ECC provides the same security as RSA with 12x smaller keys. A 256-bit ECC key = 3,072-bit RSA key. Smaller transactions, faster verification, perfect for blockchain.

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Elliptic Curves

Equation: y² = x³ + ax + b. Bitcoin uses secp256k1: y² = x³ + 7. Three operations: Point Addition, Point Doubling, Scalar Multiplication.

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Trapdoor Function

Computing Q = kP is easy (~256 operations). Finding k from Q is hard (~2^128 operations = billions of years). This asymmetry makes ECC secure!

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ECDSA Signatures

Digital signatures prove message authenticity without revealing private keys. Never reuse nonce k - this is how PlayStation 3 was hacked!

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Bitcoin Implementation

Uses secp256k1 curve with SHA-256 + RIPEMD-160 hashing. Addresses start with 1 or 3, use Base58 encoding. Compressed public keys save space.

Ethereum Implementation

Same secp256k1 curve, but uses Keccak-256 hashing. Addresses start with 0x, use hexadecimal format. Uncompressed public keys.

🚨 Critical Security Rules

1.Never reuse nonce k - Different message needs different k, or private key leaks

2.Use cryptographically secure RNG - Weak randomness = predictable keys

3.Never share private keys - Anyone with your private key can steal everything

4.Encrypt keys at rest - Hardware wallets keep keys in secure enclaves

🎯 Real-World Applications

Cryptocurrencies

Bitcoin, Ethereum, and most blockchains use ECC for wallet security and transaction signing

HTTPS/TLS

Modern web encryption uses Curve25519 or P-256 for secure connections

Mobile Devices

iOS and Android use ECC for secure boot, app signing, and encryption

Messaging Apps

WhatsApp, Signal use ECC (Curve25519) for end-to-end encryption