Browsing Computer Science by Author "Carić, Marko"
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Carić, Marko (Beograd , 2023)[more][less]
Abstract: In this dissertation, the problem of calculating the number of equiva- lence classes of Boolean functions is discussed. The difficulty of determining the number of equivalence classes increases sharply with the number of variables n. The motivation for choosing this topic lies in the fact that concrete numbers have been known so far only for relatively small values of n, although the problem itself was theoretically solved a long time ago. Let G be the group of permutations of the set Bn = {0, 1}n. The effect of the group G on scalar, Bn 7 → B1, that is, vectorial invertible Boolean functions, Bn 7 → Bn. Two scalar Boolean functions f (x) and g(x), defined on Bn, are considered equivalent with respect to the group G, i.e. f ∼ g, if for some σ ∈ G for every x ∈ Bn f (x) = g(σ(x)) holds. Two vector invertible Boolean functions f (x) and g(x), are considered equivalent with respect to the group G, i.e. f ∼ g, if for some pair (σ, ρ) ∈ G × G for each x ∈ Bn holds g(x) = ρ(f (σ(x))). The equivalence relation ∼ decomposes the set of all Boolean functions into equivalence classes. Equivalence of Boolean functions has significant applications in the logical synthesis of combinatorial circuits and in cryptography, especially in connection with the design of S-boxes. Let Un(G) and Vn(G) denote number of equivalence classes of scalar, i.e. vector invertible Boolean functions of n variables in relation to the group G. The numbers Un(G) and Vn(G) can be calculated relatively simply if the cycle index of the group G is known. The dissertation considers four groups G of permutations of the set Bn: • group S′ n induced by group Sn permutations of coordinates elements x = (x1, x2, . . . , xn) ∈ Bn, • group Gn, induced by permutations and complementations of coordinates, • group of GLn linear invertible transformations elements of the vector space Bn, i • group of AGLn affine invertible transformations elements Bn. If the permutation σ ∈ G has ik cycles of length k ⩾ 1, its cycle structure is i(σ) = (i1, i2, . . .). The cyclic index of the group G is the generatrix ZG(f1, f2, . . .) = 1 |G| X σ∈G Y k⩾1 f ik k of cycle structures of all permutations σ ∈ G. General expressions for cycle indices the four considered groups are known, but the cycle indices themselves, i.e. the numbers Un(G) and Vn(G), are practically calculated only for relatively small values, for e.g. n ⩽ 10. The dissertation presents original results in the field of enumeration of equiv- alence classes of Boolean functions in relation to these four groups of transfor- mations. A similar expression was derived for all four groups of transformations for the cycle index in the form of sum over partitions of the number n. Based on that expression and previously calculated tables, the cycle index is calculated much more efficiently. An overview of known results for relatively small n and new results in the thesis for larger n is shown in the following table: Number\ G S′ n Gn GLn AGLn Un(G) 11 → 33 10 → 32 8 → 31 10 → 31 Vn(G) 6 → 30 7 → 27 6 → 26 6 → 26 Specially, in the case of the permutation group S′ n, an effective direct procedure for calculating the number of equivalence classes that does not use a cycle index is shown, and is described in the third paper from the introductory chapter. The second part of the dissertation concerns monotone Boolean functions — scalar Boolean functions which satisfy the monotonicity condition (from x ⩽ y follows f (x) ⩽ f (y)). Let rn, i.e. dn (the n-th Dedekind number), denote the number of equivalence classes of monotone Boolean functions in relation to the group S′ n, that is, the total number of monotone Boolean functions of n variables. The difficulty of calculating the number rn increases rapidly with n, so that until recently the last calculated member of the sequence was r7. The procedure described in the dissertation is based on the Frobenius theorem, by which it was determined number r8. In doing so, the known value of the number d8 is used. The dissertation consists of the first - introductory chapter and the following three chapters. In the second chapter, theoretical terms related to the material from chapters 3 and 4 are introduced, and they refer to discrete mathematics, combinatorics and cycle indices of the considered four groups of transformations. Chapter 3 describes the procedure for calculating the cycle indices for the four considered groups of permutations, as well as numbers Un(G) and Vn(G) equivalence classes of Boolean functions in relation to these groups. First, common improvements for all four groups are considered, and then specific accelerations related to individual groups. These results are published in the second paper listed in the introductory chapter. In chapter 4, the problem of finding the number of equivalence classes of monotone Boolean functions is solved. First, a general expression for calculating the number rn is given based on the Frobenius theorem in the form of the sum (by partitions of the number n) of the number of fixed points of the permutation corresponding to the partition. After that, depending on the graphs corresponding to different partitions, different ways of calculating the number of fixed points for n ⩽ 8 are shown. The procedure based on which the number r8 was calculated, which also represents the original contribution of this dissertation is presented - see the first paper from the list from the introductory chapter. Applying a similar procedure, Pawelski [31] calculated r8 practically at the same time as the obtained result described in the dissertation. URI: http://hdl.handle.net/123456789/5792 Files in this item: 1
Disertacija_15671.pdf ( 2.414Mb ) -
Carić, Marko (Beograd , 2023)[more][less]
Abstract: This dissertation discusses the problem of calculating the number of equivalent classes of a Boolean function. The difficulty of determining the number of equivalence classes increases sharply with the number of variables sn. The motivation for choosing this topic lies in the fact that concrete numbers have been known so far only for relatively small values of n, although the problem itself has long been theoretically solved. Let G be the permutation group of the set Bn={0,1}n. Effect of Gonscalar group, Bn→B1, that is, vector invertible Boolean functions, Bn→Bn. Two scalar Boolean functions f(k)ig(k), defined by Bn, are considered equivalent. ∈G forever ∈Bnf(k )=g(σ(k)) holds. Two vector invertible Boolean functions f(k) and g(k) are considered equivalent in relation to the group G, i.e. ))).The equivalence relation∼decomposes the sets of all Boolean functions into equivalence classes.The equivalence of Boolean functions confers significant applications in the logical synthesis of combinatorial circuits and cryptography, especially in connection with the design of S-boxes. Let Un(G) and Vn(G) denote the number of scalar equivalence classes, ie. vector invertible Boolean functions of n variables with respect to the group G. The numbers Un(G) and Vn(G) can be calculated relatively simply if the cycle index from the group G to the G group is known. Induced by the group Snpermutations of coordinate elements step = (k1,k2,...,kn)∈Bn, •group Gn, induced by permutations and additions of coordinates, •group GLnlinear invertible transformations of elements of vector space Bn, and •group AGLof invertible transformations of elements Bn. If the permutation σ∈Ghasikcycles of lengthk1, its cycle structure is(σ)=(i1,i2,...).The cyclic index of the groupGisgeneratrix ZG(f1,f2,...)=1 |G|σ∈Gk1 fik k structure cycles σene structures of cycles σfraconstructionsGsides ∗pressuresGpermutations of the red group are known, but the cycle indexes itself, i.e. numbers Un(G) and Vn(G), are practically calculated only for relatively small values, for example n10. The dissertation presents original results in the field of enumeration of equivalence classes of Boolean functions in relation to these four groups of soft transformations. A similar expression is derived for all four groups of soft transformations for the cycle index in the form of the sumover partition softhennumbern. Based on the precyclical index much more. An overview of known results for relatively small and new results in the thesis for larger ones are shown in the following table: Number\GSnGnGLnAGLn Un(G)11→3310→328→3110→31 Vn(G)6→307→276→266, participates in direct effect, by special effect 26 latingthenumberofequivalenceclassesthatdoesnotuseacycleindex is shown and described in the third paper from the introductory chapter. These second parts of the dissertation refer to monotone Boolean functions — scalar Boolean functions that satisfy the condition of monotonicity (from which follows f(k)f(i)). Letrn, i.e. monotoneBooleanfunctionsofnvariables. The difficulty of calculating the number rn increases rapidly with n, so that until recently the last term of the sequence to be calculated was r7. The procedure described in the dissertation is based on Frobenius' theorem, on the basis of which the number r8 was determined. The dissertation consists of the first introductory chapter and the following three chapters. In the second chapter, theoretical concepts related to the material from chapters 3 and 4 are introduced, and they relate to discrete mathematics, combinatorics and the index of the cycle considered by the soft cycle of the 3rd form for describing the 3rd cycle. indices for the four considered groups of permutations, as well as the numbers Un(G) and Vn(G) of the equivalent class of the Boolean function with respect to these groups. First, the common improvements for all four groups are discussed, and then the specific accelerations related to individual groups are published by group. ter. In Chapter 4, the problem of finding the number of equivalence classes of a monotone Boolean function is solved. First, a general expression for calculating a number is given based on the Frobenius theorem of the form of the sum (by the soft number partition of the number) of the number from the fixed part of the graph corresponding to the point corresponding to the point. In different partitions, different ways of calculating the number of non-fixed points for n8 are shown. The procedure based on which the number r8 was calculated, which also represents the original contribution of this dissertation, is shown in the first paper from the list from pp. 1 to 3. practically at the same time the obtained result described in the dissertation. URI: http://hdl.handle.net/123456789/5571 Files in this item: 1
MarkoCaricDisertacija.pdf ( 1.467Mb )
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