Computational investment in generation of elliptic curves randomly over large prime fields

Randomly computed elliptic curves over prime fields are desirable for implementation in strategic or military grade cryptosystems. A randomly computed elliptic curve eliminates the possibility of exhibiting any pre‐studied value or special structure of the curve parameters which may be doubted to have intentionally non‐disclosed vulnerability. However, computing elliptic curves randomly over large prime fields for cryptography is an intricate task which requires reasonably high computational resources and time. These computational resources are considered in terms of the number of CPU clock cycles and the number of attempts or searches made in the security parameter space of elliptic curve. The estimate of the number of CPU clock cycles helps in determining processor requirements whereas the number of attempts or searches help to decide the number of CPU cores for speeding up the curve generation process. We present efficient statistical estimates of computational resources to randomly compute cryptographically safe elliptic curve over desired prime field size using a standard procedure.