What does a digital signature check establish?

A digital signature lets a receiver test a relationship between a message, a signature and a public verification key. A separate private key is used to sign. In a secure system, someone who lacks the private key should not be able to produce valid signatures for new messages merely by knowing the public key.

The small arithmetic example makes that relationship visible but provides no practical security. Its message is the number 2, its modulus is 33, its signing exponent is 7, and its public exponent is 3. The notation mod 33 means the remainder after division by 33. Signing computes 2⁷ mod 33. Since 2⁷ = 128 and 128 = 3 × 33 + 29, the signature is 29. The receiver checks 29³ mod 33. Here 29³ = 24,389 = 739 × 33 + 2, so the check produces 2. That matches the received message number in the first comparison.

Now keep the signature 29 but replace the received message with 3. The verification calculation still produces 2. Because 2 differs from 3, the comparison fails. The signature is linked to particular content, not simply a decorative label that remains valid beside any replacement message.

Real signature schemes do much more than this tiny calculation. They specify message processing, cryptographic hashing, encoding, secure parameter sizes and precise validation rules. The private key must remain protected. The exposed exponent and easily factored modulus here would make a usable security system fail; their purpose is to reveal arithmetic structure. A successful check also needs a trustworthy connection between the public key and the intended signer. If an attacker supplies both a message and an unrelated public key, a mathematical match alone does not establish whose key it is. Certificates and other verified key-distribution arrangements address that separate problem.

Finally, signing is not encryption. A signature need not hide the message, and an authentic signer can still make a false claim. Signature verification provides evidence about integrity and the signing key under the scheme’s assumptions. It does not decide whether the statement is true. Keeping these conclusions separate is as important as performing the arithmetic correctly.