The Fiat-Shamir transform typically works by substituting (public) coin tosses from the verifier by hashes of the prover's messages until this point, i.e.: $$H(x,\alpha_1) = \beta_1, \\ H(x,\alpha_1, \alpha_2) = \beta_2,\\H(x,\alpha_1, \alpha_2, \alpha_3) = \beta_3,\\\vdots$$ where $x$ is the pub...