Why are ECC keys smaller than RSA?

The best known algorithm for the elliptic curve discrete logarithm problem (ECDLP) is O(sqrt(n)) — square root of the group order. For RSA, the best algorithm (GNFS) is sub-exponential. So ECDLP is harder relative to key size: 256-bit ECC ≈ 3072-bit RSA, 384-bit ECC ≈ 7680-bit RSA.

Which curve should I use?

Curve25519 (Daniel J. Bernstein) for ECDH — fast, constant-time, no bad parameters. P-256 (NIST) for compatibility — widely supported in TLS and standards. secp256k1 for Bitcoin/Ethereum. Avoid curves with unexplained parameters (potential NSA backdoors). Use standardized, well-analyzed curves.

TASK

Use elliptic curves for cryptography

Elliptic curve cryptography (ECC) provides the same security as RSA with much smaller keys. Learn point operations and key exchange.

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An elliptic curve y^2 = x^3 + ax + b over a finite field forms a group under point addition. The "discrete logarithm problem" on elliptic curves (given P and kP, find k) is harder than factoring or DLP in Z_p — so ECC uses smaller keys for equivalent security. P-256 (256-bit) ≈ RSA 3072-bit. Curve25519 is popular for ECDH.
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Modular Exponentiation →
While ECC uses point multiplication (not modular exponentiation), the underlying field operations use modular arithmetic. This tool demonstrates the modular arithmetic used in ECC field operations. Point multiplication is repeated point addition, analogous to modular exponentiation being repeated multiplication.
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Prime Checker →
Verify the prime modulus of the elliptic curve field. Standard curves (P-256, Curve25519) use carefully chosen primes. The prime must be large (256+ bits) and the curve must have a large prime order subgroup to be secure.

Frequently asked questions

Why are ECC keys smaller than RSA?: The best known algorithm for the elliptic curve discrete logarithm problem (ECDLP) is O(sqrt(n)) — square root of the group order. For RSA, the best algorithm (GNFS) is sub-exponential. So ECDLP is harder relative to key size: 256-bit ECC ≈ 3072-bit RSA, 384-bit ECC ≈ 7680-bit RSA.
Which curve should I use?: Curve25519 (Daniel J. Bernstein) for ECDH — fast, constant-time, no bad parameters. P-256 (NIST) for compatibility — widely supported in TLS and standards. secp256k1 for Bitcoin/Ethereum. Avoid curves with unexplained parameters (potential NSA backdoors). Use standardized, well-analyzed curves.