pkcs_1_sha 256_with_rsa_encryption - dwilson2547/wiki_demo GitHub Wiki

PKCS #1 v1.5 with SHA-256 and RSA Encryption is a widely used digital signature scheme defined in RFC 8017 (formerly PKCS #1). It combines the SHA-256 hash function with the RSA public-key cryptosystem to provide message authentication, integrity, and non-repudiation.

Let’s break down all the parts of this algorithm in detail:

1. Overview of PKCS #1 v1.5 with SHA-256 and RSA

This scheme is used to sign messages using RSA and verify those signatures. The process involves:

  • Hashing the message with SHA-256.
  • Encoding the hash and other metadata into a structured format (ASN.1).
  • Signing the encoded data using the RSA private key.
  • Verifying the signature using the RSA public key.

2. Key Components

A. RSA Key Pair

  • Private Key (( d )): A large integer used to sign messages.
  • Public Key (( (e, n) )):
    • ( e ): Public exponent (usually 65537).
    • ( n ): Modulus, the product of two large primes ( p ) and ( q ).

B. SHA-256 Hash Function

  • Converts the message into a fixed 256-bit (32-byte) hash.
  • Ensures message integrity and provides a fixed-length input for RSA.

C. ASN.1 Encoding

  • The hash and metadata are encoded using ASN.1 (Abstract Syntax Notation One) to form a structured block.
  • This block is padded and formatted according to PKCS #1 v1.5 before signing.

3. Signing Process

Step 1: Hash the Message

  • Compute the SHA-256 hash of the message: [ h = \text{SHA-256}(m) ]
  • The hash ( h ) is a 32-byte value.

Step 2: Encode the Hash

  • The hash ( h ) is embedded into an ASN.1 DER-encoded structure called DigestInfo:

    DigestInfo ::= SEQUENCE {
    digestAlgorithm AlgorithmIdentifier,
    digest OCTET STRING
}
    • digestAlgorithm specifies the hash algorithm (SHA-256).
    • digest is the 32-byte SHA-256 hash.

  • The DigestInfo structure is encoded as a DER (Distinguished Encoding Rules) byte string.

Step 3: Pad the Encoded Hash

  • The DigestInfo byte string is padded using PKCS #1 v1.5 padding:

    0x00 || 0x01 || 0xFF || ... || 0xFF || 0x00 || ASN.1 DER-encoded DigestInfo
    • 0x00: Ensures the result is a positive integer.
    • 0x01: Block type (1 for signature).
    • 0xFF: Padding bytes (at least 8 bytes).
    • 0x00: Separator between padding and DigestInfo.
    • ASN.1 DER-encoded DigestInfo: The encoded hash.

  • The total length of the padded block must match the RSA modulus size (e.g., 256 bytes for a 2048-bit RSA key).

Step 4: Sign the Padded Block

  • The padded block is treated as a large integer and signed using the RSA private key: [ s = m^d \mod n ] where:

    • ( m ) is the padded block as an integer.
    • ( d ) is the private exponent.
    • ( n ) is the RSA modulus.

  • The result ( s ) is the signature.

4. Verification Process

Step 1: Hash the Message

  • The verifier computes the SHA-256 hash of the received message: [ h' = \text{SHA-256}(m) ]

Step 2: Encode the Hash

  • The verifier encodes ( h' ) into the same DigestInfo structure and DER-encodes it.

Step 3: Decrypt the Signature

  • The signature ( s ) is decrypted using the RSA public key: [ m' = s^e \mod n ] where:

    • ( e ) is the public exponent.
    • ( n ) is the RSA modulus.

  • ( m' ) is the recovered padded block.

Step 4: Verify the Padding and Hash

  • The verifier checks that the recovered padded block ( m' ) has the correct PKCS #1 v1.5 padding format.
  • The verifier extracts the DigestInfo from ( m' ) and compares it to the DigestInfo computed from the received message.
  • If they match, the signature is valid.

5. Security Considerations

A. PKCS #1 v1.5 Padding

  • The padding scheme is deterministic and must be strictly followed.
  • Incorrect padding can lead to security vulnerabilities (e.g., Bleichenbacher attacks).

B. Hash Function Security

  • SHA-256 is currently considered secure, but weaker hash functions (e.g., SHA-1) are vulnerable to collision attacks.

C. RSA Key Size

  • RSA keys should be at least 2048 bits for modern security. Smaller keys (e.g., 1024 bits) are considered insecure.

D. Side-Channel Attacks

  • Implementations must be constant-time to prevent timing attacks.

6. Example Workflow

Let’s say Alice wants to sign a message ( m = )"Hello, Bob!":

  1. Hashing:

    • Alice computes ( h = \text{SHA-256}("Hello, Bob!") ).

  2. Encoding:

    • Alice encodes ( h ) into DigestInfo and DER-encodes it.

  3. Padding:

    • Alice pads the DigestInfo using PKCS #1 v1.5 padding.

  4. Signing:

    • Alice signs the padded block using her RSA private key to produce the signature ( s ).

  5. Verification:

    • Bob receives ( m ) and ( s ).
    • Bob computes ( h' = \text{SHA-256}(m) ) and encodes it.
    • Bob decrypts ( s ) using Alice’s public key to recover the padded block.
    • Bob verifies the padding and compares the extracted DigestInfo with his computed DigestInfo.

7. Summary Table

Component Description
RSA Private Key Used to sign the padded block.
RSA Public Key Used to verify the signature.
SHA-256 Hashes the message to ensure integrity and fixed-length input.
ASN.1 DER Encoding Encodes the hash and metadata into a structured format.
PKCS #1 v1.5 Padding Pads the encoded hash to match the RSA modulus size.
Signature The result of RSA encryption of the padded block.

8. Comparison with Other Schemes

  • PSS (Probabilistic Signature Scheme): A more modern alternative to PKCS #1 v1.5 padding, offering better security guarantees.
  • ECDSA: Uses elliptic curve cryptography instead of RSA, providing equivalent security with smaller key sizes.

Would you like to explore the ASN.1 encoding or PSS padding in more detail, or compare this scheme with others like ECDSA?

⚠️ **GitHub.com Fallback** ⚠️