diff options
Diffstat (limited to 'docs/IMPLEMENTATIONS.md')
| -rw-r--r-- | docs/IMPLEMENTATIONS.md | 73 |
1 files changed, 47 insertions, 26 deletions
diff --git a/docs/IMPLEMENTATIONS.md b/docs/IMPLEMENTATIONS.md index 4a70b7d..f8f4e96 100644 --- a/docs/IMPLEMENTATIONS.md +++ b/docs/IMPLEMENTATIONS.md @@ -14,19 +14,11 @@ $$ y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6 $$ #### Projective $$ Y^2Z + a_1XYZ + a_3YZ^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3 $$ + ### Weierstrass (short) [EFD entry](https://www.hyperelliptic.org/EFD/g1p/auto-shortw.html) -#### Affine -$$ y^2 = x^3 + a x + b $$ - -#### Projective -$$ Y^2Z = X^3 + aXZ^2 + bZ^3 $$ - -#### Jacobian -$$ Y^2 = X^3 + aXZ^4 + bZ^6 $$ - toweierstrass weierx = x toweierstrass weiery = y a0 = 1 @@ -38,11 +30,19 @@ $$ Y^2 = X^3 + aXZ^4 + bZ^6 $$ fromweierstrass x = weierx fromweierstrass y = weiery +#### Affine +$$ y^2 = x^3 + a x + b $$ + +#### Projective +$$ Y^2Z = X^3 + aXZ^2 + bZ^3 $$ + +#### Jacobian +$$ Y^2 = X^3 + aXZ^4 + bZ^6 $$ + + ### Montgomery [EFD entry](https://www.hyperelliptic.org/EFD/g1p/auto-montgom.html) -#### Affine -$$ By^2 = x^3 + Ax^2 + x $$ toweierstrass weierx = x toweierstrass weiery = y @@ -55,11 +55,13 @@ $$ By^2 = x^3 + Ax^2 + x $$ fromweierstrass x = weierx fromweierstrass y = weiery +#### Affine +$$ By^2 = x^3 + Ax^2 + x $$ + + ### Edwards [EFD entry](https://www.hyperelliptic.org/EFD/g1p/auto-edwards.html) -#### Affine -$$ x^2 + y^2 = c^2 (1 + dx^2y^2) $$ toweierstrass u = (c+y)/(c-y) toweierstrass v = 2 c (c+y)/(x(c-y)) @@ -72,11 +74,13 @@ $$ x^2 + y^2 = c^2 (1 + dx^2y^2) $$ fromweierstrass x = 2 c u/v fromweierstrass y = c(u-1)/(u+1) +#### Affine +$$ x^2 + y^2 = c^2 (1 + dx^2y^2) $$ + + ### Twisted Edwards [EFD entry](https://www.hyperelliptic.org/EFD/g1p/auto-twisted.html) -#### Affine -$$ ax^2 + y^2=1 + dx^2y^2 $$ toweierstrass u = (1+y)/(1-y) toweierstrass v = 2 (1+y)/(x(1-y)) @@ -89,10 +93,13 @@ $$ ax^2 + y^2=1 + dx^2y^2 $$ fromweierstrass x = 2 u/v fromweierstrass y = (u-1)/(u+1) +#### Affine +$$ ax^2 + y^2=1 + dx^2y^2 $$ + + ### Hessian -#### Affine -$$ x^3+y^3+1=3dxy $$ +[EFD entry](https://www.hyperelliptic.org/EFD/g1p/auto-hessian.html) toweierstrass u = 12(d^3-1)/(d+x+y)-9 d^2 toweierstrass v = 36(y-x)(d^3-1)/(d+x+y) @@ -105,6 +112,13 @@ $$ x^3+y^3+1=3dxy $$ fromweierstrass x = (36(d^3-1)-v)/(6(u+9 d^2))-d/2 fromweierstrass y = (v+36(d^3-1))/(6(u+9 d^2))-d/2 +#### Affine +$$ x^3+y^3+1=3dxy $$ + +#### Projective +$$ X^3 + Y^3 + Z^3 = 3dXYZ $$ + + --- ## Coordinates @@ -113,9 +127,9 @@ $$ x^3+y^3+1=3dxy $$ $$ [x, y] \in \mathbb{K}^2 $$ -infinity is special cased, sometimes represented as \( [0] \). +*(Weierstrass)* infinity is special cased, sometimes represented as \( [0] \). -negation: \(-[x, y] = [x, -y] \) +*(Weierstrass)* negation: \(-[x, y] = [x, -y] \) - To Projective: \( [x, y] \rightarrow (x : y : 1) \) - To Jacobian: \( [x, y] \rightarrow (x : y : 1) \) @@ -132,9 +146,9 @@ negation: \(-[x, y] = [x, -y] \) (X : Y : Z) &= \{(λ X, λ Y, λ Z) | λ \in \mathbb{K}^* \} \end{align*} -infinity is \((0 : 1 : 0)\). +*(Weierstrass)* infinity is \((0 : 1 : 0)\). -negation: \( -(X : Y : Z) = (X : -Y : Z) \) +*(Weierstrass)* negation: \( -(X : Y : Z) = (X : -Y : Z) \) - To Affine: \( (X : Y : Z) \rightarrow [X/Z, Y/Z] \) - To Jacobian: \( (X : Y : Z) \rightarrow (X/Z : Y/Z : 1) \) ? @@ -151,9 +165,9 @@ negation: \( -(X : Y : Z) = (X : -Y : Z) \) (X : Y : Z) &= \{(λ^2 X, λ^3 Y, λ Z) | λ \in \mathbb{K}^* \} \end{align*} -infinity is \( (1 : 1 : 0) \). +*(Weierstrass)* infinity is \( (1 : 1 : 0) \). -negation: \( -(X : Y : Z) = (X : -Y : Z) \) +*(Weierstrass)* negation: \( -(X : Y : Z) = (X : -Y : Z) \) - To Affine: \( (X : Y : Z) \rightarrow [X/Z^2, Y/Z^3] \) - To Projective: \( (X : Y : Z) \rightarrow (X/Z^2 : Y/Z^3 : 1) \) ? @@ -165,9 +179,9 @@ negation: \( -(X : Y : Z) = (X : -Y : Z) \) (X : Y : Z : Z^2 : Z^3 ) &= \{(λ^2 X, λ^3 Y, λ Z, λ^2 Z^2, λ^3 Z^3) | λ \in \mathbb{K}^* \} \end{align*} -infinity is \( (1 : 1 : 0 : 0 : 0) \). ? +*(Weierstrass)* infinity is \( (1 : 1 : 0 : 0 : 0) \). ? -negation: \( -(X : Y : Z : Z^2 : Z^3) = (X : -Y : Z : Z^2 : Z^3) \) +*(Weierstrass)* negation: \( -(X : Y : Z : Z^2 : Z^3) = (X : -Y : Z : Z^2 : Z^3) \) - To Affine: \( (X : Y : Z : Z^2 : Z^3) \rightarrow [X/Z^2, Y/Z^3] \) - To Projective: \( (X : Y : Z : Z^2 : Z^3) \rightarrow (X/Z^2 : Y/Z^3 : 1) \) ? @@ -176,7 +190,14 @@ negation: \( -(X : Y : Z : Z^2 : Z^3) = (X : -Y : Z : Z^2 : Z^3) \) --- -## Addition formulas +## Formulas + + - Addition + - Doubling + - Tripling + - Differential addition + - Differential addition and doubling + - Scaling See EFD[^3]. |