We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.

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Compression is widely used in Internet applications to save communication time, bandwidth and storage. Recently invented by Jarek Duda asymmetric numeral system (ANS) offers an improved efficiency and a close to optimal compression. The ANS algorithm has been deployed by major IT companies such as Facebook, Google and Apple. Compression by itself does not provide any security (such as confidentiality or authentication of transmitted data). An obvious solution to this problem is an encryption of compressed bitstream. However, it requires two algorithms: one for compression and the other for encryption.In this work, we investigate natural properties of ANS that allow to incorporate authenticated encryption using as little cryptography as possible. We target low-level security communication and storage such as transmission of data from IoT devices/sensors. In particular, we propose three solutions for joint compression and encryption (compcrypt). The solutions offer different tradeoffs between security and efficiency assuming a slight compression deterioration. All of them use a pseudorandom bit generator (PRBG) based on lightweight stream ciphers. The first solution is close to original ANS and applies state jumps controlled by PRBG. The second one employs two copies of ANS, where compression is switched between the copies. The switch is controlled by a PRBG bit. The third compcrypt modifies the encoding function of ANS depending on PRBG bits. Security and efficiency of the proposed compcrypt algorithms are evaluated. The first compcrypt is the most efficient with a slight loss of compression quality. The second one consumes more storage but the loss of compression quality is negligible. The last compcrypt offers the best security but is the least efficient.