Protocol for proof of knowledge of $l$-th root Assume we have Group G in which the adaptive root assumption holds. 
This assumption states that if we choose an element $w$ and after that, if we receive a prime value $l$ it is hard to find the $u$ such that: $u^l = w$

Now suppose I want to prove that I know a $l\text{-}th$ root of an element $w$ without revealing it. (I don't want to reveal $u$). Is there any protocol for this?
The immediately obvious solution would be this simple cut-and-choose protocol:


Prover: selects a random value $v$ and sends the value $y = v^\ell$
Verifier: selects and sends a random bit $b$
Prover: if $b=0$, sends the value $t_0=v$.  If $b=1$, sends the value $t_1=vu$
Verifier: if $b=0$, then verify that $t_0^\ell = y$.  If $b=1$, then verify that $t_1^\ell = y w$


The standard zero knowledge logic works - if the prover knows a $y$ value for which he knows both correct responses $t_0$ and $t_1$, then (assuming that inverses are also easy to compute) he can recover the value $u$ (hence, if he succeeds with this protocol a number of times, then the probability of success without him knowing $u$ is minimal).  And, only one of the answers does not give any information to the verifier.