A large, correct key only secures the math. Real systems still fall through side-channels, badly-protected keys, software (rather than hardware) key storage, a weak first key exchange, broken authentication protocols, and attacks that never touch the wire at all.

A 256-bit AES key or a 3072-bit RSA key makes brute force hopeless — but attackers rarely fight the math. Here is how a "strong-key" system still gets broken:

• Side-channel attacks — timing, power draw or electromagnetic leakage reveal the key around the cipher, without breaking it.
• Poorly protected keys — a 2048-bit RSA private key guarded by a 6-digit password is only as strong as those 6 digits.
• Keys/algorithms in software, not a security module — if the crypto runs in the PC's memory instead of dedicated hardware (HSM, smartcard), the key can be read out.
• A weak initial key exchange — "trust on first use" (TOFU) or plain FTP lets an attacker sit in the middle during setup, before any strong key is agreed.
• Weak or wrong authentication protocols — if you can't be sure who holds the other key, its size is irrelevant.
• Attacks off the communication path — the classic Alice–Bob model only considers on-the-wire attacks; the endpoint, the user, or the supply chain are often the real target.

A sharper version of the same point: the size of a brute-force attack says nothing about an algorithm's security. 104-bit RC4 is brute-force-proof, yet it can be broken on home equipment because the algorithm is weak. Security is the weakest link in the whole chain — never just the biggest number in it.

Go deeper:

• Side-channel attack (Wikipedia) — how timing, power, and EM leakage recover a key without ever breaking the cipher.

AES 128+ bits, RSA/DH 3072+ bits, ECC 256+ bits, hash-for-signatures 256+ bits — all chosen to give roughly a 128-bit security level (the point where brute force is physically hopeless).

* For equal security, RSA/DH keys grow far faster than ECC keys (log scale). *

The BSI (Germany's Federal Office for Information Security) publishes yearly guidance in Technical Guideline TR-02102-1; these are its current recommendations. The unifying idea is that every row is dimensioned to the same ~128-bit security level — so the numbers look wildly different only because each problem (brute force vs. factoring vs. discrete log) resists attack differently.

The hash rule has a subtle floor: since 2023 the minimum hash size is 240 bits, which knocks out 224-bit hashes (and 224-bit ECC, which can only sign a 224-bit hash) — so 256 bits is effectively the practical minimum.

Equivalence table:

Key observation: Doubling the symmetric key size (128 → 256-bit security) takes RSA from 3072 → 15360 bits (≈5×) but ECC only from 256 → 512 bits (2×). This is why ECC scales much better.

Go deeper:

• keylength.com — compare key-size recommendations — pick a target year and see every standards body's recommended sizes side by side.
• Key size (Wikipedia) — the security-level equivalence between symmetric, RSA and ECC keys.