4.3 Elliptic curve cryptography

Elliptic curve cryptography (ECC) is a powerful tool in public key cryptosystems. It offers smaller key sizes and faster operations compared to traditional methods like RSA, making it ideal for resource-constrained devices. ECC's mathematical structure also enables advanced protocols and improved security.

ECC's strength lies in the elliptic curve discrete logarithm problem, which is harder to solve than factoring large numbers. This allows ECC to provide equivalent security with shorter keys, reducing storage and transmission requirements. However, proper implementation is crucial to avoid vulnerabilities and side-channel attacks.

Elliptic curves in cryptography

Mathematical foundations

Top images from around the web for Mathematical foundations

• 

  Another simple Elliptic Curve View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

• 

  Another simple Elliptic Curve View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

1 of 3

Top images from around the web for Mathematical foundations

• 

  Another simple Elliptic Curve View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

• 

  Another simple Elliptic Curve View original

  Is this image relevant?

• 

  Elliptic Curve Cryptography: finite fields and discrete logarithms - Andrea Corbellini View original

  Is this image relevant?

1 of 3

• Elliptic curves form algebraic structures defined by the equation $y^2 = x^3 + ax + b$, where a and b are constants and $4a^3 + 27b^2 ≠ 0$
• Group law for elliptic curves defines point addition and scalar multiplication operations underpinning cryptographic algorithms
• Finite field arithmetic applies to elliptic curve cryptography, typically over prime fields (Fp) or binary fields (F2^m)
• Discrete logarithm problem on elliptic curves (ECDLP) provides the security foundation for elliptic curve cryptosystems
• Elliptic curve domain parameters encompass field size, curve coefficients, base point, order of the base point, and cofactor
• Selection of appropriate curve parameters critically impacts the security and efficiency of elliptic curve cryptosystems

Cryptographic applications

• Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol establishes shared secrets over insecure channels
• Elliptic Curve Integrated Encryption Scheme (ECIES) combines asymmetric and symmetric encryption for secure message transmission
• Elliptic Curve Digital Signature Algorithm (ECDSA) creates and verifies digital signatures using elliptic curve operations
• Advanced cryptographic protocols utilize elliptic curves
  • Bilinear pairings enable novel cryptographic constructions (identity-based encryption)
  • Identity-based encryption simplifies key management in public key infrastructures
• Efficient implementation of point multiplication algorithms optimizes performance
  • Double-and-add method provides basic scalar multiplication
  • Window methods improve efficiency for larger scalar values

Security considerations

• ECDLP hardness ensures ECC security with no known sub-exponential time algorithm for well-chosen curves
• Small subgroup attacks exploit curve cofactor
  • Mitigated through cofactor multiplication
  • Selecting curves with cofactor 1 eliminates vulnerability
• Invalid curve attacks move computations to weaker curves
  • Prevention requires validating input points
• MOV attack reduces ECDLP to discrete logarithm problem in finite fields
  • Affects supersingular curves
  • Avoided by using non-supersingular curves
• Side-channel attacks exploit implementation vulnerabilities
  • Timing attacks analyze execution time variations
  • Power analysis attacks examine power consumption patterns
  • Countermeasures include constant-time algorithms and point blinding

Elliptic curve vs traditional cryptography

Performance advantages

• Smaller key sizes compared to RSA for equivalent security levels
  • 256-bit ECC key provides similar security to 3072-bit RSA key
  • Reduced storage and transmission requirements benefit resource-constrained devices (smartphones, IoT devices)
• Faster and more efficient operations than RSA, especially at higher security levels
  • ECC point multiplication outperforms RSA exponentiation
  • Improved performance in resource-constrained environments (embedded systems, smart cards)
• Enhanced scalability maintains efficiency as security requirements increase over time
  • ECC key sizes grow linearly with security level
  • RSA key sizes grow exponentially, leading to diminishing returns

Cryptographic flexibility

• Rich mathematical structure of elliptic curves enables diverse cryptographic protocols
  • Pairing-based cryptography opens new possibilities (attribute-based encryption, functional encryption)
  • Short signatures reduce bandwidth requirements in constrained environments
• Advanced protocols efficiently implemented with ECC
  • Threshold cryptography for distributed trust scenarios
  • Zero-knowledge proofs for privacy-preserving applications
• Adaptability to emerging security needs
  • Post-quantum variants of ECC under active research (supersingular isogeny-based cryptography)

Quantum resistance

• ECC generally considered stronger against quantum attacks than RSA for equivalent key sizes
  • Grover's algorithm impacts symmetric key sizes, affecting both ECC and RSA
  • Shor's algorithm more efficiently breaks RSA than ECC of comparable classical security
• Both ECC and RSA vulnerable to quantum attacks in the long term
  • Research into quantum-resistant alternatives ongoing (lattice-based, code-based cryptography)
• Hybrid schemes combining ECC with post-quantum algorithms provide transitional security

Implementing elliptic curve cryptography

Key generation and management

• ECC key generation involves selecting a random private key and computing the corresponding public key
  • Private key: random integer d within the curve's order range
  • Public key: scalar multiplication of base point G by private key (Q = dG)
• Secure random number generation crucial for private key security
  • Use cryptographically secure pseudo-random number generators (CSPRNGs)
  • Employ hardware random number generators when available
• Key sizes vary based on security requirements and application constraints
  • 256-bit keys common for general-purpose applications
  • 384-bit or 521-bit keys for high-security scenarios

Encryption and key exchange

• ECDH key exchange protocol establishes shared secrets
  • Each party generates ephemeral key pair
  • Shared secret computed through scalar multiplication of public keys
  • Key derivation function produces symmetric key from shared secret
• ECIES combines asymmetric and symmetric encryption
  • Sender generates ephemeral ECC key pair
  • Shared secret derived using recipient's public key
  • Symmetric encryption with derived key secures message
  • Ephemeral public key and ciphertext transmitted to recipient

Digital signatures

• ECDSA provides method for creating and verifying digital signatures
  • Signing process:
    1. Generate random nonce k
    2. Compute curve point R = kG
    3. Calculate signature components (r, s) using private key and message hash
  • Verification process:
    1. Compute curve point using signature components and public key
    2. Compare computed value to signature component r
• Deterministic ECDSA (RFC 6979) eliminates need for random nonce
  • Improves security by preventing nonce reuse vulnerabilities
  • Enables reproducible signatures for testing and auditing purposes

Security of elliptic curve cryptosystems

Known attacks and mitigations

• Small subgroup attacks exploit curve cofactor
  • Mitigated through cofactor multiplication in protocols
  • Selecting curves with cofactor 1 (prime order curves) eliminates vulnerability
• Invalid curve attacks move computations to weaker curves
  • Prevention requires validating input points lie on the correct curve
  • Implement efficient point validation algorithms (y^2 = x^3 + ax + b mod p)
• MOV attack reduces ECDLP to finite field discrete logarithm problem
  • Affects supersingular curves with small embedding degree
  • Mitigated by using non-supersingular curves with large embedding degree
• Side-channel attacks exploit implementation vulnerabilities
  • Timing attacks analyze execution time variations
    • Implement constant-time algorithms for all operations
  • Power analysis attacks examine power consumption patterns
    • Apply randomization techniques (point blinding, scalar blinding)
  • Fault injection attacks introduce errors to reveal secret information
    • Implement error detection and countermeasures (signature verification before release)

Implementation considerations

• Proper handling of point representation impacts security and efficiency
  • Affine coordinates (x, y) simplify implementation but are slower for some operations
  • Projective coordinates improve efficiency by eliminating expensive field inversions
• Finite field arithmetic implementation affects overall performance
  • Optimize field operations (multiplication, squaring, inversion) for target platform
  • Consider hardware acceleration for critical operations
• Side-channel attack mitigation techniques essential for secure implementations
  • Constant-time algorithms eliminate timing-based information leakage
  • Point blinding randomizes scalar multiplication to prevent power analysis
  • Regular scalar multiplication algorithms resist simple power analysis

Future-proofing and standardization

• Quantum computers pose significant threat to ECC through Shor's algorithm
  • Research into post-quantum cryptography alternatives ongoing
  • Hybrid schemes combining ECC with post-quantum algorithms provide transitional security
• Standardization efforts ensure interoperability and security
  • NIST SP 800-186 specifies approved elliptic curves for US government use
  • SECG (Standards for Efficient Cryptography Group) defines widely-used curves (secp256k1 for Bitcoin)
• Emerging ECC variants address specific security concerns
  • Edwards curves offer complete addition formulas, simplifying constant-time implementations
  • Curve25519 designed for efficient and secure Diffie-Hellman key exchange