# Discrete log hardness when secret is multiplied by public value

Given y = g ^ x is discrete log hard on some finite field, is y = g ^ (kx) also equally secure if the value k is a publicly known value which was randomly selected from a uniform distribution ?

To my understanding, if k and x are independent and chosen randomly, then the security of the discrete logarithm problem is not significantly affected as an attacker still needs to compute the discrete logarithm of y with respect to the base g, and the knowledge of k does not provide any useful information for solving the problem.

Still if there is something I'm missing, please point out. Thanks

• Could you provide us with the source of this question? Commented Jul 8, 2023 at 16:49
• I was studying Sigma Protocol and this thought randomly popped in my mind: whether the domain of possible values for secrets would be smaller if the secret is a multiple of known value. But being a finite field of prime order that wouldn't be the case. So I wondered if there was anything else. Commented Jul 8, 2023 at 21:36

is $$y = g ^ {kx}$$ also equally secure if the value $$k$$ is a publicly known value which was randomly selected from a uniform distribution ?
Here is a clearer way to look at it: suppose we have an oracle that, given $$k, y=g^{kx}$$, is able to recover $$x$$.
Then, given $$g^x$$, we can randomly select $$k$$, compute $$(g^x)^k = g^{kx}$$. We can then give $$k$$ and $$y = g^{kx}$$ to our Oracle (and note that the precondition that $$k$$ must be randomly chosen is met), which will then give us $$x$$, solving the discrete log problem.