on sagemath
-------------------------------



sage:#prime [type:integer] but prime
sage:P = 115792089237316195423570985008687907853269984665640564039457584007908834671663

sage:#Elliptic Curve y2=x3+7 for P [type:curve]
sage:E = EllitpicCurve(GF(P),[0,7])

sage:#Elliptic Curve Order [type:integer]
sage:N = E.order()

sage:#Base Point G [type:point]
sage:G= E(55066263022277343669578718895168534326250603453777594175500187360389116729240,32670510020758816978083085130507043184471273380659243275938904335757337482424)

sage:# "public_key = secret*G" or "public_key = E(pubkey_x,pubkey_y)" [type:point], we know "pubkey_x,pubkey_y"
sage:public_key = E(pubkey_x,pubkey_y)

sage:# "K=random_number*G"  [type:point] than "r = K[0]" [type:integer] , you known "r"
sage:K = E.lift_x(r)

sage:#K is_correct ? we don't know

sage:r = your_value #[type:integer]
sage:s = your_value #[type:integer]
sage:z = your_value #[type:integer]


sage:w = 1/s %N
sage:u1 = z * w %N
sage:u2 = r * w %N

sage:#correct "K" point [type:point]
sage:u2*public_key + u1*G #[type:point]

sage:+K == u2*public_key + u1*G #(true or false)
sage:-K == u2*public_key + u1*G #(true or false)

sage r == Integer(+K[0]) #(true or false) [type:integer]
sage r == Integer(-K[0]) #(true or false) [type:integer]

sage:var("k x")

sage:k*s == r*x+z


this line type: point
R = E.lift_x(r)