Elliptic Curve Cryptography, decrypting returns false result

Question

I build some application to encrypt and decrypting some text using ecc

My curve paramaters are

p = 59
a = 2
b = 2

resulting in 64 points including infinite point.

for encrypt and decrypt process i use

e1 = (3, 25)
d (private key) = 53

Here's the problem that i faced

let say i want to encrypt point (45, 48) first, i generate random number k = 47 and i calculate e1 * k

C1 = k * e1
   = 47 * (3, 25)
   = (43, 57)

C2 = P + k * e2
   = (45, 48) + 47 * (53 * (3, 25))
   = (28, 11)

But, when i decrypt the c1 and c2

M = c2 - d x c1
  = (28, 11) - 53 * (43, 57)
  = (43, 57)

I tried to check my calculation in online calculator, but the result is not even in the curve

Can someone explain to me why? where did i go wrong? Sorry for my english Thank you

Answer 1

I would imagine your calculator doesn't know how to handle the point at infinity because it relies fully on affine points. You can compute that

\begin{align*} C_2 &= P + k*e_2 \\ &= (45,48) + 47*53*(3,25) \\ &= 5*(3,25) + 47*53*(3,25) & (\text{since }(45,48) = 5*(3,25)) \\ &= 2496*(3,25) \\ &= \infty & (\text{since }2496\equiv0\bmod 64) \end{align*} This computation seems to go wrong somehow, possibly because it can't represent $\infty$ internally.