of this book. Aimed toward advanced high school students, undergraduates, and amateur mathematicians interested in cryptography, this book is suitable for independent study; - no prerequisites are assumed and it contains a large selection of problems, ranging from straightforward to -to difficult -that challenge the strongest students.