Periodic points in random substitution subshifts

We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions—random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible random substitutions to admit periodic points in their subshifts. In the case that the random substitution further has disjoint images and is of constant length, we provide a stronger criterion. A method is outlined for enumerating periodic points of any specified length in a random substitution subshift.

STUDIES ON PRACTICAL CRYPTOGRAPHIC SECURITY ANALYSIS FOR BLOCK CIPHERS WITH RANDOM SUBSTITUTIONS

In up-to-date information and communication systems (ICS) cryptography is used for ensuring data confidentiality. The symmetric block ciphers (BC) are implemented in different ICS including critical applications. Today theory of analysis and security verification of BC with fixed substitution nodes against linear and differential cryptanalysis (LDC) is developed. There are also BC with substitution nodes defined by round keys. Random substitution nodes improve security of ciphers and complicate its cryptanalysis. But through it all, quantitative assessment is an actual and not simple task as well as the derivation of formulas for practical security verification for BC with random substitution nodes against LDC. In this paper analytical upper bounds of parameters characterized practical security of BC with random substitution nodes against LDC were given. These assessments generalize known analogs on BC with random substitution nodes and give a possibility to verify security improving against LDC. By using the example of BC Kalyna-128, it was shown that the use of random substitution nodes allows improving upper bounds of linear and differential parameters average probabilities in 246 and 290 times respectively. The study is novel as it is one of the few in the cryptology field to calculate analytical upper bounds of BC practical security against LDC methods as well as to show and prove that using random substitutions allows improving upper bounds of linear and differential parameters. The security analysis using quantitative parameters gives possibility to evaluate various BCs or other cryptographic algorithms and their ability to provide necessary and sufficient security level in ICS. A future research study can be directed on improving analytical upper bounds for analyzed LDC in context to practical security against LDC, as well as practical cryptographic security assessment for other BC with random substitutions against LDC and other cryptanalysis methods including quantum cryptanalysis (Shor, Grover, Deutsch-Jozsa algorithms).

BLOCK SYMMETRIC CIPHER WITH RANDOM S-BOXES

This paper describes a new 256-bit block symmetric substitution-permutation cipher, called managed substitution cipher. This is a cipher with single-layer permutated transformations in each cycle. The management of substituted transformations occur by including them in a chain so that the output value of the previous SL transform is fed to the input of the current fortified 32-bit substituted transformation (SL conversion) along with the current value of the input data block through the adder for modulo 2. This enables to activate almost all S-blocks of the second cycle and subsequent cycles and eventually improve the dynamic indicators of the arrival of the cipher to the state of random substitution. The results of the evaluation of randomness indicators and the possibility of using random S-blocks are given. It is shown that such construction of cycled function allows us to use random substitutions without any selection in a cipher without reducing its strength.

Surface Localization in Impurity Band with Random Displacements and Long-Range Interactions

We study the random Schrödinger operators in a Euclidean space with the disorder generated by two complementary mechanisms: random substitution in a lower-dimensional layer and random displacements in the bulk, without additional assumptions regarding the reflection symmetry of the site potentials. The latter are assumed to be bounded and have a power-law decay. Complementing earlier results obtained in the strong disorder regime, we establish spectral and strong dynamical localization in the impurity zone near the bottom of spectrum for arbitrarily weak amplitudes of the random displacements, provided the concentration of impurities is sufficiently small.