Point at infinity

As explained in Real-World Cryptography

The point at infinity is an added formal point that serves as the identity element of an elliptic curve group.

The point at infinity is an invented point, usually written O, that makes elliptic-curve addition work in special cases. It acts like zero: O + O = O, and P + O = P for every point P on the curve. section 5.3.1

Geometrically, it appears when the line or tangent used for addition does not meet the curve at another ordinary point, including cases where two points cancel. Adding it completes the group structure used by elliptic-curve cryptography. section 5.3.1

Figure 5.10 Building on figure 5.9, addition on an elliptic curve is also defined when adding a point to itself or when two points cancel each other to result in the point at infinity (O).

Use the figure to see point doubling and cancellation cases that produce O. section 5.3.1

In elliptic-curve key exchanges, O also matters as a validation check:

• Reject a received public key if it is the point at infinity. section 5.4
• Check that the received coordinates describe a point on the defined curve before scalar multiplication. section 5.4

How Real-World Cryptography covers it

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 it assumes

Assumes basic elliptic-curve points and addition while introducing the identity element needed to form a group.

What you can do after

Diagnose invalid elliptic-curve public keys by checking for O and verifying coordinates against the curve equation.

How it connects

Connects elliptic-curve addition, group identity, elliptic-curve cryptography, scalar multiplication, ECDH, and invalid curve attacks.

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Also explained in

• 

  liveBookmanning.com
  Elliptic Curve Cryptography for DevelopersThe 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).
• 

  liveBookmanning.com
  Geometry for ProgrammersA point at infinity is a projective point with zero homogeneous scale that represents a direction or limiting location beyond Euclidean space.

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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.

This explainer was generated by Manning's knowledge graph from the text of Real-World Cryptography. The section links point to the parts of the book it draws on.