a and b is not given, but since we are given the two points, generator G and the public key point P, we can construct two equations in the form of $y ^ 2 = x ^ 3 + ax + b$. Solving the two linear congruent equation should give us $a, b$. Trying to do E = EllipticCurve(GF(p), [a, b]) lead to an error by Sage that the curve defined is not a elliptic curve, rather a singular curve....
Simple challenge, there are two key point to solve this. First, the order of the curve is smooth, and also the private keys are relatively small ($2 ^ {64}$). My approach is just simply relying on Sage magic to perform the heavy duty work. Sage Implementation: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 from Crypto....
An elliptic curve $E$ defined over $F_p$ whose cardinality (or order) is also $p$ is an anomalous curve, and the discrete logarithm becomes trivial. We can employ Smart’s attack, which allows us to perform the discrete log of the curve in linear time. This paper by Novotney goes over the details of why the attack works. At the time of writing, I wish I understand what’s in the paper - some very topics into group theory....