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How define Twisted Edwards elliptic curve ?

asked 2020-11-25 06:40:37 -0600

Andr gravatar image

updated 2020-11-25 13:01:06 -0600

slelievre gravatar image

In Sage, EllipticCurve takes 2 or 5 parameters.

For 5 parameters, EllipticCurve([a, b, c, d, e]) means:

y^2 + a*x*y + c*y = x^3 + b*x^2 + d*x + e.

The curve Ed25519 (for example for Monero) is defined by $$ -x^2+y^2=1-\frac{121665}{121666}x^2y^2 $$

This equation has a term in $x^2y^2$ which does not exist in Sage EllipticCurve parameters.

What does "twisted" mean?

How to define $\frac{121665}{121666}$ ?

I must multiply 121665 but inverted 121666 in field? Field is with prime p=$2^{255}-19$.

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answered 2020-11-25 13:22:58 -0600

FrédéricC gravatar image

Like this maybe

sage: x, y = polygens(GF(1931), 'x,y')                                              
sage: C = Curve(x**2-y**2-1-33*x**2*y**2)                                         
sage: Jacobian(C)                                                               
Elliptic Curve defined by y^2 = x^3 + 1078*x + 1551 over Finite Field of size 1931
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Asked: 2020-11-25 06:40:37 -0600

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Last updated: Nov 25 '20