 MPHELL  4.0.0
Edwards elliptic curves

Let be a twisted Edwards elliptic curve under affine coordinates. Under Projective coordinates becomes . The triplet represents the affine point .

Under Jacobian coordinates becomes . The triplet represents the affine point .

Under Extended twisted Edwards coordinates becomes . The quadruplet represents the affine point , the auxiliary coordinate T has the property .

Edwards elliptic curves are twisted Edwards elliptic curves with so we work here with Twisted Edwards elliptic curves.

# Projective Coordinates

Let

1. P1 = (X1,Y1,Z1)
2. P2 = (X2,Y2,Z2)

The point P3 = (X3,Y3,Z3) = P1 + P2 is given by

1. 2. 3. 4. 5. 6. 7. using 10 multiplications, 1 square, 1 times a, 1 times d, 7 additions .

This formula is taken from [BBL+08, §6] Twisted edwards curves by Daniel. J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange, and Christiane Peters (2008, June) published in International Conference on Cryptology in Africa (pp. 389-405). Springer, Berlin, Heidelberg viewable at this adress https://link.springer.com/

This formula can be viewed on this website http://www.hyperelliptic.org where it is called "add-2008-bbjlp".

It is important to notice that this formula for addition is strongly unified.

## Doubling

The point P3 = (X3,Y3,Z3) = 2 P1 is given by

1. 2. 3. 4. 5. using 3 multiplications, 4 squares, 1 times a, 6 additions, 1 times 2 .

This formula is taken from [BBL+08, §6] Twisted edwards curves by Daniel. J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange, and Christiane Peters (2008, June) published in International Conference on Cryptology in Africa (pp. 389-405). Springer, Berlin, Heidelberg viewable at this adress https://link.springer.com/

This formula can be viewed on this website http://www.hyperelliptic.org where it is called "dbl-2008-bbjlp".

# Extended twisted Edwards Coordinates becomes .

Let

1. P1 = (X1,Y1,T1,Z1)
2. P2 = (X2,Y2,T2,Z2)

This formula is taken from [HWCD08, §3.2] Twisted Edwards Curves Revisited by Hisil, Wong, Carter and Dawson published in Asiacrypt volume 5350, pages 326-343 viewable at this adress https://link.springer.com/ .

The point P3 = (X3,Y3,T3,Z3) = P1 + P2 is given by

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. using 9 multiplications, 1 times a and 7 additions.

One can notice that those formulas are independant of the curve constant d.

This formula can be viewed on the website http://www.hyperelliptic.org and is called "add-2008-hwcd-2"

## Doubling

This formula is taken from [HWCD08, §3.3] Twisted Edwards Curves Revisited by Hisil, Wong, Carter and Dawson published in Asiacrypt volume 5350, pages 326-343 viewable at this adress https://link.springer.com/ .

The point P3 = (X3,Y3,T3,Z3) = P1 + P1 is given by

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. using 4 multiplications + 4 squares + 1 times a constant 6 additions and 1 doubling.

This formula can be viewed on the website http://www.hyperelliptic.org and is called "dbl-2008-hwcd"

This formula is taken from [HWCD08, §3.1] Twisted Edwards Curves Revisited by Hisil, Wong, Carter and Dawson published in Asiacrypt volume 5350, pages 326-343 viewable at this adress https://link.springer.com/ .

The point P3 = (X3,Y3,T3,Z3) = P1 + P2 is given by

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. using 9 multiplications,1 times a, 1 times d and 7 additions.

This formula can be viewed on the website http://www.hyperelliptic.org and is called "add-2008-hwcd"