Introduction to Elliptic Curve Cryptography

Introduction to Elliptic Curve Cryptography (ECC)

• ECC is an asymmetric-key cryptography that relies on the elliptic curve discrete logarithm problem (ECDLP) for security.
• It utilizes elliptic curves over finite fields for mathematical operations.
• ECC offers similar security to other asymmetric algorithms, like RSA, while using significantly smaller key sizes.
• This leads to lower computational complexity, faster processing, and reduced key storage needs.
• ECC is suitable for resource-constrained devices and environments with limited computational power and storage.

Elliptic Curves

• Elliptic curves are smooth, algebraic curves defined by an equation.
• The general equation for an elliptic curve in a plane over a field is y² = x³ + ax + b, where 'a' and 'b' are elements in a finite field.
• These curves are defined over finite fields, enabling efficient mathematical operations.
• Essential geometric properties of elliptic curves are used for cryptographic operations.

Elliptic Curve Discrete Logarithm Problem (ECDLP)

• ECDLP is a computationally challenging problem.
• Given two points P and Q on an elliptic curve, where Q is a multiple of P, finding the scalar 'k' that transforms P into Q is computationally demanding.
• The difficulty increases with the elliptic curve's size.
• This difficulty forms the basis of ECC security.

Key Generation in ECC

• Key generation involves choosing a prime number, creating an elliptic curve over a finite field, and selecting a random private key 'd' using a secure random number generator.
• The corresponding public key 'Q' is calculated by multiplying the private key 'd' by a generator point 'G' on the elliptic curve (Q = d * G).
• 'G' is a predefined fixed point specific to the chosen elliptic curve parameters.

Encryption using ECC

• Encrypting a message with ECC uses the recipient's public key 'Q'.
• A random key 'k' is generated for encryption.
• The message is encrypted along with a specific point on the elliptic curve, using a calculation involving the recipient's public key and the ephemeral shared secret.

Decryption in ECC

• Decryption uses the recipient's private key 'd' to derive the ephemeral shared secret, identical to the sender's.
• With the shared secret, the original message data is recovered via proper calculations using the correct elliptic curve parameters.

Key Exchange with ECC

• Diffie-Hellman key exchange using ECC (ECDH) is a commonly used protocol.
• ECC Diffie-Hellman allows two parties to establish a shared secret key over an insecure channel.
• Security relies on the difficulty of ECDLP.
• The common generator 'G' is used by both parties, defined by the selected elliptic curve parameters.

Comparison with RSA

• ECC uses smaller key sizes than RSA for equivalent security.
• This leads to faster computations and reduced storage space.
• RSA security depends on integer factorization, while ECC security depends on ECDLP, considered more resilient to quantum attacks in some implementations.
• ECC's suitability for embedded systems and mobile platforms is due to its efficiency handling limited processing power and energy consumption.

Applications of ECC

• ECC is used in various applications, including digital signatures, secure communication protocols, cryptocurrencies, mobile devices, and smart cards.
• Its security, speed, and efficiency make it ideal in constrained environments demanding small hardware, fast processing, and high security standards.