STUDY OF THE MINIVERSION PROPERTIES IN THE PSEUDO-RANDOM FUNCTION PCOLLAPSER

Abstract

The aim of the work is to evaluate the cryptographic properties of the pCollapser family of pseudo-
random functions (PRF) based on the study of the properties of its mini_pCollapser_12x12 miniversion
using fixed substitutions with extremely low cryptographic properties. As a comparison element,
we used a mini-version of a typical function based on an SP-net, containing a similar number of fixed
substitutions, and having a similar input/output dimension equal to 12 bits. To achieve this goal, the
following tasks were solved: – determination of the structure of the studied functions and the number of
rounds; – definition of a model for the formation of fixed substitutions with extremely low cryptographic
properties; – generation of sets of 6-bit fixed substitutions with extremely low cryptographic properties; – inclusion of the substitutions obtained into the functions under study and determination of the main
cryptographic properties of functions – the maximum dominance value for individual key values and the
maximum dominance value averaged over the entire set of keys, the maximum and averaged over the
entire set of keys value in the difference distribution table, algebraic degree and algebraic immunity;
– analysis of the obtained results. The paper presents two models for the formation of fixed substitutions
with extremely low cryptographic properties – based on the mixing of cell values in a pre-filled table
and based on the simplest ARX function (consisting of modulo addition, cyclic shift and XOR). The use
of fixed substitutions with extremely low non-linearity makes it possible to estimate how complex (nonlinear)
the function under study is and what minimum level of non-linearity is necessary to effectively
destroy the statistical dependencies between input/output data. In addition, it becomes clear that ARX
functions can be used as non-linear elements, which often have controversial and clearly low cryptographic
properties, but allow creating high-speed software and hardware implementations. It has been
determined that the PRF pCollapser mini-version, in contrast to the typical function based on the SP
network, makes it possible to obtain a high-quality non-linear function from the set of ARX-functions
with extremely low cryptographic properties, given that no other non-linear elements are presented in
pCollapser. The obtained results reflect the existence of a fundamental difference between the
pCollapser PRF and a typical SP-network based PRF and confirm the correctness of the concept of
PD-sbox pseudo-dynamic substitutions and the pCollapser function consisting of them as a whole.