I'm reading A note on succinct representations of graphs by Papadimitriou and Yannakakis. Let me quote the following paragraph on page 183:
Formula $F$ has a highly regular structure. It has $|x|$ clauses stating that the input to the computation of $U$ is $x$, and a finite number of other types of clauses reflecting the moves of $U$, repeated in a regular way an exponential number of times (for each possible time-square combination of the computation of $U$ on $x$). In particular, it is very easy to see that, given two $c|x|$-bit integers, the indices of a literal and a clause, it can be determined in polynomial time whether the literal appears in the clause. Let us call $B$ the polynomial-time algorithm computing this literal-clause relation.
Though $B$ is polynomial-time, it's not polynomial-size because it contains an exponential amount of information, i.e., literal-clause pairs.
Then it continues:
Combining algorithms $A$ and $B$, we obtain an algorithm $C$ which, given two integers with $ck|x|$ bits, determines in polynomial time whether the two nodes are adjacent in the graph $G(F)$, based only on the bits of $x$. For fixed $x$ this algorithm can be rendered as a polynomial-size circuit $C_{G(F)}$ with $2ck|x|$ inputs, which is therefore a succinct representation of $G(F)$. Now, it is easy to see that, given $x$, we can construct $C_{G(F)}$ in polynomial time.
I don't understand how you can pack an exponential amount of information in "a polynomial-size circuit $C_{G(F)}$." Can you explain? This is the hardest part of the paper, or it is wrong.
