ELEMENTS OF NUMBER THEORY. In the following we will use mainly integers and positive integers. - the set of integers - the set of positive integers

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1 ELEMENTS OF NUMBER THEORY I the followg we wll use aly tegers a ostve tegers Ζ = { ± ± ± K} - the set of tegers Ν = { K} - the set of ostve tegers Oeratos o tegers: Ato Each two tegers (ostve tegers) ay be ae a the su s a teger (ostve teger) Subtracto Each teger b ay be subtracte fro teger a a the fferece s a teger If a>b> the a-b Multlcato Each two tegers (ostve tegers) ay be ultle a the rouct s a teger (ostve teger) Dvso Dvso of tegers s ot always feasble the set of tegers (ths s the reaso of exstece of ratoal ubers) Let a b Ζ We coser two cases tur There s a c Ζ such that a = b c We say the that b ves a a we wrte b a b s calle a vsor of a We also say that a s vsble by b ves 87 because 87 = 7 The s the vsor of s vsble by a 87 Ay teger b s a vsor of because b = Dvso by s forbe Proertes of vso of tegers If a a b the a ± b If a the b ab If b a b a the a There are ubers c Ν { K b } such that a = bc + The teger c s the calle the quotet a the reaer of the vso of a by b We wrte the a : b = c r a a c = b If b a we ay coser a = bc as a vso wth reaer PRIME NUMBERS A teger > s calle a re uber f a are the oly ostve tegers whch ve

2 If s a re uber a ab the a or b The a theore of the uber theory For each teger > there are uquely etere sequece of res < < K < a sequece of ostve tegers K such that = Corollary For the teger as above such that ff there s a sequece of oegatve tegers β K β β = K a = s 78 = 7 = β β The greater coo vsor of tegers a a b s the greatest (oegatve) teger such that a a b We eote t by gc(ab) If a = b = we efe gc( ) = Proertes a b Ζ gc( b a) = gc( a b) a Ζ gc( a) = a Let a b Ν re ubers < < K < a oegatve tegers K β K β such that a = The γ gc( a b) = β β b = γ gc(78) = = Itegers a a b are calle core ff gc(ab)= Let γ = { β} = K r

3 EUCLIDEAN ALGORITHM Euclea algorth evaluates the gc(ab) for a b > If a< or b< we aly the Euclea algorth wth a b The result of Euclea Algorth for ostve tegers a a b s always the greatest coo vsor of ab Bloc schee START Data: ab c:=a :=b c:=qrs c:= :=s o s=? yes gc(ab)= STOP F gc(78) 7 :8 = r 8 : = r : = r 9 :9 = r 9 9 : 9 = r 9 : = r 7 : 7 = r The gc(78)=7

4 EXTENDED EUCLIDEAN ALGORITHM For ay ozero tegers ab there are uque tegers uv such that Atoally hols a b u v < gc( a b) = u a + v b START Data: ab c:=a :=b u=u= v=v= c:=qrs u=u-qu v=v-qv c:= :=s u=u v=v u=u v=v o s=? yes gc(ab)= =ua+vb STOP For the revous exale 7 = 9 = 9 (9 9) = 9 9 = ( 9) 9 = 9 = ( ) = 7 = 7 (8 ) = = (7 8) = 8 7 7

5 CONGRUENCES Let us fx a ostve teger > We say that a s cogruet to b oulo ff s calle a cogruece a b We wrte a b (o ) Ths relato s (o) 9 (o7) It s easy to see that cogruece s: - reflexve e a Ζ a a (o ) - syetrc e a b Ζ a b (o ) b a (o ) - trastve e a b c Ζ a b (o ) b c (o ) a c(o ) Cogruece s the a equvalece relato The equvalece class wth resect to ths relato s the followg [ a] = { b Ζ : a b (o ) } It has all well ow roertes of equvalece relatos We ca characterze t as follows b [ a] a b (o ) a a b leave the sae reaer whe they are ve by The equvalece class s calle the resue class of a oulo The set of resue classes oulo s eote by Ζ / Ζ I ths way the whole set Ζ s ve to sot resue classes Each class cotas oe teger fro R = { K } I ths way each teger Ζ s etfe wth oe of We trouce arthetc oeratos [ a ] + [ b] = [ a + b] [ a] [ b] = [ a b] a b = a b [ ] [ ] [ ] R Ζ / Ζ : I ths way ato subtracto a ultlcato Ζ / Ζ are efe These oeratos ay be see as oeratos o R : a + b = c c a + b (o ) a b c R a so o We wll the followg coser tegers fro R as reresetatves of classes fro Ζ / Ζ It s easy to show that Ζ / Ζ wth oeratos efe above for a algebrac structure of a rg a Ζ / Ζ wth ultlcato for a structure of a segrou whch s coutatve e a b Ζ a b = b a Ths segrou cotas a eutral eleet [ ] e [ ] [ a] = [ a] [ ] = [ a] a Ζ Such a segrou s calle a oo I Ζ / Ζ we ca trouce exoetato a sle way a = a a L a tes Let a b (o ) a c (o ) The a + c b + (o ) a c b (o ) a c b (o )

6 To efe the vso oerato a verse eleet shoul be trouce Eleet [ b ] s verse to [ a ] oulo e [ b] = [ a] (o ) f a oly f [ ] [ b] = [ ] We wll sly wrte b = a (o ) It eas that e teger a For ay a Ζ there s a b Ζ verse to a oulo f a oly f a a are core Corollary If s a re uber the for ay a R a there s a eleet b R b verse to a oulo The verse theore s true e f for ay a R a there s a eleet b R b verse to a oulo the s a re uber F Euclea algorth The therefore exsts Extee Euclea algorth The Verfcato A coeffcet by a clug ts sg s a uber verse to a oulo Let a b (o ) c (o ) a gc( c ) = The a c b (o )

7 SOLVING OF CONGRUENCES We are gog to solve a cogruece a x b (o ) whe = gc( a ) Three cases are ossble: Case = I ths case there s a uque soluto x of ths cogruece the set R a other solutos are obtaable by ag ay ultle of The soluto of the cogruece s exresse by the forula x = b a (o ) Case > b The cogruece s the equvalet to the followg oe aˆ x bˆ (o ˆ ) where a ˆ = a : b ˆ = b : ˆ = : For the last cogruece Case hols Case > / b The cogruece has o solutos Solve followg cogrueces: x (o ) We see that = gc( ) = the Case hols By the above forula x (o ) We evaluate : = r = (o ) = x = = 9 7 (o ) Solutos are 7 79 K Verfcato: 7 = (o ) x (o ) = gc( ) = a the Case hols The cogruece s equvalet to x (o) We solve t aalogously : = r = (o) = x = = 9 (o) Solutos are 7 K Verfcato: = (o) = 88 (o) x (o 7) = gc( 7) = a / the Case hols The cogruece has o solutos The Chese Reaer Let K be ostve tegers whch are arwse core a let a K a be tegers The syste of cogrueces x a (o ) K x a (o ) has ftely ay solutos A fferece of ay two solutos of ths syste s a ultle of = Proof =

8 The roof here s ortat because t shows how to solve the syste of cogrueces Let M = : = = K We ultly cogrueces by M = M x am (o ) K M x am (o ) Ag the we obta M + K + M ) x ( a M + K a M ) (o ) ( + The su M +K + M a are core To see t let the ecoosto of to re ubers s If = gc( M ) > = = K the r { K } : r therefore there s a R { K } such that r R a hece r M R { K ) } above assuto les that M a there s a S { } S R a The r R K such that r S The gc( ) a they are ot core what s cotrary wth the assutos of r R S the theore The last cogruece has the a uque soluto the set obtaable by ag a ultle of Other roof ([Ko9]) R a other solutos are We eote N = M (o ) = K The execte soluto s x = am N = (o It s obvous that x a M N (o ) a (o ) because x ) a M N (o ) Uqueess ay be rove as above Ferat s Lttle If s a re uber a a s a ostve uber ot vsble by the a (o ) EULER FUNCTION The Euler fucto ϕ s efe for all ostve tegers by () = b : < b : gc( b ) = ϕ uber of eleets of the set { } Values of the Euler fucto ay be evaluate usg the followg forulas: If = s a re uber the ϕ ( ) = If = s a re uber the = ϕ ( ) If = the ϕ( ) = ϕ( ) K ϕ( ) Ferat s Lttle geeralze ϕ ( ) If a a are core ostve tegers the a (o ) Ths theore wll be of great ortace the theory of asyetrc crytosystes

9 EXPONENTIATION MODULO BY ITERATIVE SQUARING Arthetc oeratos oulo ay be colete by exoetato b b [ a ] = [ a ] Ths forula ca ot be use for evaluato a b b (o ) because a s ofte a great uber whch s calculate aroxately There s aother etho whch allows to get the esre value Bloc schee START Data: ab c:= :=a e:=b retur c o e>? STOP o e o? yes yes c:=c (o ) e:=e/ e:=(e-)/ := (o )

10 Let us f the bary reresetato of b : b = ( K ) where Σ = { } = K We have b = K+ ( a ) = K K a the a b a = ( a ) It eas that t s eough to obta a = K a ultly these ters for whch = Each ultlcato s erfore oulo the t uses oly ubers saller tha Coute 7 9 (o 7) 9 = () 7 = 7 7 = 9 8(o 7) = 8 = = (o 7) = 9 = 9 9(o 7) = 8(o 7) 9 7 = = 88 (o 7) The coutato ca be oe usg a usual calculator FINITE GROUPS AND FIELDS Let X = Z = Z / Z = { K } be the set of resue classes oulo where s a re uber Ths set wth oeratos of ato a ultlcato oulo s a algebrac structure calle a fel Z has fte uber of eleets a for ths reaso t s calle fte fel s the re uber the for each q Z q there s a q (o ) Z s a grou wth resect to ato oulo The set Z = \{} s the a grou wth Z resect to ultlcato Such a grou s calle a cyclc grou Nuber of ts eleets (-) s calle the orer of the grou By the Ferat s Lttle for each a a (o ) Let =7 We have Z = = = = = = = = = = = = = K = = = = = = = = = = = = = = = = = = = = = = = = = Orer of a eleet a of a cyclc grou a (o ) Z s the sallest ostve teger such that

11 Corollary For each eleet a of a cyclc grou Z ts orer ves the orer of the grou A eleet g of the grou whch orer s equal the orer of the grou s calle geerator of the grou Corollary If g s a geerator of the grou b { K } such that b a = g Z the for ay eleet a of ths grou there s a (cotue) I the grou Z 7 eleets a are geerators of ths grou orer of a s equal to orer of s equal to a orer of s equal to