Let $Z\sim \chi^2(k)$ be a random variable sampled from the Chi-Squared distribution with $k$ degrees of freedom.
Vague question: Conditional on the value of $Z$, how can I reconstruct a sequence of i.i.d. standard Gaussians $\{X_i\}_{i=1}^k$ such that $Z=\sum_i X_i^2$?
Precise question: Let $U$ be a uniform $[0,1]$ random variable. Is there a sequence of deterministic functions $\{f_i\}_{i=1}^k$ such that $\{f_i(Z,U)\}_{i=1}^k$ are a sequence of i.i.d. $N(0,1)$ random variables satisfying $\sum_{i=1}^k f_i(Z,U)^2=Z$ a.s.?