Elliptic curve
An elliptic curve is a nonsingular algebraic curve defined by a cubic equation whose points, including a point at infinity, support a group operation.
Elliptic Curve Cryptography for Developers
Elliptic curve mathematics is the study of algebraic curves defined by nonsingular cubic equations and the mathematical structures on their points.
- How it is used
- Supports public-key cryptography with smaller numbers, including key exchange, digital signatures, pairings, and zero-knowledge proofs.
- How deep it goes
- Develops visual point algebra through finite-field implementation, then connects complex-plane mappings and cyclic groups to pairing-oriented curves.
- Examples
- Uses y² = x³ − 5x + 5 and y² = x³ + 23x + 42 mod 43, with enumerated point orders.
- What you can do after
- Build it from scratch for elliptic-curve key transfer, pairing-based signatures, and tests of cryptographic code.
Read how Elliptic Curve Cryptography for Developers explains elliptic curve →
Software Security for Developers
An elliptic curve is an algebraic curve defined by a cubic equation that supports cryptographic constructions.
- How it is used
- Applies elliptic curves to public-key cryptography, especially TLS key exchange and secure communication.
- How deep it goes
- Defines the curve, illustrates point operations and one-way behavior, then stops short of mathematical proofs and implementation internals.
- Examples
- Uses the equation y² = x³ + ax + b, geometric point addition, named curves, and color mixing as a key-exchange analogy.
- What you can do after
- Configure ECC by selecting approved curves and recognize how TLS uses them to establish shared secrets.
Read how Software Security for Developers explains elliptic curve →
Hacking Cryptography
An elliptic curve is a set of points satisfying a specified cubic equation and equipped with a defined point-addition operation.
- How it is used
- Supports cryptographic constructions, specifically DUAL_EC_DRBG, through structured point arithmetic.
- How deep it goes
- Defines the curve and operations, delegates arithmetic details to Go, and demonstrates coordinate validation during an attack.
- Examples
- Uses plotted curves, P256 operations, truncated-coordinate guesses, modular square roots, and Go exploit tests.
- What you can do after
- Apply it via Go's crypto/elliptic package to add points, perform scalar multiplication, and validate candidate coordinates.
Real-World Cryptography
An elliptic curve is a nonsingular algebraic curve that, with a specified point at infinity, supports a group operation.
- How it is used
- Explains the mathematical group behind elliptic-curve cryptography and Diffie–Hellman key exchange.
- How deep it goes
- Builds from curve equations through geometric addition to prime-field coordinates and named standardized curves.
- Examples
- Uses geometric point addition, the point at infinity, modular-prime graphs, P-256, and Curve25519.
- What you can do after
- Evaluate how elliptic-curve groups support ECDH and distinguish ordinary geometric curves from cryptographic prime-field curves.
Related concepts
- elliptic-curve cryptographyused by · Elliptic-curve cryptography is public-key cryptography based on algebraic operations over elliptic curves, typically defined over finite fields.
- Dual ECused by · Dual EC is an elliptic-curve-based deterministic pseudorandom number generator whose design permits state recovery if its generator points have a hidden mathematical relation.
- P-256includes · P-256, also called secp256r1, is a standardized elliptic curve over a 256-bit prime finite field used in public-key cryptography.
- ECDSAused by · ECDSA is a public-key digital signature algorithm that uses elliptic-curve arithmetic to generate and verify signatures.
- field primeconfigured by · A field prime is the prime modulus that defines the elements and arithmetic of a prime field used for elliptic-curve computations.
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