# A Map to the Mathematical Wonderland: Demystifying Cryptography Bryant Nielson | October 18, 2023

Blockchain owes its cryptographic security to mathematic magic that may seem puzzling. However, peer behind the curtain and you find concrete mathematical foundations underpinning its wonders. This guide serves as a map explaining how computationally hard math problems enable the asymmetric cryptography securing transactions.

The one-way functions underlying cryptography rely on trapdoor problems – calculations easy in one direction but difficult reversing. Multiplication of two large prime numbers creates immense products trivial to multiply but astronomically difficult to factor back into the original primes. This factoring difficulty powers RSA encryption.

Discrete logarithms also pose a one-way trapdoor underpinning Diffie-Hellman key exchange. Taking a number to a power is simple. But deducing the original number given only the exponent result proves virtually impossible at large scales. Computational intractability makes brute forcing the discrete log infeasible.

Elliptic curve cryptography depends on a different hard problem – deriving a point’s discrete logarithm on a twisty, complex elliptic curve. The mechanics behind elliptic curve points form a cryptographic wormhole. Determining a random point’s scalar multiplier is an intractable puzzle.

These mathematically hard problems share qualities that make them ideal cryptographic foundations. They rely on mathematical conjectures unproven but widely accepted as true. No efficient solving algorithm exists, forcing adversaries to rely on brute computational power. Adding complexity rapidly outpaces solving speed. Even quantum algorithms offer little advantage.

But how do math puzzles enable key generation? The secret lies in asymmetric key pairs. Take RSA, which uses randomly generated prime numbers as secret keys, multiplying them to form a public modulus. Reversing this multiplication and factoring the public modulus back to discover the private primes is considered intractable. The private key holder retains the prime factors needed to decrypt while the public key enables encryption.

Elliptic curve cryptography relies on similar principles but based on the complex multiplication of elliptic curve points plotted on a graph. Multiplying a random private key scalar with the curve’s base point produces a seemingly random public point. Reversing this to deduce the private scalar from the public point poses a hard problem.

Together, asymmetric key pairs allow creating mathematical puzzles only the holder of the private key can solve. Communications can be securely transmitted using the public key, but decrypting the contents requires the difficulty of the hard problem only the private key overcomes.

This leaves the secrecy of private keys paramount. Compromising the trapdoor secrets circumvents cryptography’s mathematical magic. Thankfully, cryptographers have honed key generation, encryption protocols and ciphers to minimize this risk over decades of research.

While the math underlying cryptography appears complex, it rests on understandable principles of hard problems. One-way functions based on factoring, discrete logs, and elliptic curves underlie rock-solid asymmetric encryption. Though the specifics take study to master, you can take comfort blockchain’s security relies on proven mathematical foundations. The magic dissolving these traps requires control of private keys, leaving messages securely scrambled for everyone else without the secrets. 