  We use Hidden Markov Models to motivate a quantitative compositional
semantics for noninterference-based security with iteration, including a
refinement- or "implements" relation that compares two programs with respect to
their information leakage; and we propose a program algebra for source-level
reasoning about such programs, in particular as a means of establishing that an
"implementation" program leaks no more than its "specification" program.
<p>This joins two themes: we extend our earlier work, having iteration but only
qualitative, by making it quantitative; and we extend our earlier quantitative
work by including iteration. <p>We advocate stepwise refinement and
source-level program algebra, both as conceptual reasoning tools and as targets
for automated assistance. A selection of algebraic laws is given to support
this view in the case of quantitative noninterference; and it is demonstrated
on a simple iterated password-guessing attack.
