Public Key Cryptography

In this chapter, you will be introduced to the concepts and practical aspects of public key cryptography, also called asymmetric cryptography or asymmetric key cryptography. We will continue to use OpenSSL, as we did in the previous chapter, to experiment with some applications of cryptographic algorithms so that you can gain hands-on experience. We will start with the theoretical foundations of public key cryptography and will gradually build on the concepts with relevant practical exercises. After this, we will introduce some new and advanced cryptography constructs.

Before discussing cryptography further, some mathematical terms and concepts need to be explained in order to build a foundation for fully understanding the material provided later in this chapter. The next section serves as a basic introduction to these concepts. An explanation with proofs and relevant background for all of these terms would require somewhat complex mathematics, which is...

Mathematics

As the subject of cryptography is based on mathematics, this section will introduce some basic concepts that will help you understand the concepts presented later.

Modular arithmetic

Also known as clock arithmetic, numbers in modular arithmetic wrap around when they reach a certain fixed number. This fixed number is a positive number called modulus (sometimes abbreviated to mod), and all operations are performed concerning this fixed number.

Modular arithmetic is analogous to a 12-hour clock; there are numbers from 1 to 12. When 12 is reached, the numbers start from 1 again. Imagine that the time is 9:00 now; 4 hours from now, it will be 1:00 because the numbers wrap around at 12 and start from 1 again. In normal addition, this would be 9 + 4 = 13, but that is not the case on a 12-hour clock; it is 1:00.

In other words, this type of arithmetic deals with the remainders after the division operation. For example, 50 mod 11 is 6 because 50 / 11 leaves a remainder...

Asymmetric cryptography

Asymmetric cryptography refers to a type of cryptography where the key that is used to encrypt the data is different from the key that is used to decrypt the data. These keys are called private and public keys, respectively, which why asymmetric cryptography is also known as public key cryptography. It uses both public and private keys to encrypt and decrypt data, respectively. Various asymmetric cryptography schemes are in use, including RSA (named after its founders, Rivest, Shamir, and Adelman), DSA (Digital Signature Algorithm), and ElGamal encryption.

An overview of public-key cryptography is shown in the following diagram:

Figure 4.1: Encryption/decryption using public/private keys

The preceding diagram illustrates how a sender encrypts data P using the recipient's public key and encryption function E, and produces an output encrypted data C, which is then transmitted over the network to the receiver. Once it reaches the receiver...

Cryptographic constructs and blockchain technology

Now, we'll present some advanced topics in cryptography that not only are important on their own, but are also relevant to blockchain technology due to their various applications in this space.

Homomorphic encryption

Usually, public key cryptosystems, such as RSA, are multiplicative homomorphic or additive homomorphic, such as the Paillier cryptosystem, and are called partially homomorphic systems. Additive Partially Homomorphic Encryptions (PHEs) are suitable for e-voting and banking applications.

Until recently, there has been no system that supported both operations, but in 2009, a fully homomorphic system was discovered by Craig Gentry. As these schemes enable the processing of encrypted data without the need for decryption, they have many different potential applications, especially in scenarios where maintaining privacy is required, but data is also mandated to be processed by potentially untrusted parties,...

Summary

This chapter started with an introduction to some basic mathematics concepts and asymmetric key cryptography. We discussed various constructs such as RSA and ECC. We also performed some experiments using OpenSSL to see how theoretical concepts can be implemented practically. After this, hash functions were discussed in detail, along with their properties and usage. Next, we covered concepts, such as Merkle trees, that are used extensively in blockchains and in fact are at its core. Other concepts such as Patricia trees and hash tables were also introduced. We also looked at some advanced and modern concepts such as zero-knowledge proofs and relevant constructions, along with different types of digital signatures.

In the next chapter, we will explore the captivating world of distributed consensus, which is central to the integrity of any blockchain and is a very hot area of research.