  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, we first consider a
scenario where the effect of the physical layer is omitted and all the channels
between the involved parties are assumed to be noiseless. We introduce a model
for threshold-secure coding, where the legitimate parties communicate using a
shared key such that an eavesdropper 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 from 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. It is
shown that the encoder and the decoder of the scheme can be implemented
efficiently with quasi-linear time complexity. In particular, a successive
cancellation decoder is shown for the RM-based coding scheme. Then we extend
the setup to the scenario where the channel between the legitimate parties is
no longer noiseless. The reliability condition for noisy channels is then
modified accordingly, and a method is described to construct codes attaining
threshold security as well as desired reliability. Also, we propose a coding
scheme based on RM codes for threshold security and robustness designed for
binary erasure channels along with a unified successive cancellation decoder.
The proposed threshold-secure coding schemes are flexible and can be adapted
for different key lengths.
