Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent smooth group order which can then be factorized, and the discrete log problem can be solved by Pohlig-Hellman algorithm.

How can the same be accomplished for a group of points on an elliptic curve over finite field?


Make sure the group order is not smooth
Find an element of large prime order $q$.
For the curves used in cryptography today, the order of the group of points is typically 256 bits, sometimes 512. Such numbers are within the range of current factoring algorithms (indeed, this is why you don't use 512-bit RSA keys), and so typically the order of the group can be factored and it is easy to see whether it has a large prime factor.

Moreover, standard curves such as the NIST curves or Curve25519 are chosen so that the order of the group of points is either prime (as in the NIST curves) or is a large prime multiplied by a small cofactor ($8$ in the case of Curve25519), so if you use a standard curve you know there is a subgroup of large prime order for you to work in. (It is important that you work in a group of prime order to prevent other types of attacks.)