  We consider a Shannon cipher system for memoryless sources, in which
distortion is allowed at the legitimate decoder. The source is compressed using
a rate distortion code secured by a shared key, which satisfies a constraint on
the compression rate, as well as a constraint on the exponential rate of the
excess-distortion probability at the legitimate decoder. Secrecy is measured by
the exponential rate of the exiguous-distortion probability at the
eavesdropper, rather than by the traditional measure of equivocation. We define
the perfect secrecy exponent as the maximal exiguous-distortion exponent
achievable when the key rate is unlimited. Under limited key rate, we prove
that the maximal achievable exiguous-distortion exponent is equal to the
minimum between the average key rate and the perfect secrecy exponent, for a
fairly general class of variable key rate codes.
