Guesswork forms the mathematical framework for
quantifying computational security subject to brute-force determination
by query. In this paper, we consider guesswork
subject to a per-symbol Shannon entropy budget. We introduce
inscrutability rate to quantify the asymptotic difficulty of guessing
U out of V secret strings drawn from the string-source and
prove that the inscrutability rate of any string-source supported
on a finite alphabet X, if it exists, lies between the per-symbol
Shannon entropy constraint and log |X|. We show that for a
stationary string-source, the inscrutability rate of guessing any
fraction (1 - ϵ) of the V strings for any fixed ϵ > 0, as V
grows, approaches the per-symbol Shannon entropy constraint
(which is equal to the Shannon entropy rate for the stationary
string-source). This corresponds to the minimum inscrutability
rate among all string-sources with the same per-symbol Shannon
entropy. We further prove that the inscrutability rate of any
finite-order Markov string-source with hidden statistics remains
the same as the unhidden case, i.e., the asymptotic value of hiding
the statistics per each symbol is vanishing. On the other hand, we
show that there exists a string-source that achieves the upper limit
on the inscrutability rate, i.e., log |X|, under the same Shannon
entropy budget
