A large deviations approach to secure lossy compression

A Shannon cipher system for memoryless sources is considered, 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. The perfect secrecy exponent is defined as the maximal exiguous-distortion exponent achievable when the key rate is unlimited. Under limited key rate, it is proved 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.

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