Crypto Transactions: Public and Private Keys

Public and Private Keys: Foundations of Cryptographic Security

Introduction

Public and private keys are fundamental components of asymmetric cryptography, which underpins the security and functionality of cryptocurrencies. This section explores the intricacies of public and private keys, their generation, their relationship, and their role in cryptographic operations. Through detailed explanations and illustrative examples, we will delve into the concepts of key pairs, encryption, digital signatures, and address generation.

Key Pairs: Public and Private Keys

Public Key

A public key is a cryptographic key that is freely shared and known to all participants in a cryptographic system. It is derived from a complex mathematical algorithm and acts as a one-way function. The public key can be used for encryption and verifying digital signatures. Example: In the RSA algorithm, the public key consists of a modulus (n) and an exponent (e), which are used to encrypt messages.

Private Key

A private key is a secret component of asymmetric cryptography known only to the key owner. It corresponds to the public key and enables decryption and the creation of digital signatures. Private keys must be kept secure and not disclosed to anyone else. Example: In RSA, the private key consists of the modulus (n) and an exponent (d), which is used for decryption and signing.

Key Generation

Public and private key pairs are generated using complex mathematical algorithms, typically based on number theory and prime numbers. The key generation process involves the following steps:

Prime Number Selection

The algorithm selects two large prime numbers, p and q, that are kept secret.

Key Calculation

The algorithm calculates the modulus (n = p * q) and the Euler's totient function (φ(n) = (p - 1) * (q - 1)).

Public Key Derivation

The algorithm selects a public exponent (e) that is coprime with φ(n). The public key is the combination of the modulus (n) and the public exponent (e).

Private Key Calculation

The algorithm computes the private exponent (d) using the extended Euclidean algorithm or other mathematical techniques. The private key is the combination of the modulus (n) and the private exponent (d).

Encryption and Decryption

Encryption

To encrypt a message using the public key, the sender converts the message into a numerical representation and applies a mathematical operation involving the public key. The result is a ciphertext that can only be decrypted by the corresponding private key. Example: In the RSA encryption scheme, the sender uses the recipient's public key to encrypt a message, resulting in a ciphertext that can only be decrypted by the corresponding private key.

Decryption

The recipient of an encrypted message uses their private key to decrypt the ciphertext and recover the original message. The private key's mathematical properties allow the decryption operation to reverse the encryption process. Example: Using the recipient's private key in the RSA scheme, the ciphertext is decrypted, revealing the original message.

Digital Signatures

Signature Creation

To create a digital signature, the signer applies a cryptographic operation to the message using their private key. This generates a unique signature that is specific to the message and the signer's private key. Example: In the ECDSA algorithm, the signer uses their private key to generate a digital signature for a given message.

Signature Verification

To verify the digital signature, the recipient applies a verification algorithm using the corresponding public key and the original message. If the verification process is successful, it confirms that the signature is valid and the message is authentic and unchanged. Example: In ECDSA, the recipient verifies the digital signature using the signer's public key and the original message.

Address Generation

Cryptocurrency addresses, such as Bitcoin addresses, are derived from public keys. They provide a convenient way to identify recipients and allow others to send funds securely. Address generation involves hashing the public key and applying additional transformations. Example: Bitcoin addresses are generated by hashing the public key using the SHA-256 and RIPEMD-160 algorithms, followed by encoding the hash with Base58.

Conclusion

Public and private keys form the foundation of asymmetric cryptography, providing secure encryption, decryption, and digital signatures. The generation of key pairs involves complex mathematical algorithms and prime number selection. Public keys are freely shared, while private keys must be kept confidential. Through encryption and decryption operations, messages can be securely transmitted between parties. Digital signatures verify the authenticity and integrity of digital documents. Address generation allows for secure identification and convenient transactional interactions in the world of cryptocurrencies. Understanding the concepts of public and private keys is crucial for ensuring the security and privacy of cryptographic systems and designing robust cryptocurrency architectures.

This article takes inspiration from a lesson found in 15.S12 at MIT.