Elliptic Curve Cryptography (ECC) Cheat Sheet

Basics of Field Theory

• Field: A set with two operations (addition and multiplication) satisfying specific properties (closure, associativity, commutativity, distributivity, existence of identity elements, and inverses).
• Finite Field (Galois Field): A field with a finite number of elements, denoted as GF(p) where p is a prime number.

Group Theory in ECC

• Group: A set with an operation (e.g., addition or multiplication) that satisfies closure, associativity, identity, and inverse properties.
• Elliptic Curve Group: Defined by points on the curve and an operation (usually point addition).
• Base Point (G): A generator point on the curve that generates the entire group when combined with the point addition operation.

ECC Basics and Mathematical Representation

• Elliptic Curve Equation: y^2 = x^3 + ax + b.
• Properties: Non-linear, continuous, and symmetrical.
• Finite Field Usage: ECC operates on points within a finite field GF(p).

Key Generation and Operations

• Curve Selection: Choose an elliptic curve defined over a finite field.
• Point Generation:
  • Base Point (G): A predefined point on the curve.
  • Scalar Multiplication: Repeated addition of the base point.
    • Notation: xP = P + P + P ... + P (x times), where x is an integer.
• Private Key (d): Randomly chosen integer within the field.
• Public Key (Q): Computed as Q = d * G.

ECC Encryption and Decryption

• Encryption:
  • Recipient's Public Key: Q = d * G.
  • Random Number (k): Generated for each message.
  • Ciphertext:
    • Calculate C1 = k * G and C2 = M + k * Q.
• Decryption:
  • Use the recipient's private key to derive Q.
  • Compute M = C2 - d * C1.

Advantages of ECC

• Security: Stronger security with smaller key sizes compared to RSA.
• Efficiency: Requires fewer computational resources.
• Applications: Secure communication, digital signatures, and key exchange.

Conclusion

• Significance: ECC leverages field and group theory concepts, incorporating properties like closure, identity, inverse, and scalar multiplication, to offer robust security with smaller key sizes, crucial in modern cryptography.