Neri Merhav — Page 9

arXiv2005

On the Shannon cipher system with a capacity-limited key-distribution channel

We consider the Shannon cipher system in a setting where the secret key is delivered to the legitimate receiver via a channel with limited capacity. For this setting, we characterize the achievable region in the space of three figures of merit: the security (measured in terms of the equivocation), the compressibility of the cryptogram, and the distortion associated with the reconstruction of the plaintext source. Although lossy reconstruction of the plaintext does not rule out the option that the (noisy) decryption key would differ, to a certain extent, from the encryption key, we show, nevertheless, that the best strategy is to strive for perfect match between the two keys, by applying reliable channel coding to the key bits, and to control the distortion solely via rate-distortion coding of the plaintext source before the encryption. In this sense, our result has a flavor similar to that of the classical source-channel separation theorem. Some variations and extensions of this model are discussed as well.

cs.IT↗

arXiv2005

On the Wyner-Ziv problem for individual sequences

We consider a variation of the Wyner-Ziv problem pertaining to lossy compression of individual sequences using finite-state encoders and decoders. There are two main results in this paper. The first characterizes the relationship between the performance of the best $M$-state encoder-decoder pair to that of the best block code of size $\ell$ for every input sequence, and shows that the loss of the latter relative to the former (in terms of both rate and distortion) never exceeds the order of $(\log M)/\ell$, independently of the input sequence. Thus, in the limit of large $M$, the best rate-distortion performance of every infinite source sequence can be approached universally by a sequence of block codes (which are also implementable by finite-state machines). While this result assumes an asymptotic regime where the number of states is fixed, and only the length $n$ of the input sequence grows without bound, we then consider the case where the number of states $M=M_n$ is allowed to grow concurrently with $n$. Our second result is then about the critical growth rate of $M_n$ such that the rate-distortion performance of $M_n$-state encoder-decoder pairs can still be matched by a universal code. We show that this critical growth rate of $M_n$ is linear in $n$.

cs.IT↗