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1

Random generation of combinatorial sets with special properties

Grebennik I., Lytvynenko O.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2016

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Vol. 5, No 4

43--48

EN

General approach for solving the problem of random generation of compositional k - images of combinatorial sets (k-sets) has been proposed. K-sets are powerful apparatus that can be applied for solving many scientific and applied problems. Though many literature is dedicated to the problem of generating combinatorial configurations, existing studies deals mostly with simple combinatorial configurations like combinations, permutations etc. The algorithms of generation both basic combinatorial sets and k-sets have been described. Algorithm for random generation of basic sets allows generating various combinatorial sets, and laws of constructing basic combinatorial sets can be pre-set. If identification of the laws fails, the algorithm allows using other algorithms to generate basic sets. Complexity of described algorithms has been evaluated. The complexity of the algorithm of generation k-sets is determined by the complexity of generation of basic sets, as well as the complexity of operations of nsubstitution and a number of levels of a certain k-set. The described approach to the random generation is very flexible since it allows obtaining various results by varying algorithm parameters. In its turn, it allows adjusting the number of elements for both basic sets and k-sets. The developed software allows solving the described problems of random generation of k -sets and basic combinatorial sets.

2

Random Generation of hv-Convex Polyominoes with Given Horizontal Projection

Hantos N., Balázs P.

Fundamenta Informaticae

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2014

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Vol. 135, nr 1/2

103--115

EN

We provide a quadratic-time algorithm for generating hv-convex polyominoes according to a given horizontal projection. The method can be used to generate hv-convex polyominoes with the prescribed projection and with a fixed or arbitrary horizontal dimension, from a uniform random distribution.

3

Random generation of Boolean functions with high degree of correlation immunity

Grocholewska-Czuryło A.

Journal of Telecommunications and Information Technology

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2006

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nr 3

14-18

EN

In recent years a cryptographic community is paying a lot of attention to the constructions of so called resilient functions for use mainly in stream cipher systems. Very little work however has been devoted to random generation of such functions. This paper tries to fill that gap and presents an algorithm that can generate at random highly nonlinear resilient functions. Generated functions are analyzed and compared to the results obtained from the best know constructions and some upper bounds on nonlinearity and resiliency. It is shown that randomly generated functions achieve in most cases results equal to the best known designs, while in other cases fall just behind such constructs. It is argued that the algorithm can perhaps be used to prove the existence of some resilient functions for which no mathematical prove has been given so far.

4

A study of differences between bent functions constructed using Rothaus method and randomly generated bent functions

Grocholewska-Czuryło A.

Journal of Telecommunications and Information Technology

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2003

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nr 4

19-24

EN

Bent functions, having the highest possible nonlinearity, are among the best candidates for construction of S-boxes. One problem with bent functions is the fact that they are hard to find among randomly generated set of Boolean functions already for 6 argument functions. There exist somealgorithms that allow for easy generation of bent functions.The major drawback of these algorithms is the fact that they rely on deterministic dependencies and are only able to generate bent functions belonging to one specific class. In our paper we present an efficient generator of random bent functions of more than 4 arguments. Resulting functions are not bounded by constraints described above. The generator operates in algebraic normal form domain (ANF). We also present our result on comparing the performance of S-boxes build using our bent function generator versus a standard method of bent function construction. We also give some directions for further research