Digital Signatures Fundamentals

Digital signatures leverage cryptography to ensure the authenticity, integrity, and non-repudiation of digital messages or documents. Acting as electronic fingerprints, they confirm the sender’s identity and protect against tampering. This MathMultiverse guide explores their mechanics, RSA-based examples, security features, and real-world applications.

How Digital Signatures Work

Digital signatures rely on asymmetric cryptography, such as RSA:

  1. Hashing: The message \( M \) is hashed: \( H = \text{hash}(M) \).
  2. Signing: The hash is encrypted with the sender’s private key: \( S = H^d \mod n \).
  3. Verification: The receiver decrypts with the sender’s public key: \( H' = S^e \mod n \), and checks \( H' = \text{hash}(M) \).
\[ S = H^d \mod n \] \[ H' = S^e \mod n \]

The signature ensures the message is from the claimed sender and unaltered.

RSA Signature Example

For RSA with \( p = 3 \), \( q = 11 \), \( n = 33 \), public key \( e = 7 \), private key \( d = 3 \), message \( M = 5 \):

\[ H = 5 \text{ (simplified hash)} \] \[ S = 5^3 \mod 33 = 125 \mod 33 = 26 \] \[ H' = 26^7 \mod 33 = 5 \]

Verification succeeds as \( H' = H \). Modern systems use SHA-256 for robust hashing.

Security Features

  • Authenticity: Only the private key holder can generate a valid signature.
  • Integrity: Any alteration invalidates the hash comparison.
  • Non-repudiation: The sender cannot deny signing the message.

Security hinges on strong key management and robust hash functions.

Applications

Digital signatures are critical in:

  • E-Contracts: Legally binding digital agreements.
  • Software Updates: Ensuring trusted sources (e.g., Microsoft, Apple).
  • Secure Email: PGP and S/MIME for encrypted communication.
  • Blockchain: Verifying transactions in cryptocurrencies.