Publicly verifiable secret sharing scheme

In https://www.win.tue.nl/~berry/papers/crypto99.pdf, Schoenmakers proposes a publicly verifiable secret sharing scheme, that uses a non-interactive DLEQ proof to allow any participant to verify the shares of the secret (section 3.1 of the paper).

In "Distribution of the shares", it says "Applying Fiat-Shamir’s technique, the challenge $$c$$ for the protocol is computed as a cryptographic hash of $$X_i , Y_i , a_{1i} , a_{2i} , 1 ≤ i ≤ n$$."

And later, "Using $$y_i , X_i , Y_i , r_i , 1 ≤ i ≤ n$$ and $$c$$ as input, the verifier computes $$a_{1i} , a_{2i}$$ as

$$a_{1i} = g^{ri} X_i^c,$$ $$a_{2i} = y_i^{ri} Y_i^c ,$$

and checks that the hash of $$X_i , Y_i , a_{1i} , a_{2i} , 1 ≤ i ≤ n$$, matches $$c$$."

My question is: how can the challenge $$c$$ be used as input of the hash that computes itself (the challenge $$c$$), or am I misunderstanding?

• The input of the hash is $$x_i,y_i,a_{1i},a_{2i}$$.
• The output of the hash is $$c$$.
The wording is '$$c$$ is computed as ...'
And later the verifier checks if $$c$$ matches the Output of the hash function used on the same input variables. If different values were used in the original run, the hash would differ.
• The problem is that $a_{1i}$ and $a_{2i}$ are computed using $c$, I edited the question to add that. – Fiono Feb 20 '20 at 14:22