Introduction

Cryptography is not “Secret Writing”. It is Mathematical Warfare. It allows a lone individual to hide a secret that no government, army, or supercomputer can uncover.

It relies on Hard Problems: Math that is trivial to compute one way, but impossible to reverse without a “Trapdoor”. This lesson explores the Physics of the Impossible.

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The Physics: Symmetric Encryption (AES-256)

AES is the standard for locking data. It Shuffles, Substitutes, and Mixes bytes in 14 “Rounds”.

The Physics of Brute Force: To crack AES-256, you must check $2^{256}$ keys. The Landauer Limit states the minimum energy to flip 1 bit is $k T \ln 2$. To count to $2^{256}$, you would need to harness the energy of a Dyson Sphere around the Sun for 3 billion years. AES is not just secure. It is Thermodynamically Secure.

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Deep Dive: Asymmetric Encryption (RSA vs ECC)

Symmetric keys are great, but how do I send you the key? Public Key Cryptography.

RSA (The Factoring Problem):

• Easy: $13 \times 17 = 221$.
• Hard: What are the factors of $239847129...?$
• Key Size: Requires 3072 bits to be secure. Slow.

ECC (Elliptic Curve Cryptography):

• Easy: $P + P + P... = Q$.
• Hard: Given $Q$, how many times did I add $P$? (Discrete Log).
• Key Size: Only 256 bits for same security. 1000x faster.
• Physics: Used in Bitcoin, TLS 1.3, Signal.

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Strategy: Diffie-Hellman (Paint Mixing)

How do two people agree on a secret color in public without revealing it?

1. Public: Yellow Paint.
2. Alice: Adds Secret Red. Sends Orange Mixture.
3. Bob: Adds Secret Blue. Sends Green Mixture.
4. Alice: Adds Secret Red to Bob’s Green -> BROWN.
5. Bob: Adds Secret Blue to Alice’s Orange -> BROWN.

Result: Both have the same Shared Secret (Brown). An eavesdropper sees Orange and Green but cannot separate the colors to find the secret.

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Code: ECDSA Signing

import ecdsa
import hashlib

# 1. Generate Key Pair (Curve secp256k1)
sk = ecdsa.SigningKey.generate(curve=ecdsa.SECP256k1)
vk = sk.verifying_key

# 2. Sign a Message
message = b"Attack at Dawn"
signature = sk.sign(message)

# 3. Verify the Signature
try:
    assert vk.verify(signature, message)
    print("Signature Valid!")
except ecdsa.BadSignatureError:
    print("WARNING: Forged Signature")

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Practice Exercises

Exercise 1: Hash Collision (Beginner)

Task: Find two strings that produce the same MD5 hash. Result: Easy. You can do it in seconds on a laptop. MD5 is broken. Task: Do it for SHA-256. Result: Impossible.

Exercise 2: RSA Key Size (Intermediate)

Scenario: You use 1024-bit RSA. Risk: This can be factored by a Nation State. You must use 2048 or 3072.

Exercise 3: Quantum Threat (Advanced)

Scenario: Shor’s Algorithm runs on a Quantum Computer with 4000 Qubits. Result: It solves Factoring and Discrete Logs instantly. RSA and ECC are dead. AES-256 survives (only weakened to AES-128 via Grover’s Algo).

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Knowledge Check

1. Why is AES-256 considered thermodynamically secure?
2. What is the Hard Problem behind RSA?
3. Why do we prefer ECC over RSA today?
4. What does Diffie-Hellman achieve?
5. Are Hashes reversible?

Answers

1. Energy limits. There isn’t enough energy in the solar system to check all keys.
2. Integer Factoring. Finding prime factors of a huge number.
3. Efficiency. Smaller keys, faster computation, same security.
4. Key Exchange. Shared secret over an insecure channel.
5. No. They are lossy compression (Pigeonhole Principle).

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Summary

• Symmetric: Fast, Unbreakable.
• Asymmetric: Solves Key Exchange (Slow).
• Hashing: Digital Fingerprint.

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