Complete arithmetic implementation for bitcoin_key types

[dr-orlovsky]

Definition	Operation	Fails if	Trait	Return type
neg: Scalar -> Scalar	x -> (0 - x) mod n	-	Neg	Self
checked_add: Scalar, Scalar -> Scalar	x, y -> (x + z)	x + z > n	n/a	Result<Self, OverflowError>
checked_sub: Scalar, Scalar -> Scalar	x, y -> (x + (0 - y) mod n)	y > x	n/a	Result<Self, OverflowError>
checked_mul: Scalar, Scalar -> Scalar	x, y -> add(x, x) y times	x * y > n	n/a	Result<Self, OverflowError>
neg: Scalar -> Scalar	P -> ∞ + P	-	Neg	Self
checked_add: Point, Point -> Point	P, Q -> P + Q	Q = -P	n/a	Result<Self, PointAtInfinity>
checked_sub: Point, Point -> Point	P, Q -> P + ∞ + Q	Q = P	n/a	Result<Self, PointAtInfinity>
checked_add_exp: Point, Scalar -> Point	P, y -> P + yG	yG = -P	n/a	Result<Self, PointAtInfinity>
checked_sub_exp: Point, Scalar -> Point	P, y -> P + ∞ + yG	yG = P	n/a	Result<Self, PointAtInfinity>
mul: Point, Scalar -> Point	P, y -> yP	-1	Mul<Self>	Self

where

• x, y, z are scalars with the value up to n,
• n is a order of the secp256k1 generator point G,
• P, Q, R are points on the elliptic curve secp256k1 generated from G via xG, yG and zG multiplication.
• ∞ is a point on elliptic curve at infinity, corresponding to G * 0

Footnotes

1. allows overflow over G order ↩