This book provides a broad overview of cryptography and enables cryptography for trying out. It emphasizes the connections between theory and practice, focuses on RSA for introducing number theory and PKI, and links the theory to the most current recommendations from NIST and BSI. The book also enables readers to directly try out the results with existing tools available as open source. It is different from all existing books because it shows very concretely how to execute many procedures with different tools. The target group could be self-learners, pupils and students, but also developers and users in companies. All code written with these open-source tools is available. The appendix describes in detail how to use these tools. The main chapters are independent from one another. At the end of most chapters, you will find references and web links. The sections have been enriched with many footnotes. Within the footnotes you can see where the described functions can be called and tried within the different CrypTool versions, within SageMath or within OpenSSL.

Table of Contents:
Introduction   1 Ciphers and Attacks against Them 1.1 Importance of cryptology 1.2 Symmetric encryption 1.3 Asymmetric encryption 1.4 Hybrid procedures 1.5 Kerckhoffs’ principle 1.6 Key spaces – theoretical and practical view 1.7 Attack types and security definitions 1.8 Best known attacks on given ciphers 1.9 Algorithm types and self-made ciphers 1.10 Further references / Recommended books 1.11 AES visualizations/implementations 1.12 Educational examples for symmetric ciphers using SageMath   2 P&P and Pre-Computer Ciphers 2.1 Transposition ciphers 2.2 Substitution ciphers 2.3 Combining substitution and transposition 2.4 Further P&P methods (including new ones) 2.5 Hagelin machines as sample for pre-computer ciphers 2.6 Ciphers defined by ACA 2.7 Samples of open-access publications on cracking classical ciphers 2.8 Examples using SageMath 3 Historical Cryptology 3.1 Introduction 3.2 Analyzing historical ciphers – from collection to interpretation 3.3 Collection of manuscripts and creation of metadata 3.4 Transcription 3.5 Cryptanalysis 3.6 Contextualization and interpretation: Historical and philological analysis 3.7 Conclusion   4 Prime Numbers 4.1 What are prime numbers? 4.2 Prime numbers in mathematics 4.3 How many prime numbers are there? 4.4 The search for extremely large primes 4.5 Prime number tests 4.6 Special types of numbers and the search for a formula for primes 4.7 Density and distribution of the primes 4.8 Outlook 4.9 Notes about primes 4.10 Number of prime numbers in various intervals 4.11 Indexing prime numbers (n-th prime number) 4.12 Orders of magnitude / dimensions in reality 4.13 Special values of the binary and decimal system 4.14 Visualization of the quantity of primes in higher ranges 4.15 Examples using SageMath   5 Introduction to Elementary Number Theory with Examples 5.1 Mathematics and cryptography 5.2 Introduction to number theory 5.3 Prime numbers and the first fundamental theorem of elementary number theory 5.4 Divisibility, modulus and remainder classes 5.5 Calculations with finite sets 5.6 Examples of modular calculations 5.7 Groups and modular arithmetic in Zn and Z 5.8 Euler function, Fermat’s little theorem and Euler-Fermat 5.9 Multiplicative order and primitive roots 5.10 Proof of the RSA procedure with Euler-Fermat 5.11 Security aspects regarding the security of practical RSA implementations 5.12 Considerations regarding the security of the RSA algorithm 5.13 Applications of asymmetric cryptography using numerical examples 5.14 The RSA procedure with actual numbers 5.15 Didactic comments on modulo subtraction 5.16 Base representation of numbers, estimation of length of digits 5.17 Examples using SageMath   6 The Mathematical Ideas behind Modern (Asymmetric) Cryptography 6.1 One way functions with trapdoor and complexity classes 6.2 Knapsack problem as a basis for public-key procedures 6.3 Decomposition into prime factors as a basis for public-key procedures 6.4 The discrete logarithm as basis for public-key procedures 6.5 The RSA plane 6.6 Outlook   7 Hash Functions, Digital Signatures, and PKIs 7.1 Hash functions 7.2 Digital signatures 7.3 RSA signatures 7.4 DSA signatures 7.5 Public-key certification   8 Elliptic-Curve Cryptography (ECC) 8.1 Elliptic-curve cryptography – a high-performance substitute for RSA? 8.2 Elliptic curves – history 8.3 Elliptic curves – mathematical basics 8.4 Elliptic curves in cryptography 8.5 Operating on the elliptic curve 8.6 Security of elliptic-curve cryptography: the ECDLP 8.7 Encryption and signing with elliptic curves 8.8 Factorization using elliptic curves 8.9 Implementing elliptic curves for educational purposes 8.10 Patent aspects 8.11 Elliptic curves in use   9 Foundations of Modern Symmetric Encryption 9.1 Boolean functions 9.2 Block ciphers 9.3 Stream ciphers 9.4 Table of SageMath examples in this chapter   10 Homomorphic Ciphers 10.1 Origin of the term “homomorphic” 10.2 Decryption function is a homomorphism 10.3 Classification of homomorphic methods 10.4 Examples of homomorphic pre-FHE ciphers 10.5 Applications 10.6 Homomorphic methods in CrypTool   11 Lightweight Introduction to Lattices 11.1 Preliminaries 11.2 Equations 11.3 Systems of linear equations 11.4 Matrices 11.5 Vectors 11.6 Equations – revisited 11.7 Vector spaces 11.8 Lattices 11.9 Lattices and RSA 11.10 Lattice basis reduction 11.11 PQC standardization   12 Solving Discrete Logarithms and Factoring 12.1 Generic algorithms for the discrete logarithm problem in any group 12.2 Best algorithms for prime fields Fp 12.3 Best known algorithms for extension fields Fpn and recent advances 12.4 Best known algorithms for factoring integers 12.5 Best known algorithms for elliptic curves 12.6 Possibility of embedded backdoors in cryptographic keys 12.7 Conclusion: Advice for cryptographic infrastructure   13 Future Use of Cryptography 13.1 Widely used schemes 13.2 Preparing for tomorrow 13.3 New mathematical problems 13.4 New signatures 13.5 Quantum cryptography – a way out of the dead end? 13.6 Post-Quantum Cryptography (PQC) 13.7 Conclusion