HNDJ-Νί, IfiJRLDUCmLE P O L Y N O M I A L S O V E I l F i N J T E P I E L 0 S
Hendrik W Lcnstri, Jr MiÜumittsch Instituut UnmrMtoit van Amstcrdim nd M Hin πι itit il Saenrts Research Institute
Irr f j/iU | r l j π n1 ιυ ί K' in us( ] ίο < i m
Λ Mi inil n ti i» ri<! l M n ι n f Ϊ C urnput iL» ns
n t ' n\iui MI ι r u ' \i\i-, h >rv ,2] f jrn| k\il\ tu' (Kj irid i r) pu grij li\ [ i[ K uuiorri ροί> norjn ii M t]f r l)jiiis \i l f r ί ι linr irr !u il»l· p l MI π Μ! E l " v r I !„ x n t t i | r ι U ι null r l· [ r< ' l n is ' ( i II \\ < \ ι r U dt U nininstio
l i! f Ii | r H i I s \ i r l ι i' lixfi i
Ii N o t a t i o n
If K is i ficld K \viil dcuotc in aigciiriic \V)K n p is i ntHJiiil primc- Ι ) ν,ιϋ donote ilu fiold
p iipinf tifs If K is ι numbir ficki (ι Ρ finitt cxtLiibio O) O. will d'iiotc iLs nn^ of intigcr1; \Vhin ι Γΐΐκ
l riiiu p nid i nuinhtr f» Id K irc fix« d llun Tor r, -a will ilit:ol( ή + ρ Οκ ίί ι ratinnil pruia p ts fix' d
J h t o r i m l l In r ι- ι c / „ «n l in il nlim \ irh ι h it n inf il ι l / ^ v. ilh p | πι n
l Ο ι ut f I itl iMlh Γ irr( luubl in i
M lor n 7 ^ , s. «ill do mir p
Alxorithm Λ
(0) Jnpiitpd 7.>l
(1) Cjieill ite π ι t / 0 such ib ]t
D i t o r c m 2 Jb ti ι ; L , /, ^ IM) in ilgrnlhln li (J) ( i k u h t i ,, , 7> ( ) such (11 U
i l l'nl n m p i t i l / , , · . Μ ΐ Ι p | rilm q r, tho Ir-vst pnm( »ith q s : l(mod n) uxl |) im rt in (In biiljlidd K C <J((q)
with (K OJ - n (1) ( iliiililc E I ! (x| such Hut
Π
U ir <. ;> *> ) Jl Jrt il j p m l l) s nilLf j is j,r inltd 1UI < ^d ai Ην, ί \i M p is n i u u UK ! K 1l le ^f Iht l U l i ! ι d u | ι Ι ι l er s ^ u i i h u L n\iri) ^ h\ Γ- ι . V Jl l n l , , M u. n m l i,,.j
(l) l or i — \ 2 d calcuUtc i j i 0 a . t I such
1 Ξ E j L i <ΊΡΕ W « * Pa) * n h c£ ({01 p 1}
Tor (=1 2 i ι ΞΞ (fmod n) with e c {0 l n 1]
,, f Γρ n i rcxit of f, - χΡ χ l < I p[x]
/* i I is i root üf R
(5) C ilciilite f ( l [\] such t tut f is tlif ' li ir if tf n s ü e | x > h n o r m L l « > f ( i) J)(\ i(( Ip)
(l>) Output f»
fV P r o o f of Cprrectn_es9-Ajgpri_thrn A
\\ L bcgm bj irßinng l!) i L Mgr rithtrt Λ r uns in po]j jonml diiu TAMI tauig (.xU lutid [in, m um h\j >llnsis
Mcp (1) cleirly is pohnurnnl timn I or Sttp (2) wt, iisod α \ incnt of l hcorom 2 in [l]
Proposition 3 Assuming extendul Ricrmnn h\pothcsis thcrc is a c t Z Q such tfnt for al! p n f /> Q wilh p
primc and (p n) = l Llitro is α pnmo cj ™ l(mod n) with q < (n^logfiip))2 such (Int p is in. rt m thc (umqut)
«ubfipld k C Q(i(() v, ah [K Qj = n
Htncc th( q rcquired in Step (2) n sufficn n(ly btii^ü that 1 1 ein bc found nid t r sied for prnn ihn in polynomml hm* Smrt p is uurt in \\ iff (<f I/f n) = l *hcrc f tf the order of p in (Z/ qZ)* tins too can be twted in polj nomnl limo
For Step (3) \\c bigm by notinj; (Int it follous frotn Gku« thcor) of pmods (sct, [()]) t h i t Πσ ι 6 (x O <
7 ( l ' ^ /<?
Λ l^l w the difinitmn of f; inakis senst Smtc
uhich are pol>nomn] in q log p H follou·- thit Sttp (3) ein bc don<- in polynoniiil time
I or Step (1) U -jtifficcs to show tlt it for ι f m d i tlu c ikuLiUon of Iht i s takes iiol\t\omiil tstm l in tileul ition proctcds in i + t bligr^ In Stigo l powtrs of ·? grc Htr (hin n l are reducetä u^ing g In sii^c ι = _} 3 i + l po\virs of a 4 ( groitir t h in p l i n nducf l
nsing f ( l )u rt miinirig ci( tiits ir( striitihtfor» ir l
I IH lim] ition of thf ch-iricU rMic p K m ini-il <if i imlrix of \ | j(l ) cm b< doiic in Um« polj tionuat in (l !<κ ρ ΙΙΜΠΚ s! mdird im l lind* {10 pp i > i i » 110-tllj
\\ t rif w irguf thil the pul\ noinn] f r l J\] prodiiT i l\ Alpirithiu \ is irr«. Ιικ il)|( of (U „n i \S t htgin b\ ciinsi icruiii (ho f II \MIIC; l )wu of ful
J-\V hcre a. a »^ ff arf äs m the ilgonthm \Ve will pro\e tfic fullouing cliims
Cl-um I [l· I , jj = p for i — l 2 a Chim II μ F J = n
Clum III Ι Γ, , ( ^ +Λ αί
[L follmvs from ihe tUmis Ih il H f,
.C χ >ίί — Τ1σί(. ίχ ι ) l l i c
C M l bc dont usmg (In ruift oponiions in Z
calculilion of g
Sinti the \( of tht VIgonlliin irt cl( irlv a basis fcr i /I
it, OÜOMS frofii Standard reiiilt (src for cv\ni( k [7] p i £t s
" 8) i h i t f i·, the diMred poKnomiil
C Kim I will follow from t h» n<\l l wo l· nima-s \ arnnts of ν,ίικίι m·*} bc- fouiid in J7] ί >r ex im p) o
L e m m a Ί l et K be i fidd of ihirictcristic |> yf· ü ind i t K Ιίκη fithor f — λρ λ ι is irn dunblt or tt ho.·- i
F r o o f l f . i t k »r. r ols )f f llicn (n /i)p — > ί ίο ο ß i I h r Γ rf K( i) ~ K()) lud « md ß li i\* ÜK s-ime
| ((r r i r K R H!< « , Ih it »II im dildbh f .rl ,rs of f m
} !\j Im L ΙΙΐί smii degrt But f h us prnm dcgrcc so ull rf irr liuill r r ill lls (icfors i n Im« ir Π
Ι'τοοΓ Ι I r i uou UIL tnr-r fum tion from \\(f) to )\ Λ ι li iilculiU I r( , ' ^ Ί Ην ι sum[ hon
(') ' , n - 0
M .In > n; C) b> » ' g.vis· <'"' - i . ' f l so Ι φ , " · ' ) = ΓΠ ' < ! ) i l r h ' j + J ' O ) - i l r ( n ' ) Mulliplying ' ' ) l» ( ' ) ' i ' ' » " . . ' ( . ' K « ' Γ1 ( ')!' ~ 0 fr m,
Ί ι ο ' ι il f M>«s I h n i ' + 1 'x^' + x11 is Ihc monir
irr d. n l pil\n'inial fc r o ' ind ! r( ') -= , ' I h n « · l M -,'* ) l
Vs Ι.ΠΗ g s rulti ib! th n bv lonmii l Iti τ>. is i h H «i h fi" l-i »-' 0 I ilmg I M « (l m|
il i-rung llut r-mms t j h pd l p< \u r (•nrnmuli's «ilh
lil.n? Iri-r) M, IN H b i -- 0 for b _ lr(i) r K >· /u'radKtir·' ihr hypolhcsis 1h .1 f , Klx] is
~r< <1 .eil k n
Il> I i-inini 1 Γ! - χ' χ l I |x| l s , , i |1 ( r
ir diifib1 jr h i.s 1 rrx)t in I In t!n l i l l i r r i s t (χ'1 χ l rx ) / l wlu^i , , Ir i r| ) not irrrtfl I hus [l· ] I J =_
p l'κ r ^L of C hini E IK w füllt us froin I cinrni ")
Π
P r o o f U i·, will linran [l)| Uni S = ( , , " , , C,K / Q) is
i bisis for OK ovtr 7. Ilona T =. (.,"» , C'K / (jl ι·, ι
bi.sis for tiie nt h dcgr«; cxtcnsion ficld OK/ p OK ^s i
\trlor spicc ovtr I ( II follous thit S ]ii^ n duncut
[ lu fichi mtomorp'iisrns of ( \ / p f >K inducid b\ tln
b i n , - I r(Jj ^Kd ) llnn (ζ is irn hi ibl< in I Jx
(Uinuils of S in ro'ijügiti o \ ( r I R fillous Ihii g i irn ducibl< in Ϊ [xj u
\oticf ih it smcc f is irrodiK ibli i ! , ( · , ' , » , , Ι Ν 't Mlows- ü u l 1^ = 1ρ(η >) \ ο
Clinn III follcms from
7 let „ , , lp « „ h ( Ir( „ | fp] ind [I p( i | f f
P r o o f I n d„ = | ί , , ( · | 1ρ| <lj = Ι Ι , Ι Ό Ί , Ι ' l , n ~
[ lp| H 1)1; )] Sinci l p C I p(0) C Ι μ(Μη+5) « Ρ h ^ o
< i , | [ lp( i , + , J l Ip| Also |I p( , „ + l ) Γ J!d; )d(,+ , so do ld< m ^miilarlv d ^ i ^ ^ lioiuc d(_d.j[di ( + / j fk-irlv
* p^1"4 ^) £ ^ (α ^) ί·0 l'1 ( l v mu^i bt oqml D
R e m a r k Assiirno the txLcnded Rieininn hypolhosis thr-n tht polynoniu] f tliit is caiculated bv AlKonthin \ I m 'sinal! cofffmonls " if r is- sin il] lud p Is l irg( Mort prmsdv f is of tiu form f = xd+ I l f ^ i l,x'' ' w l"r l
i r / Sll IsflCS
l",l ^^^'(ht^df))^ for ι = l 2 rf, with c äs in Proposilion 3
VVithout liu txtpudtd Ritminn i^po^h^sis it is not <!<ir how lo provt du (x,st(nc< )f an irrodiicibU poljnomiil in i ' J x ] of Uns form
V Aloiorithm Π
(0) I n p u l p d , 7.>l
(2) lor . = 12 , cil.ul.1 llnl
q ~ li^sl positive pnino witli q i squ ir< fr«
Ί, T1 l> >n<l Ί, / P ' ' l
αι — li ist positiif mitgtr Mit)] p ' ^ l inod q
R t I M such llntRU) = Π (χ ,"|
1 1 " ' h/tj
K IS 11|0 UlllqllL sllbfllhl of Ö d )
l( Willi [ K Q J = b ,
2 ^= Icist positive intcgtr such tlnt e ^ d
5) C ilcilhtc I C (0 I 7} äs follons
S a lt =_ l
l or J - / ι l l
(i) Jf l b ·£ <1 tlan s<t l , = Ib ])iit
(b) ΙΓ t b > d ind t c j < d Üi* n spt t0 —
1/j , put 01 j l <1 goto(J)
(r) If t b > il md tc: f ^ tl tlicn sei
VI Proof of Corr^ctncsa Algorithm Π
\\ ( IHKIII t ) prmuig tln fotJouuiR clviin (tli
mitation is that uf Algtinlliin B)
Claim IV l lun- txiits ·> t t / -> 0 such lli il for U) p <1 ,
l> } " l "1 P pnrno H» l),s and qt * producul m Algonthm
II tm iniMils p <i Ιι-ut (ht. /oJlowing proportici.
(0 J :£. l', < i, ü c o, ι ' " ί Ρ < cdlogp ι l i '
(b) (b, bj) - I ior l ^ K J <: '
NVi will nicd die folloumg well known rcsult wlncli
is ι dirort corntquincc of Tlitorim 2 m |S] (pul k = 2
ind t = 1)
Proposition 8 1 1« re oxis ] \ 7. \Mlh ρ iinsn*1 ind \ ^
g- l iitiore frf
Proof of Claim IV Ckarly l ^ . t), <^ a! Assume b, =
l tlun a |c so p ' 1 Ξ l mod ij( and conlrir) to out
Kinslruilnin (||p ' ' I Bi l (rnnt s l mlc I lnoreni ^ <^
<1 l < <i so b < c\ B) J'rnposjlion ft
;/ <) > /'"' > /'"'-i
ι s f ) i(MOn \ ) willi PV < (0 l l)k ,!
[or k , l
?k < l p κ ι root of Rk ( l (j\] for k t I
(r>) Cnlculitc f ι ί (v) sucli tlnt
f 11 rhirictcHhlii· pol)nonnil of (i(j) ι N^ (l ^1
(d) Output f
so <| ^ . er, , log p u (liimid rcr ,log |> < <dlog p for
ι -= l 2 z b> Iht choire of z m Step (2) Thcrtforc (i) li AI |b) 1s rlcir from l In rnnstriiodou 0
H follous from (·ν) of Claim I\ thjt z <^ log d and Uiit eich q ein bi. fouiid in polvnomiil linic b^ niivc snrdi Ί(ιρ (ΐ\ is n«w ( tsil) si cn to bi rompiitiblc in
pul) iiunn t! t um algonflim for f nid in» qn , lr l l l f,
notirtsidnt*. ind t iktng sqti ire roots m iF
it ι-j rlf*ir fr< πι ttit conilruetton (h) of C Htrn IV •M1 l In ΐΓίΜίΠΗίιΙ- f r Mp >nthm x t l l v t t l i f l oulpni f of
\ ]ih n K ι irr In .1 I. of !, r !(J ( \ , U in
j 1} r r u. ; , ι π u r κ 1) f ι Ι*κ firld-, K (Krümlig i i ^ f ι Jl ι)^<· t Min squirr- frcuuss of q l u n p h o t h i t
< r Μ ι Μ] Γ 1h i!r >n!hm \l lln Uarl v t
l \sMimi 1h vt \l M ii . j u( 1t IM l (
11 ^ : ', , X ' " I1"1" ( ) > ' · < ' " " ' U l l " «
l , , l , i i Ί Ii f .11 » s llnl if " | Hon (l>) ι Ι Κ Κ Γ i«-, l il ii ([ (0 0 ^ i On llii. utlur li ind if for
11 j 11 'κ u Ib) is ( i . o j i ] ih«n UMiis|i).,r (Kim l\ v.t
M l -- t ^ ,- (( ; , log p - 10 , k ε p so
i ^ ' l' - r. in in·. ' IM sine' it l1· cir j r from SU p t i)
\ ritifrnniii ι c Ο^οηίΐιηι for findmg ,m irrtduriblc l <jl\ i' π ι J ff d trt Ί dt^rrc n Γ (\j is prcscnted rl hc
ι! Ε, π'ί in r u ii κι ι> Λ\ siounal turn if in t* A U nsiosi of H κ ι u n i v j otli'M1- i, i w i m H Λ S^OIK! i jonliirn li h - u s κ i n n m m h c polynornul luni vtiLliout
In f thc«,ΐ1· an ) ρτοΊικΡΗ in irrcdij^iljU pe>l> nornnl of
^ frc u m i t ' h (h* dtM t ! d< gr< Λ is ahn pres* n t cd Γίιι ob\tous r'iniining probli ΠΊΊ ire lo rtmove the n i ' d f >r (xtPödf l R n i m r i n hypotl-csn in Al^ontlmi Λ or <; ji it 111M 11! v impr ΛΛ !h· ij j>roxnri ition ΐ(.1ικ\( l in
\|?Λ r lljin Ii Ι ΙΗΊ l olti miv In difficult sine P tln '.flulion tr n l h t r ^( iilii nnply tli* soiiilinn H) otin r ^ill l i >v. n μ r ' κ τη (n n u in l·* r ihct τ< t K roll \> il Ujoni)
r )[,li Λ|ί\ [ f r ' x unpli1 r (nun il of tlu u* « i für \ l i n i l Ui m u.» hypMh, H tu MV nlhin Λ « ul l
(rf ι !r i m ins f t fiii'lifi(r irr-diiulil' jindr Hir
I ih u n ι ι N wln li 111 l n r n v. uld ρ π η ι 11 ι l· h riniin l κ
\\( wouM iik< to tlunk I Von zur Gült ι fr
hnngmg Uu^ proUcin to our ittuntiou for itliiniMi' nsulls on t Ins problcm st p Von zur G ithon j-lj
Rf-fercnces
fl| I ilirii IIK) ) Sliilln M utorins wifh < \<Uoiüi IMvrionn\ls" l'r.x. tliiiRs Jl.llt I J J I ^ >^Γ» l'l· (Π 110
[ij II ( lior ind R Rjvc,t, ' λ K m p ^ i c k T>po l'iiblii KL\ C r\ptos> stcm I h s i d on \nthrnetic in i initc I itlds " \dv inpcs in Crv ptocripln (Ld , (Ϊ Goos and J Hartmams) Springor Verlag, N(W "lork, pp 5 l 65
|-1] J λοη zur Gathi'n "Irnducible Pol>nomials Oxcrfiiiiti luids," Minusonpt IOSj
(r)] L L Kimuncr Ϊ her die Divisortn gcuis^r
Formen der /alilp» wflchc aus der Theorie dtr Krcisthcdung entstehen I reine angew Math 30 (J8IC), 107-116 ρρ Ηϋ-202 m Collerled papcrs, ^pnngcr-VcrKg Berlin 1075
[G) l Mirskv ' Ihc Numbir of Rt pnstntitioiis of in Int igt r -is liio S u n of i Priint -ind ι \ frre Infcßtr * Amer M Uli Monliilv 30 (Ifllfl) 17 10
[8] M O Ralun -IVobiliilutio \lgontlim-i ι» Finite 1VMV SIAM l Comput , \ol 9, (1880), pp 273-280
[i)j L Washington, 'CjdoUmiic l nIds,· Sprmgcr-\ t r h g , New York l OSO
JSOJ J II \\ilkmson, "Th<? Algfbraic Ligenvatuo {'roliicin,· Oxforrf Chrendon Pnss, 10l)r) pp
