Journal of Theoretical and Applied Information Technology 10 th January Vol. 47 No JATIT & LLS. All rights reserved.

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1 Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: E-ISSN: RESEARCH ON PROPERTIES OF E-PARTIAL DERIVATIVE OF LOGIC FUNCTIONS WANG FANG Dpt. o Computr and Inormaton Tchnology Zhang Changzhng Vocatonal and Tchncal Collg Hangzhou 33;.Dpt. o Inormaton and Elctroncs Engnrng Zhang Unvrsty Hangzhou 38 E-mal: @qq.com ABSTRACT E-drvatv [] has accssd a wd rang o applcatons n ths aras and causd acadma concrnd [-4] such as dtctng aults n combnatonal crcuts dscussng on cryptographc proprts o H- Boolan uncton and rvalng th ntrnal structur o Boolan uncton togthr wth Boolan drvatv thus mor ctvly analyzng th proprty o Boolan unctons. Rrrng to th dscusson on Boolan drvatv and Boolan partal drvatv th concpt o -partal drvatv s prsntd. Th dntons and proprts o -partal drvatv and hgh ordr -partal drvatv ar gvn. W also gv thr proos or som proprts. Th work mad n ths papr s th complmnt and mprovmnt o th rsarch on th -drvatv o logc unctons. y words: Logc Functon; -Drvatv; -Partal Drvatv; Spcal Opraton O Logc Functon. INTRODUCTIONS Logc uncton - drvatv s a nw spcal opraton. Snc th documnt [] puts orward - drvatv du to th unqu charactrstcs o - drvatv t has accssd a wd rang o applcatons n ths aras and causd acadma concrnd [-4] such as dtctng aults n combnatonal crcuts dscussng on cryptographc proprts o H-Boolan uncton and rvalng th ntrnal structur o Boolan uncton togthr wth Boolan drvatv thus mor ctvly analyzng th proprty o Boolan unctons. Howvr th documnt [-4] smply dscusss dnton proprts and applcaton o thrst ordr - drvatv o logc uncton th hgh ordr - drvatv and -partal drvatv s a lack o rsarch. Rrrng to th dscusson o Boolan drvatv and Boolan partal drvatv o logc uncton ths artcl wll study th logc uncton - partal drvatv and ts proprts and proo o som proprts ar gvn out thus maks urthr complmnt and prcton o -drvatv study n ordr to promot rsarch on spcal opratons.. DEFINITION AND RELEVANT PROPERTY OF - PARTIAL DERIVATIVE Documnt [] ntroducs dntons and proprts o -drvatv and ths artcl ust taks som rlvant proprts or ampl. Dnton st ( n ) as varabl n ully dnd logc unctons dnton o -drvatv to th varabls ( ) n s: ( n) ( n) (.) Proprty. (.) ( ) ( ) n Proprty (.3 ) Proprty.3 ( ) n g g (.4)

2 Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: E-ISSN: Proprty.4 d d (.5) Formula d s th Boolan Drnc rst ordr d -partal drvatv. Proprty.5 ( g) ( g) (.6) Formula mans XOR(clusv OR) Proprty.6 d( g) dg (.7) d d Proprty.7 IF has no rlatonshp wth (.8)Proprty.8 IF varabl (.9) s th lnar varabl o Proprty.9 IF d d (.) Proprty. IF d d (.) 3. DEFINITION AND RELEVANT PROPERTY OF -PARTIAL DERIVATIVE Dnton st ( n ) as varabl n ully dnd logc unctons rst ordr -partal drvatv to th varabls s -drvatv. Dnton 3 st ( n ) as varabl n ully dnd logc unctons dnton o scond ordr - partal drvatv to th varabls s: (.) Dnton 4 st ( n ) as varabl n ully dnd logc unctons dnton o ordr - partal drvatv to th varabls s: (.) Many proprts o th -partal drvatvs can b drctly dducd accordng to th dnton o -partal drvatvs and rlatd proprts n hr only gvs proo o som proprts. Proprty. Dducton. Proprty. Dducton. Proprty.3 (.3) (.4) (.5) ( ) ( ) ( ) ( ) (.6) (.7) ( ( ) ( ) ) ( ) ( ) ( ) ( ) Dducton.3 ( ) ( ) ( ) ( ) Proprty.4 Formula drvatv ( ) ( ) (.8) (.9) s th scond ordr -

3 Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: E-ISSN: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Dducton.4 k ( ) ( ) ( ) ( ) Proprty.5 Formula Partal drvatv (.) (.) s th Boolan scond ordr - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Dducton.5 Proprty.6 Dducton.6 k (.) (.3) (.4) Proprty.7 IF a ( constant ) a (.5) Dducton.7 IF a ( constant ) Proprty.8 IF a (.6) (.7) Dducton.8 Proprty.9 ( ) g g (.8) (.9) Dducton.9 ( g) g (.) Proprty. ( g) g (.) g ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ g g ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g( ) g g [ g ( ) g( ) g ] Dducton. g ( g) g Proprty. ( + g) ] (.) (.3) 3

4 Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: E-ISSN: ( g) + ( ) ( ) ( ) ( ) ( g g ) ( ) ( ) ( ) ( ) [ g g ( ) ( ) ( ) ( ) ( ) g( ) ( ) g( ) ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g( ) ( ) g( ) ( ) ( ) ( ) ( ) ( ) g( ) ( ) g( )] [ g g g g Dducton. ( + g) wth (.4) Proprty. IF ( n ) has no rlatonshp wth (.5) Dducton. IF ( n ) has no rlatonshp (.6) s th lnar Proprty.3 IF on o varabl (.7) St as lnar varabl ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Dducton.3 IF on o s th lnar varabl Proprty.4 IF (.8) (.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) So ( ) ( ) ( ) ( ) Dducton.4 IF Proprty.5 IF (.3) (.3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4

5 Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: E-ISSN: Dducton.5 IF Proprty.6 IF ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Dducton.6 4. CONCLUSIONS IF (.3) (.33) (.34) ( ) Ths artcl has studd th logc uncton -partal drvatv and ts proprts and proo o som proprts ar gvn out thus maks urthr complmnt and prcton o -drvatv study and promotd rsarch on spcal opratons. Obvously -drvatv dnd n documnt [] s th spcal cas o -partal drvatv whn. Proprts o -partal drvatv n ths papr ar all sutabl to -drvatv. Howvr som o th proprts o -drvatv ar not applcabl to th hgh ordr -partal drvatv. For ampl: + d d drvatv and Proprts and.6 plans rlatonshp btwn hghr ordr -partal drvatv and Boolan partal drvatv. (3) Introducton o -partal drvatv mprovs rsarch on th drvatv hlps to rval th proprty o Boolan unctons. Applcaton domans lk -drvatv hgh ordr -drvatv and - partal drvatv nd to burthr wdnd. REFERENCES [] LI W W WANG z. Th drvatv o Boolan unctons and ts applcaton n thault dtcton and cryptographc systm [J]. ybrnts 837(): [] LI W W WANG z.zhang z J. Th applcaton o drvatv and -drvatv on H-Boolan unctons [J]. CHINA SCI-TEC 8():67-7. [3] DING Y JWANG z YE J H. Intal-valu problm o th Boolan uncton s prmary uncton and ts applcaton n cryptographc systm [J]. ybrnts 39(6):9-96. [4] HE Lang WANG zhuo LI W-w. Algorthm o rducng th balancd H-Boolan uncton corrlaton-masur and rsarch on corrlatv ssu[j]. Journal o communcatons 3(): [5] OU Ha-wnZHANG Yu-uan. On algbrac mmunty o a class o spcal Boolan unctons[j].applcaton Rsarch o Computrs. [6]LIU Nan-nanZHAO Fng.Th rlatonshp btwn th algbrac mmunty and nonlnarty o Boolan unctons[j].huab coal ndustry tachrs Collg (Natural Scnc). [7] LIU Guan-shng LIAN Y-qun CHEN Xong. Tabular mthod o calculatng Boolan partal drvatv and drnc o th OC typ logc uncton[j]. Journal o zh ang Unvrsty: Scnc Edton 734(): () Proprty.4 plans rlatonshp btwn hgh ordr -partal drvatv and hgh ordr 5