Book Image

TLS Cryptography In-Depth

By : Dr. Paul Duplys, Dr. Roland Schmitz
Book Image

TLS Cryptography In-Depth

By: Dr. Paul Duplys, Dr. Roland Schmitz

Overview of this book

TLS is the most widely used cryptographic protocol today, enabling e-commerce, online banking, and secure online communication. Written by Dr. Paul Duplys, Security, Privacy & Safety Research Lead at Bosch, and Dr. Roland Schmitz, Internet Security Professor at Stuttgart Media University, this book will help you gain a deep understanding of how and why TLS works, how past attacks on TLS were possible, and how vulnerabilities that enabled them were addressed in the latest TLS version 1.3. By exploring the inner workings of TLS, you’ll be able to configure it and use it more securely. Starting with the basic concepts, you’ll be led step by step through the world of modern cryptography, guided by the TLS protocol. As you advance, you’ll be learning about the necessary mathematical concepts from scratch. Topics such as public-key cryptography based on elliptic curves will be explained with a view on real-world applications in TLS. With easy-to-understand concepts, you’ll find out how secret keys are generated and exchanged in TLS, and how they are used to creating a secure channel between a client and a server. By the end of this book, you’ll have the knowledge to configure TLS servers securely. Moreover, you’ll have gained a deep knowledge of the cryptographic primitives that make up TLS.
Table of Contents (30 chapters)
1
Part I Getting Started
8
Part II Shaking Hands
16
Part III Off the Record
22
Part IV Bleeding Hearts and Biting Poodles
27
Bibliography
28
Index

7.2 Groups

Groups are the most basic mathematical structure in which public-key cryptography can take place. So, let’s plunge right into the math and explain the properties of a so-called abelian group.

Let M be a nonempty set, and let be an operation on M, which means is a function M × M M, which maps pairs of elements of M to elements of M. The pair (M,⋆) is called an abelian group 𝔾 if the following properties hold:

  • (G1) is an associative operation that means for all cases of a,b, and c M, we have


    a ⋆(b⋆ c) = (a ⋆b)⋆ c.
  • (G2) In M exists a neutral element e with the property that for all cases of a M,


    a ⋆ e = e ⋆a = a.
  • (G3) For all cases of a M exists an inverse element a−1M with the property a−1 ⋆ a = a ⋆ a−1 = e.

  • (G4) is a commutative operation, which means for all cases of a,b M, we have


    a ⋆ b = b ⋆a.

The number of elements of M is called the order of the group 𝔾...