Elligator-2 against curves over Fq, q mod 4 = 3
Most probably, the elligator (hiding in here: Elligator: Elliptic-curve points indistinguishable from uniform random strings – PDF) will jump onto the attackers and bite them, so that they will not risk any further attack. 

As an additional security, as described in section 5.1 of the paper, the definition of Elligator 2 is parametrized with a non-square element $u\in \mathbb{F}_q$. So if I understand this correctly, an attacker will not know this parameter $u$ and will thus not be able to decode and encode points of the curve. All the attacker can do is observe a uniformly distributed sequence of bits.