Point at infinity
A point at infinity is a formal point added to a geometric space to represent behavior at infinity, such as a direction, limiting location, or elliptic-curve identity.
Elliptic Curve Cryptography for Developers
The point at infinity is the distinguished ideal point included in an elliptic curve's group structure and represented here by the coordinate pair (0, 0).
- How it is used
- Provides the identity and exceptional-case marker used by elliptic-curve addition, multiplication, and pairing routines.
- How deep it goes
- Defines the identity geometrically, encodes it as (0, 0), and traces input, inverse-output, and pairing checks in code.
- Examples
- Uses R plus −R, a 54-point tiny curve group counted as 55, and point-addition listings.
- What you can do after
- Implement it as a (0, 0) sentinel and handle identity and inverse-point cases without invalid slope division.
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Real-World Cryptography
The point at infinity is an added formal point that serves as the identity element of an elliptic curve group.
- How it is used
- Completes elliptic-curve addition as the identity element and serves as a public-key validation condition in key exchanges.
- How deep it goes
- Defines the point geometrically and algebraically, then applies it to public-key validation without detailing implementation internals.
- Examples
- Uses O + O = O, P + O = P, geometric cancellation, and public-key rejection checks as concrete examples.
- What you can do after
- Diagnose invalid elliptic-curve public keys by checking for O and verifying coordinates against the curve equation.
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Geometry for Programmers
A point at infinity is a projective point with zero homogeneous scale that represents a direction or limiting location beyond Euclidean space.
- How it is used
- Represents directions and limiting locations so Euclidean geometry can be extended for projective computation.
- How deep it goes
- Explains zero-scale coordinates through a numerical limiting sequence and stops short of formal projective constructions.
- Examples
- Uses shrinking homogeneous scales, a zero-scale coordinate belt, and central-versus-parallel projection figures.
- What you can do after
- Evaluate homogeneous coordinates with zero scale and understand why such points support projective transformations.
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Related concepts
- identity elementkind of · An element that, under a specified binary operation, leaves every element in the operation’s domain unchanged when combined with it.
- elliptic curve point additionused by · Elliptic-curve point addition combines two curve points according to the curve's group law to produce another curve point.
- point ordermeasured by · The order of a point is the smallest positive integer scalar whose multiple of that point is the point at infinity.
- group orderpart of · The order of a finite group is the number of elements it contains.
- elliptic curve point multiplicationproduced by · Elliptic-curve point multiplication multiplies a point by an integer scalar through repeated point addition and doubling.
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