Digital Signature

A digital signature is the cryptographic counterpart of a handwritten signature — but far stronger, because it is mathematically bound to the exact document it signs. It proves three things at once:

• Authenticity — the message really came from the claimed sender.
• Integrity — not one bit has been altered since signing.
• Non-repudiation — the signer cannot later deny having signed it.

It is Public-Key Cryptography run in reverse. Encryption uses the public key to lock and the private key to unlock. Signing swaps the roles: the private key signs, and the matching public key verifies. Only the key's owner can produce a signature, but anyone can check it.

How signing works

You never sign the whole message directly — that would be slow and unwieldy. Instead you sign its hash, a short fingerprint of the document.

Because the signature is tied to H(m), the avalanche effect does the heavy lifting: change a single character of m and its hash changes completely, so the recomputed h no longer matches h' and verification fails. The signature guards every bit of the document.

Why the hash matters

Hashing first is not just an optimization — it is what makes signatures safe and practical.

• Speed. Public-key operations are expensive; signing 32 bytes of digest instead of a gigabyte file is enormously faster.
• Integrity coverage. A good Hash Function makes it infeasible to find a different message with the same digest, so an attacker cannot lift your signature onto forged content.
• Identity binding. Verification ties the message to a specific public key; certificate authorities then vouch that the key belongs to a real person or server.

This is the missing piece for Diffie–Hellman, which agrees a shared key but cannot by itself confirm who is on the other end. Signing the exchange defeats the man-in-the-middle and is exactly how TLS authenticates the websites you visit — the padlock in your browser is a verified digital signature.

See also

• Public-Key Cryptography
• Hash Function
• RSA