# A Cryptographer’s Guide: Demystifying Keys and Cryptography

Bryant Nielson | November 1, 2023

Cryptography plays a crucial role in blockchain technology, yet the mathematical principles involved can seem esoteric. By decoding key concepts in public key cryptography and cryptographic hash functions, professionals can gain insight into how mathematics enables blockchain security.

Public key cryptography relies on asymmetric key pairs made up of a private key and a public key. These keys are mathematically related and used for different purposes. A user’s private key allows them to digitally sign transactions or messages proving authenticity and integrity. The corresponding public key is used to verify the signature. However, it cannot be used to derive the private key. This one-way relationship enables secure communication.

The keys are generated using mathematical one-way functions that are easy to compute in one direction but practically impossible to reverse. Multiplication of two large prime numbers is a common method. While easy to perform, factoring the product back into the two primes is extremely difficult. Computational hardness assumptions around prime factorization and discrete logarithms underpin the security of schemes like RSA and Elliptic Curve.

Blockchains utilize digital signatures based on public key cryptography to authenticate transactions. Users sign transactions with their private key. Network nodes verify the signatures using the corresponding public keys. Valid signatures prove the transactions originate from addresses owning the associated key pairs. This prevents forgery and tampering.

Public keys also enable user addresses on blockchains. A blockchain address is derived by hashing a public key and encoding the result. Users can share addresses publicly to transact without revealing their private keys. Addresses provide pseudonymity by being unrelated to user identities.

In addition to digital signatures, cryptographic hash functions are integral to blockchain security. Hash functions take an input and generate a fixed-size output known as a hash value or digest. They exhibit useful security properties:

1. One-wayness – infeasible to determine the input from a hash value
2. Determinism – same input always produces the same hash output
3. Collision resistance – impractical to find inputs with the same hash value

These attributes enable hash functions to operate as fingerprinting mechanisms. Minute differences in inputs lead to wildly different hashes. Blockchains apply hashes extensively, including in digital signatures, address generation, and linking blocks through chains.

For digital signatures, public keys are hashed before encoding into addresses. One-way hashing prevents reconstructing private keys from addresses. In proof-of-work consensuses, mining nodes race to find valid block hashes below a target threshold by iterating nonce values. The deterministic property ensures the same block generates the same hash.

Each new block contains the hash of the previous block. This creates an immutable chain since altering any block would invalidate subsequent hashes. Combining public key cryptography and hash functions enables blockchain capabilities from ownership representation to immutable recordkeeping.

While mathematically complex, public key cryptography and hash functions essentially rely on hard computational problems. Future mathematical or quantum breakthroughs could potentially undermine these hardness assumptions. Ongoing research on post-quantum cryptography aims to avert this risk.

By grasping the basics of keys, signatures, and hashes, professionals can better understand blockchain security foundations. Mathematics provides the crucial tools for trust minimization in decentralized networks. Demystifying cryptography is key to engineering robust blockchains and safely unlocking their potential across industries.