algebraic structures (groups, rings, fields) and some...
TRANSCRIPT
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Lecture7AlgebraicStructures(Groups,Rings,Fields)
andSomeBasicNumberTheory
Read:Chapter7and8inKPS
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FiniteAlgebraicStructures• Groups• Abelian• Cyclic• Generator• GroupOrder
• Rings• Fields• Subgroups• EuclidianAlgorithm• CRT(ChineseRemainderTheorem)
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GROUPsDEFINITION:AnonemptysetGandoperator@,(G,@),isagroupif:
• CLOSURE:forallx,yinG:• (x@y)isalsoinG
• ASSOCIATIVITY:forallx,y,zinG:• (x@y)@z=x@(y@z)
• IDENTITY:thereexistsidentityelementIinG,suchthat,forallxinG:
• I@x=xandx@I=x
• INVERSE:forallxinG,thereexistinverseelementx-1inG,suchthat:
• x-1@x=I=x@x-1
DEFINITION:Agroup(G,@)isABELIANif:
• COMMUTATIVITY:forallx,yinG:
• x@y=y@x
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Groups(contd)DEFINITION:AnelementginGisagroupgeneratorofgroup(G,@)if:forallxinG,thereexistsi≥0,suchthat:
x=gi=g@g@g@…@g(itimes)Thismeanseveryelementofthegroupcanbegeneratedbygusing@.Inotherwords,G=<g>
DEFINITION:Agroup(G,@)iscyclicifagroupgeneratorexists!DEFINITION:Grouporderofagroup(G,@)isthesizeofsetG,i.e.,|G|or#{G}orord(G)
DEFINITION:Group(G,@)isfiniteiford(G)isfinite.
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RingsandFieldsDEFINITION:Astructure(R,+,*)isaRingif(R,+)isanAbeliangroup(usuallywith
identityelementdenotedby0)andthefollowingpropertieshold:• CLOSURE:forallx,yinR,(x*y)inR• ASSOCIATIVITY:forallx,y,zinR,(x*y)*z=x*(y*z)• IDENTITY:thereexists1≠0inR,s.t.,forallxinR,1*x=x• DISTRIBUTION:forallx,y,zinR,(x+y)*z=x*z+y*z
Inotherwords(R,+)isanAbeliangroupwithidentityelement0and(R,*)isaMonoid
withidentityelement1≠0.AMonoidisasetwithasingleassociativebinaryoperationandanidentityelement.
TheRingiscommutativeRingif
• COMMUTATIVITY:forallx,yinR,x*y=y*x
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RingsandFields
DEFINITION:Astructure(F,+,*)isaFieldif(F,+,*)isacommutativeRingand:• INVERSE:allnon-zeroxinR,havemultiplicativeinverse.i.e.,thereexistsaninverseelementx-1inR,suchthat:x*x-1=1.
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Example:IntegersUnderAddition
G=Z=integers={…-3,-2,-1,0,1,2…}
thegroupoperatoris“+”,ordinaryaddition
• integersareclosedunderaddition• identityelementwithrespecttoadditionis0(x+0=x)• inverseofxis-x(becausex+(-x)=0)• additionofintegersisassociative• additionofintegersiscommutative(thegroupisAbelian)
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Non-ZeroRationalsunderMultiplication
G=Q-{0}={a/b}wherea,binZ*
thegroupoperatoris“*”,ordinarymultiplication
• ifa/b,c/dinQ-{0},then:a/b*c/d=(ac/bd)inQ-{0}• theidentityelementis1• theinverseofa/bisb/a• multiplicationofrationalsisassociative• multiplicationofrationalsiscommutative(thegroupisAbelian)
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Non-ZeroRealsunderMultiplication
G=R-{0}thegroupoperatoris“*”,ordinarymultiplication
• ifa,binR-{0},thena*binR-{0}• theidentityis1• theinverseofais1/a• multiplicationofrealsisassociative• multiplicationofrealsiscommutative(thegroupisAbelian)
Remember:
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IntegersmodNUnderAddition
G=Z+N=integersmodN={0…N-1}thegroupoperatoris“+”,modularaddition
• integersmoduloNareclosedunderaddition• identityis0• inverseofxis-x(=N-x)• additionofintegersmoduloNisassociative• additionintegersmoduloNiscommutative(thegroupisAbelian)
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Integersmod(p)(wherepisPrime)underMultiplication
G=Z*pnon-zerointegersmodp={1…p-1}
thegroupoperatoris“*”,modularmultiplication
• integersmodpareclosedunder“*”(whereGCD=GreatestCommonDivisor):becauseifGCD(x,p)=1andGCD(y,p)=1 thenGCD(xy,p)=1(NotethatxisinZ*PiffGCD(x,p)=1)
• theidentityis1• theinverseofxisus.t.ux(modp)=1
• ucanbefoundeitherbyExtendedEuclidianAlgorithmux+vp=1=GCD(x,p)
• OrusingFermat’slittletheoremxp-1=1(modp),u=x-1=xp-2• “*”isassociative• “*”iscommutative(sothegroupisAbelian)
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PositiveIntegersunderExponentiation?
G={0,1,2,3…}thegroupoperatoris“^”,exponentiation
• closedunderexponentiation• theidentityis1,x^1=x• theinverseofxisalways0,x^0=1• exponentiationofintegersisNOTcommutative,x^y≠y^x(non-Abelian)
• exponentiationofintegersisNOTassociative,(x^y)^z≠x^(y^z)
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Z*N:PositiveIntegersmod(N)RelativelyPrimetoN
• Groupoperatoris“*”,modularmultiplication• Grouporderord(Z*N)=numberofintegersrelativelyprimetoNdenotedby
phi(N)
• integersmodNareclosedundermultiplication:ifGCD(x,N)=1andGCD(y,N)=1,GCD(x*y,N)=1
• identityis1• inverseofxisfromEuclid’salgorithm: ux+vN=1(modN)=GCD(x,N) so,x-1=u(=xphi(N)-1)• multiplicationisassociative• multiplicationiscommutative(sothegroupisAbelian)
G=Z*Nnon-zerointegersmodN={1…,x,…n-1}suchthatGCD(x,N)=1
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Non-AbelianGroupExample:2x2Non-SingularRealMatricesunder
MatrixMultiplication
• if A and B are non-singular, so is AB • the identity is I = [ ] • Inverse:
= / (ad-bc)
• matrix multiplication is associative • matrix multiplication is not commutative
GL(2) = {[ ], ad-bc = 0} a b c d
1 0 0 1
[ ] a b c d
-1 [ ] d -b -c a
Recall:asquarematrixis non-singular if itsdeterminant is non-zero. A non-singularmatrixhasaninverse.
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Non-AbelianGroups(contd)
[ ] 2 5 10 30
-1 [ ] 3 -0.5 -1 0.2 =
[ ] 2 5 10 30
[ ] 3 5 1 2
[ ] 11 20 60 110
=
[ ] 3 5 1 2
[ ] 2 5 10 30
[ ] 56 165 22 65
=
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Subgroups
DEFINITION:(H,@)isasubgroupof(G,@)if:
• HisasubsetofG• (H,@)isagroup
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SubgroupExample
Let(G,*),G=Z*7={1,2,3,4,5,6}LetH={1,2,4}(mod7)Notethat:• Hisclosedundermultiplicationmod7• 1isstilltheidentity• 1is1’sinverse,2and4areinversesofeachother• Associativityholds• Commutativityholds(HisAbelian)
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Let(G,*),G=R-{0}=non-zerorealsLet(H,*),Q-{0}=non-zerorationalsHisasubsetofGandbothGandHaregroupsintheirownright
SubgroupExample
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OrderofaGroupElement
Letxbeanelementofa(multiplicative)finiteintegergroupG.Theorderofxisthesmallestpositivenumberksuchthatxk=1Notation:ord(x)
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Example:Z*7:multiplicativegroupmod7Notethat:Z*7=Z7ord(1)=1because11=1ord(2)=3because23=8=1ord(3)=6because36=93=23=1ord(4)=3because43=64=1ord(5)=6because56=253=43=1ord(6)=2because62=36=1
OrderofanElement
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Theorem(Lagrange)
Theorem (Lagrange): Let G be a multiplicative group of order n. For any g in G, ord(g) divides ord(G).
element!any of order largestn *
nG oforder - )(Φ
n modg
m
m 1
such that integer smallest :g oforder
≡
11 :thus
//)ord)(
)ord :because
mod1
:1 COROLLARY
/1/)(
*)(
===
Φ==
=Φ
∈∀≡
ΦΦ
Φ
kk(n)n
*n
*n
nn
bb
k(n)k(Zbord
(Z(n)
Zb n b
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Example:inZ*13primitiveelementsare:
{2,6,7,11}
element primitive
)( )2
)1
then prime is p if:2 COROLLARY
*
−
−=∍∈∃
≡
∈∀
a
1p aord Z a and
p mod bb
Zb
p
p
p
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EuclidianAlgorithmPurpose: compute GCD(x,y)
GCD = Greatest Common Divisor
1),gcd(
mod1*
, of
1
1
1
=⇔∃Ζ∈∀
≡
−
−
−
−
nbb b nbb
bsetive invermultiplicab
n
Recall that:
11),( −∃⇒= bbn Euclidian
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EuclidianAlgorithm(contd)
init : r0 = x r1 = y
q1 = r0 / r1⎢⎣ ⎥⎦ r2 = r0 mod r1
...= ...qi = ri−1 / ri⎢⎣ ⎥⎦ ri+1 = ri−1mod ri
...= ...qm−1 = rm−2 / rm−1
⎢⎣ ⎥⎦ rm = rm−2mod rm−1
(rm == 0)?OUTPUT rm−1
Example:x=24,y=151. 192. 163. 134. 20
Example:x=23,y=141. 192. 153. 144. 115. 40
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ExtendedEuclidianAlgorithmPurpose:computeGCD(x,y)andinverseofy(ifitexists)
init : r0 = x r1 = y t0 = 0 t1 =1
q1 = r0 / r1⎢⎣ ⎥⎦ r2 = r0 mod r1 t1 =1
...= ...qi = ri−1 / ri
⎢⎣ ⎥⎦ ri+1 = ri−1mod ri ti = ti−2 − qi−1ti−1 mod r0
...= ...qm−1 = rm−2 / rm−1
⎢⎣ ⎥⎦ rm = rm−2mod rm−1 tm = tm−2 − qm−1tm−1 mod r0
if (rm =1) OUTPUT tm else if (rm = 0) OUTPUT "no inverse"
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ExtendedEuclidianAlgorithm(contd)
Theorem: )1(1 >= i rtr ii rtm 11 =
I R T Q
0 87 0 --
1 11 1 7
2 10 80 1
3 1 8 --
Example: x=87 y=11
! " r mod tqtt r modrr rrq 0iiiiii1iiii 11211 / −−−−+− −===
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I R T Q__
0 93 0 --
1 87 1 1
2 6 92 14
3 3 15 2
4 0 62 --
Example: x=93 y=87
ExtendedEuclidianAlgorithm(contd)
! " r mod tqtt r modrr rrq 0iiiiii1iiii 11211 / −−−−+− −===
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ChineseRemainderTheorem(CRT)The following system of n modular equations (congruences)
nn
1
m mod ax
m mod ax
≡
≡...
1
Has a unique solution:
ii
i
n1
n
ii
ii
m mod mM
y
mm M
Mmod ymM
ax
1
1
*...*:where
−
=
""#
$%%&
'=
=
""#
$%%&
'=∑
(all mi-s relatively prime).
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CRTExample
!!"
#$$%
&
≡
≡
11375 mod x mod x
4777 modx mod y
7 mod mod y
mMmM M
MmodymMymMx
=+=
==
===
=
=
=
+=
−
−−
)8*7*32*11*5(8117
24711
7/11/
77])/(3)/(5[
12
111
2
1
2211