x9.62_ecdsa_with_sha 256 - dwilson2547/wiki_demo GitHub Wiki
The X9.62 ECDSA (Elliptic Curve Digital Signature Algorithm) with SHA-256 is a widely used digital signature scheme that combines elliptic curve cryptography with the SHA-256 hash function. Here’s a detailed breakdown of its components and how they work together:
- EDCSA
- Role of SHA-256
ECDSA is a variant of the Digital Signature Algorithm (DSA) that uses elliptic curve cryptography (ECC) instead of modular arithmetic. It provides the same level of security as DSA but with smaller key sizes, making it more efficient for resource-constrained environments.
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Elliptic Curve Parameters:
- Curve Equation: ( y^2 = x^3 + ax + b ) (Weierstrass form)
- Base Point (G): A point on the curve that generates a subgroup of prime order ( n ).
- Order (n): The number of points in the subgroup generated by ( G ).
-
Private Key (( d )):
- A randomly chosen integer in the range ( [1, n-1] ).
-
Public Key (( Q )):
- Computed as ( Q = d \cdot G ), where ( \cdot ) denotes scalar multiplication on the elliptic curve.
The signing process involves the following steps:
-
Hash the Message:
- Compute the SHA-256 hash of the message: ( h = \text{SHA-256}(m) ).
- Convert the hash ( h ) to an integer ( e ) (leftmost bits of ( h ) are used if ( h ) is longer than the bit length of ( n )).
-
Generate a Random Nonce (( k )):
- Choose a random integer ( k ) in the range ( [1, n-1] ).
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Compute the Signature Components (( r ) and ( s )):
- Compute ( (x_1, y_1) = k \cdot G ).
- Let ( r = x_1 \mod n ). If ( r = 0 ), restart the process with a new ( k ).
- Compute ( s = k^{-1}(e + d \cdot r) \mod n ). If ( s = 0 ), restart the process with a new ( k ).
-
Signature Output:
- The signature is the pair ( (r, s) ).
The verification process ensures the signature is valid:
-
Hash the Message:
- Compute the SHA-256 hash of the message: ( h = \text{SHA-256}(m) ).
- Convert the hash ( h ) to an integer ( e ).
-
Check Signature Validity:
- Ensure ( r ) and ( s ) are in the range ( [1, n-1] ). If not, the signature is invalid.
-
Compute the Verification Value (( w )):
- Compute ( w = s^{-1} \mod n ).
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Compute Two Points on the Curve:
- Compute ( u_1 = e \cdot w \mod n ).
- Compute ( u_2 = r \cdot w \mod n ).
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Compute the Point ( (x_1, y_1) ):
- Compute ( (x_1, y_1) = u_1 \cdot G + u_2 \cdot Q ).
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Check the Signature:
- If ( x_1 \mod n = r ), the signature is valid. Otherwise, it is invalid.
- SHA-256 is a cryptographic hash function that produces a 256-bit (32-byte) hash value. It is part of the SHA-2 family and is widely used for its security and efficiency.
- In ECDSA, SHA-256 is used to hash the message before signing, ensuring the message's integrity and providing a fixed-length input for the signing algorithm.
- Random Nonce (( k )): The security of ECDSA relies on the randomness of ( k ). If ( k ) is reused or predictable, the private key can be compromised.
- Key Size: ECDSA with a 256-bit curve (e.g., secp256r1) provides security equivalent to RSA with a 3072-bit key.
- Side-Channel Attacks: Implementations must be constant-time to prevent timing attacks.
- X9.62: The standard that defines ECDSA, including key generation, signing, and verification.
- FIPS 186-4: The U.S. government standard that approves ECDSA for digital signatures.
- NIST SP 800-186: Provides guidelines for elliptic curve cryptography.
Let’s say Alice wants to sign a message ( m ):
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Key Generation:
- Alice generates a private key ( d ) and computes her public key ( Q = d \cdot G ).
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Signing:
- Alice computes ( h = \text{SHA-256}(m) ) and converts it to ( e ).
- She picks a random ( k ), computes ( (x_1, y_1) = k \cdot G ), and sets ( r = x_1 \mod n ).
- She computes ( s = k^{-1}(e + d \cdot r) \mod n ).
- The signature is ( (r, s) ).
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Verification:
- Bob receives ( m ), ( (r, s) ), and Alice’s public key ( Q ).
- Bob computes ( h = \text{SHA-256}(m) ) and converts it to ( e ).
- He computes ( w = s^{-1} \mod n ), ( u_1 = e \cdot w \mod n ), and ( u_2 = r \cdot w \mod n ).
- He computes ( (x_1, y_1) = u_1 \cdot G + u_2 \cdot Q ).
- If ( x_1 \mod n = r ), the signature is valid.
| Component | Description |
|---|---|
| Private Key | Random integer ( d ) in ( [1, n-1] ). |
| Public Key | ( Q = d \cdot G ). |
| Message Hash | SHA-256 hash of the message, converted to integer ( e ). |
| Nonce | Random integer ( k ) in ( [1, n-1] ). |
| Signature | Pair ( (r, s) ), where ( r = x_1 \mod n ) and ( s = k^{-1}(e + d \cdot r) \mod n ). |
| Verification | Check if ( x_1 \mod n = r ) for ( (x_1, y_1) = u_1 \cdot G + u_2 \cdot Q ). |
The SHA-256 hash function plays a critical role in the X9.62 ECDSA signature scheme, ensuring the integrity, security, and efficiency of the digital signature process. Let’s break down its role in detail:
- SHA-256 is a cryptographic hash function that is part of the SHA-2 (Secure Hash Algorithm 2) family, designed by the NSA and standardized by NIST.
- It takes an input message of any length and produces a fixed-size 256-bit (32-byte) hash value, often represented as a 64-character hexadecimal string.
- SHA-256 is designed to be deterministic (same input always produces the same output), fast to compute, and computationally infeasible to reverse (one-way function).
- Before signing, the message is hashed using SHA-256. This ensures that even a tiny change in the message will produce a completely different hash, making it impossible for an attacker to alter the message without invalidating the signature.
- Example: If Alice signs a message ( m ), and Bob receives ( m' ) (a tampered version), the hash of ( m' ) will not match the hash of ( m ), and the signature verification will fail.
- ECDSA operates on integers and requires the message to be represented as a number. SHA-256 converts any message (regardless of length) into a fixed 256-bit hash, which is then converted to an integer ( e ).
- This ensures that the ECDSA algorithm can process messages of any size uniformly.
- SHA-256 is designed to be collision-resistant, meaning it is computationally infeasible to find two different messages ( m_1 ) and ( m_2 ) that produce the same hash.
- This property is crucial for preventing attackers from creating a fraudulent message that hashes to the same value as a legitimate one, which could otherwise allow them to reuse a valid signature.
- SHA-256 is optimized for performance, making it suitable for resource-constrained environments (e.g., IoT devices, mobile applications).
- It strikes a balance between security and computational efficiency, which is essential for real-world applications of ECDSA.
- The message ( m ) is passed through the SHA-256 algorithm to produce a 256-bit hash: [ h = \text{SHA-256}(m) ]
- The hash ( h ) is then converted to an integer ( e ). If the hash is longer than the bit length of the curve order ( n ), only the leftmost bits are used to form ( e ).
- The integer ( e ) (derived from the hash) is used in the ECDSA signing process:
[
s = k^{-1}(e + d \cdot r) \mod n
]
where:
- ( k ) is a random nonce,
- ( d ) is the private key,
- ( r ) is derived from the x-coordinate of ( k \cdot G ).
- During verification, the same hash ( e ) is recomputed from the message ( m ).
- The verifier checks if the signature ( (r, s) ) is valid for the hash ( e ) and the public key ( Q ).
- SHA-256 is designed to be preimage-resistant, meaning it is computationally infeasible to reverse the hash function and recover the original message from its hash.
- This ensures that an attacker cannot derive the original message from the signed hash.
- SHA-256 is also second preimage-resistant, meaning it is computationally infeasible to find a second message that hashes to the same value as a given message.
- This prevents attackers from creating a fraudulent message that could be verified with an existing signature.
- While collisions (two different messages producing the same hash) are theoretically possible, SHA-256 is designed to make finding such collisions computationally infeasible with current technology.
- This property is critical for ensuring that signatures cannot be forged by finding two messages with the same hash.
Suppose Alice wants to sign a message ( m = )"Hello, Bob!":
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Hashing:
- Alice computes ( h = \text{SHA-256}("Hello, Bob!") ).
- The hash ( h ) is a 256-bit value, e.g.,
1a2b3c4d5e6f7890123456789abcdef0123456789abcdef0123456789abcdef0. - She converts ( h ) to an integer ( e ).
-
Signing:
- Alice uses ( e ), her private key ( d ), and a random nonce ( k ) to compute the signature ( (r, s) ).
-
Verification:
- Bob receives ( m ), ( (r, s) ), and Alice’s public key ( Q ).
- Bob recomputes ( h = \text{SHA-256}("Hello, Bob!") ) and converts it to ( e ).
- He verifies the signature using ( e ), ( (r, s) ), and ( Q ).
| Role | Description |
|---|---|
| Message Integrity | Ensures the message cannot be altered without invalidating the signature. |
| Fixed-Length Input | Converts messages of any length to a fixed 256-bit integer for ECDSA. |
| Collision Resistance | Prevents attackers from finding two messages with the same hash. |
| Preimage Resistance | Prevents attackers from reversing the hash to recover the original message. |
| Efficiency | Provides a balance between security and computational efficiency. |
- Using a weaker hash function (e.g., SHA-1) could expose the signature scheme to collision attacks, where an attacker finds two messages with the same hash and forges a signature.
- SHA-256 is currently considered secure and is widely adopted in standards like FIPS 180-4 and NIST SP 800-186.
Would you like to explore how SHA-256 compares to other hash functions (like SHA-3 or BLAKE2) in the context of ECDSA, or dive deeper into the mathematical properties of hash functions?