  The survey presents the evolution of Short Weierstrass elliptic curves after
their introduction in cryptography. Subsequently, this evolution resulted in
the establishment of present elliptic curve computational standards. We discuss
the chronology of attacks on Elliptic Curve Discrete Logarithm Problem and
investigate their countermeasures to highlight the evolved selection criteria
of cryptographically safe elliptic curves. Further, two popular deterministic
and random approaches for selection of Short Weierstrass elliptic curve for
cryptography are evaluated from computational, security and trust perspectives
and a trend in existent computational standards is demonstrated. Finally,
standard and non-standard elliptic curves are analysed to add a new insight
into their usability. There is no such survey conducted in past to the best of
our knowledge.
