Threshold-Secure Coding with Shared Key

Cryptographic protocols are often implemented at upper layers of communication networks, while error-correcting codes are employed at the physical layer. In this paper, we consider utilizing readily-available physical layer functions, such as encoders and decoders, together with shared keys to provide a threshold-type security scheme. To this end, the effect of physical layer communication is abstracted out and the channels between the legitimate parties, Alice and Bob, and the eaves-dropper Eve are assumed to be noiseless. We introduce a model for threshold-secure coding, where Alice and Bob communicate using a shared key in such a way that Eve does not get any information, in an information-theoretic sense, about the key as well as about any subset of the input symbols of size up to a certain threshold. Then, a framework is provided for constructing threshold-secure codes form linear block codes while characterizing the requirements to satisfy the reliability and security conditions. Moreover, we propose a threshold-secure coding scheme, based on Reed-Muller (RM) codes, that meets security and reliability conditions. Furthermore, it is shown that the encoder and the decoder of the scheme can be implemented efficiently with quasi-linear time complexity. In particular, a low-complexity successive cancellation decoder is shown for the RM-based scheme. Also, the scheme is flexible and can be adapted given any key length.