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.