class

Ed25519::ExtendedPoint

Inherits Reference < Object

Extended Point works in extended coordinates: (x, y, z, t) ∋ (x=x/z, y=y/z, t=xy). Default Point works in affine coordinates: (x, y) https://en.wikipedia.org/wiki/Twisted_Edwards_curve#Extended_coordinates

Constants

BASE = ExtendedPoint.new(Curve::Gx, Curve::Gy, One, Ed25519.mod(Curve::Gx * Curve::Gy))

ZERO = ExtendedPoint.new(Zero, One, One, Zero)

Constructors

from_affine(p : Point) : ExtendedPoint

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new(x : BigInt, y : BigInt, z : BigInt, t : BigInt)

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Class methods

normalize_z(points : Array(ExtendedPoint)) : Array(ExtendedPoint)

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to_affine_batch(points : Array(ExtendedPoint)) : Array(Point)

Takes a bunch of Jacobian Points but executes only one invert on all of them. invert is very slow operation, so this improves performance massively.

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Instance methods

==(other : ExtendedPoint) : Bool

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add(other : ExtendedPoint)

Fast algo for adding 2 Extended Points when curve's a=-1. http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-4 Cost: 8M + 8add + 2*2. Note: It does not check whether the other point is valid.

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double

Fast algo for doubling Extended Point when curve's a=-1. http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd Cost: 3M + 4S + 1a + 7add + 12.

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equals(other : ExtendedPoint) : Bool

Compare one point to another.

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multiply(scalar : Int, affine_point : Point | Nil) : ExtendedPoint

Constant time multiplication. Uses w_naf method. Windowed method may be 10% faster, but takes 2x longer to generate and consumes 2x memory.

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multiply_unsafe(scalar : Int) : ExtendedPoint

Non-constant-time multiplication. Uses double-and-add algorithm. It's faster, but should only be used when you don't care about an exposed private key e.g. sig verification. Allows scalar bigger than curve order, but less than 2^256

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negate

Inverses point to one corresponding to (x, -y) in Affine coordinates.

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small_order?

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subtract(other : ExtendedPoint) : ExtendedPoint

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t=(t : BigInt)

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to_affine(inv_z : BigInt = Ed25519.invert(@z)) : Point

Converts Extended point to default (x, y) coordinates. Can accept precomputed Z^-1 - for example, from invert_batch.