Algorithm of Superposition of Boolean Functions Given with Truth Vectors

Transcription

1 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: wwwijcsiorg 9 Algorithm of Superpositio of Boolea Fuctios Give with Truth Vectors Aatoly Plotikov, Aleader Petrov, Ato Petrov 3 Departmet of Computer Systems ad Networks, Volodymyr Dalh East-Ukraiia Natioal Uiversity Luhask, 934, Ukraie Departmet Applied Iformatics, AGH Uiversity of Sciece ad Techology Krakow, Polad Departmet of Computer Systems ad Networks, the Dalh East-Ukraiia atioal uiversity 3 Departmet of Computer Systems ad Networks, Volodymyr Dalh East-Ukraiia Natioal Uiversity Luhask, 934, Ukraie Abstract I this paper are eamied the practical problems of costructio of arbitrary superpositio of Boolea fuctios whe all fuctios are give with truth vectors Keywords: Boolea Fuctio, True Vector, Truth Table, Superpositio Problem statemet Let there be a set E = {;} The the mappig of f: E E is called a Boolea fuctio A Boolea fuctio f (,,, E ( i E, i =,,, ca be completely defied with truth table: Table : Truth table f (,,,, f(,,,, f(,,,, f(,,,, f(,,,, f(,,,, f(,,,, The first colums of this table cotai the leicographically ordered value sets of Boolea variables ad the last colum of this table is the value of the give fuctio of every set This last colum is called truth vector of the Boolea fuctio f (,,, It is obvious that it is ot ecessary to write all sets of values of Boolea variables for determiig the Boolea fuctio It is eough to submit the truth row that correspods with it Eample Let the Boolea fuctio of three variables be represeted with the truth vector: ( The we ca write the appropriate truth table: 3 f (,, 3 Let there be a system of Boolea fuctios: f (,,, f(,,, m f(,,, m f(,,, m ( The the fuctio f ( f (,,,, f (,,,,, f (,,, m m m F(,,, m is called a superpositio of fuctios of the system ( I simple terms superpositio is the costructio of ew fuctio by meas of replacemet of some variables of the iitial fuctio by the appropriate fuctios Copyright (c Iteratioal Joural of Computer Sciece Issues All Rights Reserved

2 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: wwwijcsiorg The fuctio f is called basic fuctio ad the fuctio f k is called substitutio fuctio Suppose we wat to perform superpositio of Boolea fuctios Such problem arises, for eample, while aalyzig cryptosystems (see [,, 5] ad Boolea multipoles [3, 4] To cosider the practical problems of costructio of superpositio of Boolea fuctios, we assume that every Boolea fuctio of the system ( is give with truth vector Thus, the purpose of this paper is to develop a algorithm of costructio of truth vector of fuctio, which is the result of superpositio of fuctios, which are also give with truth vectors Let us cosider reasoability of represetatio of a Boolea fuctio as a truth vector There are differet forms of represetatio of Boolea fuctios [3] Thus, the tabular represetatio i form of S-blocks is used to defie the system of Boolea fuctios i cryptosystems This represetatio is absolutely icoveiet for aalysis whe it is ecessary to retrace the chages of every bit i process of cipherig The other most popular form of represetatio of Boolea fuctios is represetatio of fuctio i disjuctive ormal form (DNF Let oly oe bit be used for represetatio of every literal (complemeted or ucomplemeted variable of DNF The for represetatio of every elemetary cojuctio of DNF bits are required Assumed that the fuctio f takes the uit value i half of sets, which occurs i cryptosystems, we come to coclusio that it is required to eped bits to represet the fuctio At the same time it is required bits to represet Boolea fuctio with truth vector It is obvious that if 3 the > =, thus the represetatio of fuctio with truth vector is more efficiet I geeral, other forms of represetatio of Boolea fuctio are ot so efficiet as a truth vector Essetial ad fictitious variables Let us cosider the cocepts of essetial ad fictitious variables of the fuctio f Let there be the Boolea fuctio f (,,, Let us desigate the set of values of variables,,, of the fuctio f as (,,, It is said that the fuctio f heavily depeds upo the variable k (k =,,, if two sets (,, k,, k,, (,, k,, k,, ca eist ad f( f( Such variable is called essetial variable Otherwise the variable k is called fictitious variable It is easy to see that if the variable k i fuctio f is fictitious, the truth table of this fuctio ad, accordigly, its truth vector ca be halved by deletig, for eample, all sets, i which the variable k = O the cotrary, if it is ecessary to eter the fictitious variable i fuctio f (,,,, the its truth table will double ( i, i,, i Let there be the Boolea fuctio which is give with truth vector Let us cosider the procedure of eterig of the fictitious variable k ( k{, i i,, i } ito the fuctio As a result, we obtai truth vector of the f ( i,,, i i Boolea fuctio which is equivalet to the fuctio We will cosider the truth vector of the fuctio as array with legth ad the truth vector of the fuctio f as array F with legth Theorem Let the Boolea fuctio, which depeds upo variables, is give with the array If the fictitious variable k, which is etered ito, takes k- place i the set of variables of the fuctio f ( i,,, i i (if couted from left to right, the it is ecessary to copy k cells of the array i cosecutive order to costruct the truth vector of the fuctio f (the array F ad doubly place the same value i the cells of the array F i cosecutive order Copyright (c Iteratioal Joural of Computer Sciece Issues All Rights Reserved

3 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: wwwijcsiorg Provig I fact, the procedure of costructio of the array F depeds upo the umber k of the iput fictitious variable k Let this umber take k-place (if couted from left to right i set of values of variables of the fuctio f ( i,,, i i Assumig that the sets of values + of variables of the fuctio f are ordered leicographically, we ca see that the variable k i the truth table f possess the same value i successio i k sets (k =,,, + This proves Theorem QED Thus, the procedure of eterig of oe fictitious variable i the Boolea fuctio, which is give with truth vector, is completely defied with the Theorem Eample Let it be ecessary to eter the fictitious ( variable ito the fuctio, 3 = ( = Thus, =, k = It is easy to see that the truth table of the f (,, 3 fuctio will be the followig: 3 f 3 (,, I this case it is obviously that it is ecessary to copy k = = cells of the array successively ad doubly place them successively ito the array F It is clear, that if it is ecessary to eter several fictitious variables ito the fuctio, it ca be doe by meas of eterig fictitious variables oe by oe ito the iitial fuctio, ad the i every fuctio, which would obtaied at the previous stage Theorem Let there be the fuctio (,,,, which is defied with truth vector, ad p fictitious variables should be etered ito it The the time of f( costructio of the fuctio,,, p is equal to p Provig Let us defie the time required for eterig of fictitious variable ito the fuctio I accordace with Theorem it is ecessary to copy every k bits of truth vector ad doubly place them ito the truth vector of the fuctio f(,,, The total umber of the bits, placed ito the truth vector of the fuctio f, obviously is equal to, ad the total umber of copied bits is equal to Cosequetly, the time of eterig of the first fictitious variable is equal to O( +O( = O( It is easy to see that the time of eterig of the secod fictitious variable is equal to O( Cosequetly, the time required for eterig of the first two fictitious variables will be equal to O( + O( = O( Similarly we come to coclusio that it is ecessary to eped p time uits for eterig of p fictitious variables ito the fuctio QED 3 Algorithm of costructio of superpositio Thus, let there be a system of Boolea fuctios ( Let us say, that X {,, } is the set of variables of the basic fuctio f, ad Xi {,, m} is the set of variables of substitutio fuctio f i (i =,,, Further, we suppose that X i X ( i It is clear that, i geeral, the relatio ( may be ot satisfied Therefore we suppose that the ecessary fictitious variables are beforehad etered ito the basic fuctio We shall carry out the costructio of truth vector f of the resultig fuctio r bit by bit The umber of the bit, which is costructed, will determie couter cotet (, which has positios The couter cotet is ( (,,, writte as sequece I fact, the couter cotet determies the set of variables of the Boolea fuctio f Copyright (c Iteratioal Joural of Computer Sciece Issues All Rights Reserved

4 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: wwwijcsiorg Firstly let us cosider the substitutio of the fuctio f(,, m ito the basic fuctio f istead of the variable k (k =,,,, m We suppose that we have three arrays: F, F ad F r, which represet the truth vectors of the basic fuctio, the substitutio fuctio ad the resultig fuctio respectively, ad three couters C(, ( C(m ad Let us write C( =, whe the couter cotet is zero, ad C( := C(+, whe the couter cotet is icreased by Fially, we will desigate the cotet of the array F r ( with umber (address as Fr( ( Algorithm A Step Fi ( : Step Copy to couter C(m those positios of the couter ( which correspod to variables of the fuctio f Also, copy all positios of the couter ( to C( Step 3 Fid the value of F ( C( m i the array F Step 4 Substitute the value F ( C( m for the k th positio of the couter C( Step 5 Fid the value of FC ( ( i the array F Step 6 Write Fr( ( F( C(, that is, write the value of the bit with umber ( i the array F r Step 7 Fi ( : ( Step 8 If C ( r the go to Step Step 9 The array F r is completely formed Eample 3 Let it be ecessary to substitute the Boolea f(,, 3 fuctio = ( for the variable i (, 3, 4 the fuctio = ( To satisfy the relatio ( we eter the fictitious variable ito the basic fuctio Usig the procedure, determied by Theorem, we obtai ew epressio for f (,, 3, 4 the basic fuctio: = ( Now we ca use the algorithm A to costruct the truth fr vector of the resultig fuctio : ( C (4 Step Fi = r := Step Have: Cm ( = C(3 :=, C(4 := Step 3 Fid F ( C( m = f ( := Step 4 Obtai C( = C(4 := 8 (high-order positio ( of the couter chages by Step 5 Fid FC ( ( = FC ( (4:= Step 6 Write Fr( ( = F r ( := C (4 Step 7 Fi r := + = C (4 4 Step 8 As r = < the go to Step Step Have: C(3 :=, C(4 := Step 3 Fid F ( := Step 4 Obtai C(4 := Step 5 Fid f( := f ( Step 6 Write r := C (4 Step 7 Fi r := + = C (4 4 Step 8 As r = < the go to Step Step Have: C(3 :=, C(4 := Step 3 Fid F ( := Step 4 Obtai C(4 := Step 5 Fid F ( := Step 6 Write F r ( := ad so o Cosequetly we obtai the truth vector of the fr (,, 3, 4 resultig fuctio = ( Theorem 3 The algorithm A fids the truth vector of the resultig fuctio Provig Let there be the basic fuctio f (,,, ad the substitutio fuctio f(,, m (m We ca suppose without loss of geerality that the first m variables of the fuctios f ad f coicide ad the fuctio f is substituted for the variable Suppose that it is ecessary to fid the value of the resultig Boolea fuctio fr( f,,, i the set (,,, Let (,,, m be the subset of the first m values of variables Accordig to the algorithm A the value of the substitutio fuctio f ( should be foud at Step 3 It is obvious that the value of the resultig fuctio i the set will deped upo which value the fuctio f takes i the set As this fuctio replaces the variable, the resultig fuctio f r takes the same value i the set as the fuctio f i the set ( f(,,, Copyright (c Iteratioal Joural of Computer Sciece Issues All Rights Reserved

5 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: wwwijcsiorg 3 which should be foud at Step 5 of the algorithm A QED Theorem 4 The ruig time of algorithm A is equal to O(( m Provig I fact, carryig out of Step takes uits of time Carryig out of Step takes, obviously, m uits of time Each of Steps 3 6 takes oe uit of time I the geeral case, Step 7, as Step, takes uits of time Step 8 also takes oe uit of time Thus, o-time eecutio of Steps 8 takes m++5 uits of time, therefore, it is ecessary to sped (m = O(( m uits of time for costructio of the truth vector of the resultig fuctio QED Refereces [] AD Plotikov, Logical cryptoaalysis o the eample of the cryptosystem DES [] Data Ecryptio Stadard (DES Federal Iformatio Processig Stadards Publicatio (FIPS PUB 46-3 Natioal Bureau of Stadards, Gaithersburg, MD (999 [3] AD Zakrevskij, Solvig large systems of Boolea equatios Iformatics, 4, 4 pp 4 53 [4] S Rudeau, Boolea fuctios ad equatios Amsterdam- Lodo-New York: North- Hollad ad America Elsevier, 974 [5] JA Clark,, J L Jacob, S Maitra, ad P Staica, Almost Boolea Fuctios: the Desig of Boolea Fuctios by Spectral Iversio Computatioal Itelligece, Volume, Number 3, 4 pp Aatoly Plotikov has got his PhD i 969, He is a professor of Volodymyr Dalh East-Ukraiia Natioal Uiversity, Luhask He is also a associate member of AMS (USA ad a reviewer of the joural Mathematical Reviews His scietific iterests iclude algorithms, combiatorial optimizatio, compleity theory Aleader Petrov is a professor of Departmet Applied Iformatics, AGH Uiversity of Sciece ad Techology, Krakow, Polad Ato Petrov is a professor of Volodymyr Dalh East-Ukraiia Natioal Uiversity, Luhask His scietific iterests iclude algorithms ad Boolea fuctios Copyright (c Iteratioal Joural of Computer Sciece Issues All Rights Reserved