Elliptic-curve point
An elliptic-curve point is a solution to an elliptic-curve equation over a specified field, participating in the curve's group operation.
Hacking Cryptography
An elliptic-curve point is a coordinate pair satisfying the equation of an elliptic curve and participating in its group operation.
- How it is used
- Supports generator-point operations and state recovery in the DUAL_EC_DRBG backdoor attack.
- How deep it goes
- Explains curve membership, modular square roots, coordinate validation, truncation recovery, and scalar multiplication, but not point-addition internals.
- Examples
- Uses P-256 validation, truncated output reconstruction, Go implementations, and a test that clones 100 subsequent outputs.
- What you can do after
- Diagnose/debug point reconstruction by validating coordinates and testing scalar-multiplication candidates against observed outputs.
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Elliptic Curve Cryptography for Developers
An elliptic curve point is a solution point of an elliptic curve equation over a specified field.
- How it is used
- Supports elliptic-curve algebra and cryptographic operations over finite fields and field extensions.
- How deep it goes
- Descends from geometric intuition to point structures, addition, multiplication, infinity handling, embedding, and random-point routines.
- Examples
- Uses the curve y² = x³ − 5x + 5, subgroup factorization, Schnorr verification, and polynomial random-point generation.
- What you can do after
- Implement point addition, multiplication, identity handling, random generation, and coordinate-based Schnorr verification.
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Related concepts
- elliptic curve data embeddingproduced by · Elliptic curve data embedding is a technique for mapping arbitrary data to a point on an elliptic curve by adjusting a candidate x-coordinate until the curve equation has a square-root solution.
- elliptic curve point additionproduced by · Elliptic-curve point addition combines two curve points according to the curve's group law to produce another curve point.
- Schnorr signatureused by · A Schnorr signature is a public-key digital signature scheme based on a proof of knowledge of a discrete logarithm.
- ECDSAused by · ECDSA is a public-key digital signature algorithm that uses elliptic-curve arithmetic to generate and verify signatures.
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