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??? ????? ??????? ???N? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? c????Z? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? {...}? ??? ???????E={a,b,c,d}? x????? ??? ?? ??????? ??E?? ????x /?E ????5/?E? :p?Z,q?Z?} ????? ??P? ??????? ?? ????? ?? ?????? ??P VF FV VFV FVF ??????? ?????x?R,x2≥0? ????? ?????? ?? ?????x?E??? ??????P(x)? ??????? ?????x?R:x2= 2? ???? ????? ?????? ?? ????x??? ???P(x)? ??????? ?????!x?R+:x2= 2? ??????? ?????x?R,?y?R:x < y??? ?????? ????? ????y?R:?x?R,x < y??? ??????? 24)
? ?PetQ≡QetP ? ?(PetQ)etR≡Pet(QetR) ? ?PetP≡P?
P??Q??? ????? ???PouQ???
? ?PouQ≡QouP? ? ?(PouQ)ouR≡Pou(QouR)? ? ?PouP≡P
PouQ≡PetQ
?????? ???P?Q??? ? ?P?Q≡PouQ? ? ?(P?Q)≡PetQ ????? ??? ???????(PetQ)?R?(P?Q)?P? ????
Q??? ??????
?x2+x≥1 + 1 = 2? ????x2+x-2≥0 ????x2= (2k+ 1)2= 4k2+ 4k+ 1 = 2k?+ 1? ??k?= 2k2+ 2k? ????x2??? ??????? ???x?Q?x2?Q? ???P??? ??????? ????P??? ?????? ????? ? ???? ???⎷2/?Q?? ?????? ?? ?????? ????x:P(x)????? ??? ??????? ?? ???n≥n0,P(n)?P(n+ 1)? ??????n≥n0?P(n)??? ??????
P(n) :?n?N,Sn=n(n+ 1)2
S n+1=Sn+n+1 =n(n+1)2 +n+1 =n(n+1)+2(n+1)2 =(n+1)(n+2)2 ? ????P(n+1)??? ?????? ?? ?? ?????? ??????? ??? ????? ???? ????n? 2 n+1≥n2+ 2n+ 1 = (n+ 1)2? ??????n≥n0,P(n)??? ?????? ????? ?? ??? ?? ???? ????n?
E?F? ??
?x, x?E?x?F
E??F? ?x?E:x /?F
??????? ?????E={0,1,2}, F={1,2,3}, G={0,1,2,4}?? ?E?G????E??F? (E=F)?(E?F??F?E) ????{x,y}? ?????? ????? ??x??y? ??????? ???????E={a,b,c}? ?????P(E) ={∅,{a},{b},{c},{a,b},{a,c},{b,c},E}? ?? ??????? ???E??F?E?F={x:x?E??x?F}?????E?????F?? ? ?A?A=A?A?B=B?A?A?(B?C) = (A?B)?C)? ?? ?A∩A=A?A∩B=B∩A?A∩(B∩C) = (A∩B)∩C)? ??? ?A∩(B?C) = (A∩B)?(A∩C)?A?(B∩C) = (A?B)∩(A?C)? ? ?E\F=E∩F? ? ?A?B=A∩B? ?? ?A∩B=A?B?
EΔF= (E?F)\(E∩F)?
?? ??????? ??E? ?A ? P(E)? ????? ??? ? A? A A A A (x,y) = (x?,y?)?x=x???y=y? ??? ???????(x,y)???? ???x?E??y?F?
E×F={(x,y) :x?E??y?F}
??E=F?E×E??? ???? ???????E2? ??????? ????E={0,1,2,3}?F={a,b,c}?G={(0,b),(0,c),(2,a),(3,a)}? ?x?E,?!y?F: (x,y)? G? ???? ???? ??????? ??E??????? ??? ?? ??? ????? ??????
1|x-1|,??????
f g◦f(x) =x2+ 1?f◦g(x) = (x+ 1)2=x2+ 2x+ 1? ????? ???f◦g?=g◦f? -1f({-1}) =∅? g(y) =?x,??y=f(x)?G a,??x?H ???????x?E?g(f(x)) =x? ???g◦f=IE? ??????x,x??E? ???? ???f(x) =f(x?)? ?? ??????? ????? ? ?????? ???g? ?? ? ????? y?F??x?E?????? ??? ???f(x) =y? ?????? ?????g(y) =x? ?? ? ?????f(g(y)) = y,?y?F? ????f◦g=IF? f -1◦g-1? f◦h=IF? ?? ? ?????h=IE◦h= (g◦f)◦h=g◦(f◦h) =g◦IF=g? ????h=g? i?IAi={x?E:x?A1??x?A2}=A1?A2? i?IAi={x?E:x?A1??x?A2}=A1?A2? n?N]-n,n[=R? n?N?[-1n ,1n ] ={0}? ????(E)? R? ?x,y?Z, x|y? ?k?Z:y=kx k?Z? ?? ?¯k=k+nZ? z?¯x????xRz??z?¯y? ????zRy? ?? ?? ??????? ???xRy? ????¯x= ¯y?? R? ?x,y?E, xRy?f(x) =f(y) ¯f:E/R →F?????? ????? ???f=¯f◦π? ??π:E→E/R E?F? E/R? ????πff ??A? ??E? inf(A)? ??? ????? ??????? ???N n
E??? ?????
??? ???xn=xn0??n≥n0 ,...,> x b? ???b >0??r > b? ?? ?a-b(q+ 1) =a-bq-b=r-b >0? ????a-b(q+ 1)?E?? ???b <0??r >-b? ?? ?a-b(q-1) =a-bq+b=r+b >0? ????a-b(q-1)?E?? ???-23 = 6· -4 + 1? ? ?a|b?bZ?aZ? ? ??a?Z?a|a? ? ??a,b?Z?a|b??b|a?b=±a? ? ??a,b,c?Z?a|b??b|c?a|c? ? ??a,b,c?Z? ??a|b??a|c??????α,β?Z?a|αb+βc? ??????? ????D12={1,2,3,4,6,12}? ?? ??????? ???? ??a??b????a?b? ?? ???? ????? ??????? ??Da∩ Db? ????? ????m?n? ?? ???? ?? ?? ???????a1,a2,...,an? ????? ????a1?a2?...?an? ??? ?? ???? ????? ??????? ??Da1∩ Da2∩...Dan? ?? ???? ????a1?a2?...?an??? ?? ???? ????? ??????? ??a1N?∩a2N?∩...∩anN?? ??a??b???? ? ??-1? ????? ? ????a?b= 1 ??D12={1,2,3,4,6,12}?D15={1,3,5,15}?D12∩ D15={1,3}? ???? ?? ?12?15 = 3?
12N?={12,24,36,48,60,72,...}?15N?={15,30,45,60,75,90,...}? ?? ?12N?∩15N?=
{60,...}? ????12?15 = 60? ? ?a?b=d?d? Da∩ Db,et?c? Da∩ Db,onac|d? ? ?a?b=a?a?b=b?a|b ? ?aZ+bZ= (a?b)Z? ? ?aZ∩bZ= (a?b)Z? ????? ???? ?? ??????x= 1??y= 0? ?? ???????a?cZ? ????c|a? ??????? ????? ?? ??????? x= 0??y= 1? ?? ???????b?cZ? ????c|b? ?? ?? ??????? ???c|d? a|s??b|s? ????m|s?
αa+βb=a?b= 1?
a?b= 1?? ?????? ?x?? ?y? ?? ?d|9x-11y= 45-44 = 1 ? ? ??????a,b,c?N?? ?????ac?bc=c(a?b)? ? ? ??????a,b?N???s? Da∩ Db? ?????as ?bs =a?bs ? ? ??????a,b?N???d? Da∩ Db? ????? ?a?b=d?ad ?bd = 1? α,β?Z:d=αa+βb? ????dc=αac+βbc? ??? ?????dc|ac?bc? ? ? ??????d=a?b? ?????s|d??ds |as ?bs ????uas +vbs =ds ? ??? ?????as ?bs |ds = 1?ad ?bd = 1?? ?????a?(b1b2···bn) = 1 ???a?b1=a?b2= 1?αa+βb1= 1? ????αab2+βb1b2=b2? ?? ??????u,v?Z:ua+vb2= 1? ????ua+v(ab2+vβb1b2) = 1?(u+b2)a+vβb1b2= 1? ????a?b1b2= 1? ?i= 1,...,n? ?????a1·a2·...·an|b? a m=ka?b=ka?b?d? ??? ??????a?b?d|m? ????m=a?b?d?md=a?db?d=|ab| r
0=a?r1=b?
?? ???????rn-2??rn-1?????? ? ??rn-1= 0?? ????rn= 0? ?? ?? ??????k??? ???rk= 0? ?? ?? ??????? ????? ??? ??? ??? ???? ?? ???? ??a??b? r ? ? ????rn?? ??????? ????? ??? ???? ??????? ?? ?????? ?? ?a?b=r0?r1=r1?r2= a?b=rn?? ??????a,b?Z? ?? ???? ? r
0=a?r1=b?
r k-1 ?? ?????? ??? ??????uk??vk??? ? u
0= 1,u1= 0??uk=uk-2-qkuk-1
v
0= 0,v1= 1??vk=vk-2-qkvk-1
r n=a?b ??k= 0? ?? ?u0= 1??v0= 0?u0a+v0b=a=r0? ??k= 1? ?? ?u1= 0??v1= 1?u1a+v1b=b=r1? u ka+vkb=uk-2-qkuk-1a+vk-2-qkvk-1b= (uk-2a+vk-2b)-qk(uk-1a+vk-1b= r k-2-qkrk-1=rk? ?? ? ????224?175 = 7??7 = (9×175)-(7×224) ??b?=ba?b? h=cd
32?b?=1757
k?Z? p?n= 1? ...·an? ????? ?? ??????i??? ???p|ai? ???p?(a1a2···an) = 1?? ??? ?????p?(a1a2···an)? ... < p ???? ?? ?????a=pm11pm22...pmkk? ??????? ?????1260 = 2·630 = 22·315 = 22·3·105 = 22·32·35 = 22·32·5·7? p a?b=pl11pl22...plkk??li= min(si,ti)? a?b=ph11ph22...phkk??hi= max(si,ti) u k i=1psi+ti-min(si,ti) i=?k i=1pmax(si,ti) i? ??????? ?????180 = 22·32·5?42 = 2·3·7? ?? ? ?180?42 = 2·3 = 6?180?42 = 2
2·32·5·7 = 1260?
x≡y[n]?n|x-y? ?k?Z:x-y=k·n {¯0,¯1,...,n-1} x-z? ? ?y?¯x?n|y-x? ?k?Z:y=x+kn?y?x=nZ? ? ? ?? ?{¯0,¯1,...,n-1} ?Z/nZ?
¯x+ ¯y=x+y
¯x·¯y=xy
?(Z/nZ,+)??? ?? ?????? ??????? ? (x+y) +z= (x+y+z) =(x+y) +z==x+ (y+z) =x+y+z=x+ (y+z)? (x×y)×z= (x×y×z) =x×y×z=x×y×z=x×(y×z)? ?x,y,z?Z?x×(y+z) =x×(y+z) =xy+xz=xy+xz= (x×y) + (x×z)? (Z/nZ,+,·) ??????un= 32n+1+ 2n+2? ????Z/7Z? ?? ?¯un=¯32n+1+¯2n+2=¯9n·¯3 +¯2n·¯4? ¯9 =¯2? ????¯un=¯2n·¯3 +¯2n·¯4 =¯2n·(¯3 +¯4) =¯0? i=0?i2i? ???i? {0,1}? ? ?ak=? i:?i?=0ai[n]? 6
2= 36[100]?64= 362=-4[100]?68= (-4)2= 16[100]?616= 56[100]?632= 562= 36[100]?
6
64= 362=-4 = 96[100]? ????673= 6×16× -4 = 16[100]?
?? ??????x??? ???x≡aimodmi??i= 1,...,s? m=m1m2...ms? u kmk+vkhk= 1? ??????a=?s i=1aivihi? ????? ??????mk? ?? ?
¯a=s
i=1a ivihi??? ?????mk|ki,?i?=k? ?? ? ?
¯a=a
kvkhk=a k(1-ukmk) =a k??? (2×7)-11?1 = 4-3 = (11-7)-(2×7) + 11 = (2×11)-(3×7)? ??? ???km≡1??????n? ????n|km-1? ???? ?? ??????s?Z??? ???sn=km-1? ?? ??????mk-sn= 1? ?? ??? ????n|mk-1? ??? ??????km≡1??????n?? U
18={1,5,7,11,13,17}?φ(18) = 6
U
15={1,2,4,7,8,11,13,14}?φ(15) = 8
????? ?? ?ak+bn= 1? ????ka=1 ?? ?x= 7a+ 4 = 11b+ 2? ????11b-7a= 2? ???? ?????? ?? ?? ?11b≡2? ????b≡4? ??? ?????b= 7k+ 4? ????x= 11×(7k+ 4) + 2 = 46 + 77k? U ??????Un={x1,x2,...,xm}? ??m=φ(n)? ????? ?? ?f(x1)f(x2)...f(xm)≡x1x2...xm ??????n? ????amx1x2...xm≡x1x2...xm??????n? ??????y=x1x2...xm? ????? n|y(1-am)? ?????n?y= 1? ?? ?? ??????? ???am≡1??????n? ??a???n? ?????a=qn+k?k?Un? ????? ???????a≡k??????n? ?? ?aφ(n)≡kφ(n)≡1? ??????n?? U p={1,2,...,p-1}?φ(p) =p-1?? ?? ? ?a?Z,p|ap-a n? ?? ?? ???? ???n(a)? n= min{k?N?:n|ak-1} ? ? ??????9? ?? ?22≡4?23≡8?24≡7?25≡5?26≡1? ???? ???9(2) = 6? ? ? ??????15? ?? ?42≡1? ???? ???? ???15(4) = 2? a u n= 7(7n)??? ??? 7
3≡63≡3[10]?74≡21≡1[10]? ???? ??????? ??7?????? ?? ??? ???? ? ??
??????7n= 4k+r? ?????un≡74k+r≡7r[10]? ??r? {0,1,2,3}? ??n??? ?????7n≡1[4]?7n= 4k+ 1?un≡74k+1≡7[10]? ??n??? ???????7n≡3[4]?7n= 4k+ 3?un≡74k+3≡3[10]? ???? ? ? ??n??? ??????? ??????n???? ?d? ????? ?k?N,ak??? ???????d??????n??k?d= 1 ??????? ???d|km? ??? ?????(ak)m≡1? ?????ak??? ???????d? ?? ?d|m=dk?d? ?? ??? φ(m1m2···mk) =φ(m1)φ(m2)···φ(mk) ? ? ??????p?? ?????? ??????? ??k?N?? ?????
φ(pk) =pk-pk-1
? ? ??????n=?s
φ(n) =s?
k=1(pαkk-pαk-1 k) =ns? k=1(1-1p k) d|nφ(d) =n? m ?? ?ri=x-qimi??mi?ri=mi?qimi+ri=mi?x? ?????m?x= 1? ??m=m1m2? ?? ?mi?x= 1? ????ri?Umi? ????(r1,r2)?Um1×Um2? ?? ??????x?N?????? ??? ???x < m??x≡ri[mi]? ????? ????φ(m) =????(Um) =????(Um1)????(Um2) =φ(m1)φ(m2)? ? ? ????E={1,2,...,pk}?F={m?E:m?pk= 1}?G={m?E:m|pk}?? ?
φ(pk) =????(F) =????(E)-????(G) =pk-pk-1?
? ? ????n=?s
φ(n) =?s
k=1φ(pαkk) =?s k=1(pαkk-pαk-1 k) =?s k=1pαk(1-1p k) =n?s k=1(1-1p k)? ? ? ??????E={1n ,2n ,3n ,...,kn ,...nn = 1}?? ???? ????d|n?Fd={kd 1}? d|nFd=En? ????d|n??kd ?Fd? ??????m=nd ? ?? ?kmdm =kmn F d?E??? d|nFd?E? kn ?E? ??????m=k?n?k?=km ??d=nm ? ?????k??d= 1?? k ?d ?Fd? ????E?? d|nFd? kd =k?d ??Fd∩Fd?? ?????kd?=k?d? ?? ?d|kd??????k?d= 1? ?????d|d?? ?? ???? ?? ?d?|d? ????d=d???Fd=F?d?? ????? ??? ???? ?p-1 a?Up?? ??????? ???????d? ??????H={1,a,a2,...,ad-1}??????p? ?????(ak)d≡1 d|nφ(d) =n? ????? d|n????(Ed) =? d|nφ(d)? ?? ??????? d|n(φ(d)-????(Ed)) = 0? ??? a???????n?? ????p? ??? ??????? ??n= 8?U8={1,3,5,7}??φ(8) = 4? ?????? ? ?? ??32= 9≡1?52= 25≡1? 7 f:{0,1,...,p-2} -→Up k?-→αk??????p ?? ????k=????α(x)? ??x???p? ?? ? ?????x?Z??k? {0,1,...,p-2} k=????α(x)?αk=x??????p
α(xy) =????α(x) +????α(y)??????p-1?
{0,1,...,p-2}???? ???αm=αn? ?????αm-n≡1??????p? ?????α??? ???????p-1? 3
7≡11?????? ??? ???? ????¯3(11) = 7?????? ???
? ??????αn? ?? ? ???????(αn)m=αnm? ? ??????αm? ?? ? ???????(αm)n=αnm? p??q????? ?? ??? ??????? ??????? ?? ??????? ???? ???????n=p.q? ???? ???? ??????? ?? (RSA,n,e)? ???? ???????d??? ???ed≡1??????(p-1)(q-1)? ???? ?? ?????? ???d?????
883,207≡4102? ?? ??????? ??????? ?y= (0928,0001,4179,0883,4102) = (y1,y2,y3,y4,y5)?
y di= (xei)d=xedi=xkφ(n)+1 i= (xφ(n) i)k.xi??????n ??xφ(n)quotesdbs_dbs14.pdfusesText_20
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